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For Exercises 1–4, refer to .

1. Name the . SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a b. a c. a SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in . Therefore, DN = 8 cm.

10-1 and ANSWER: 8 cm

For Exercises 1–4, refer to . 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

Therefore, the diameter is 26 ft. 1. Name the circle. ANSWER: SOLUTION: 26 ft The center of the circle is N. So, the circle is The of , , and are 8 ANSWER: inches, 18 inches, and 11 inches, respectively. Find each measure.

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: 5. FG a. A chord is a segment with endpoints on the circle. So, here are chords. SOLUTION: b. A diameter of a circle is a chord that passes Since the diameter of B is 18 inches and A is 8 through the center. Here, is a diameter. inches, AG = 18 and AF = (8) or 4. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: Therefore, FG = 14 in. a. b. ANSWER: c. 14 in.

3. If CN = 8 centimeters, find DN. 6. FB SOLUTION: SOLUTION: Here, are radii of the same circle. So, Since the diameter of B is 18 inches and A is 4

they are equal in length. Therefore, DN = 8 cm. inches, AB = (18) or 9 and AF = (8) or 4. ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? Therefore, FB = 5 inches. SOLUTION: Here, is a radius and the diameter is twice the ANSWER: radius. 5 in. 7. RIDES The circular ride described at the beginning eSolutionsTherefore,Manual -thePowered diameterby Cognero is 26 ft. of the lesson has a diameter of 44 feet. What arePage the1 radius and circumference of the ride? Round to the ANSWER: nearest hundredth, if necessary. 26 ft SOLUTION: The diameters of , , and are 8 The radius is half the diameter. So, the radius of the circular ride is 22 feet. inches, 18 inches, and 11 inches, respectively. Find each measure.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 5. FG 22 ft; 138.23 ft SOLUTION: 8. CCSS MODELING The circumference of the Since the diameter of B is 18 inches and A is 8 circular swimming pool shown is about 56.5 feet. inches, AG = 18 and AF = (8) or 4. What are the diameter and radius of the pool? Round to the nearest hundredth.

Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB SOLUTION: SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches.

ANSWER:

5 in. The diameter of the pool is about 17.98 feet and the 7. RIDES The circular ride described at the beginning radius of the pool is about 8.99 feet. of the lesson has a diameter of 44 feet. What are the ANSWER: radius and circumference of the ride? Round to the nearest hundredth, if necessary. 17.98 ft; 8.99 ft SOLUTION: 9. SHORT RESPONSE The right shown is The radius is half the diameter. So, the radius of the inscribed in . Find the exact circumference of circular ride is 22 feet. .

Therefore, the circumference of the ride is about 138.23 feet. SOLUTION: ANSWER: The diameter of the circle is the of the 22 ft; 138.23 ft .

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π centimeters. SOLUTION: ANSWER:

For Exercises 10–13, refer to .

10. Name the center of the circle.

The diameter of the pool is about 17.98 feet and the SOLUTION: radius of the pool is about 8.99 feet. R ANSWER: ANSWER: 17.98 ft; 8.99 ft R

SHORT RESPONSE 9. The right triangle shown is 11. Identify a chord that is also a diameter. inscribed in . Find the exact circumference of . SOLUTION: Two chords are shown: and . goes through the center, R, so is a diameter.

ANSWER:

SOLUTION: The diameter of the circle is the hypotenuse of the 12. Is a radius? Explain. right triangle. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord.

The diameter of the circle is 4 centimeters. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: The circumference of the circle is 4π Here, is a diameter and is a radius. centimeters.

ANSWER:

For Exercises 10–13, refer to . Therefore, RT = 8.1 cm. ANSWER: 8.1 cm

For Exercises 14–17, refer to .

10. Name the center of the circle. SOLUTION: R

ANSWER: 14. Identify a chord that is not a diameter. R SOLUTION: 11. Identify a chord that is also a diameter. The chords do not pass through the center. So, they are not diameters. SOLUTION: ANSWER: Two chords are shown: and . goes

through the center, R, so is a diameter.

15. If CF = 14 inches, what is the diameter of the circle? ANSWER: SOLUTION:

Here, is a radius and the diameter is twice the radius. . 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on Therefore, the diameter of the circle is 28 inches. the circle, so it is a chord. ANSWER: ANSWER: 28 in. No; it is a chord. 16. Is ? Explain. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: SOLUTION: All radii of a circle are congruent. Since Here, is a diameter and is a radius. are both radii of , .

ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? Therefore, RT = 8.1 cm. SOLUTION: ANSWER: Here, is a diameter and is a radius. 8.1 cm

For Exercises 14–17, refer to .

Therefore, EF = 3.7 cm.

ANSWER: 3.7 cm

14. Identify a chord that is not a diameter. Circle J has a radius of 10 units, has a SOLUTION: radius of 8 units, and BC = 5.4 units. Find each The chords do not pass through the measure. center. So, they are not diameters.

ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: 18. CK Here, is a radius and the diameter is twice the SOLUTION: radius. . Since the radius of K is 8 units, BK = 8.

Therefore, the diameter of the circle is 28 inches. Therefore, CK = 2.6 units.

ANSWER: ANSWER: 28 in. 2.6

16. Is ? Explain. 19. AB SOLUTION: SOLUTION: All radii of a circle are congruent. Since Since is a diameter of circle J and the radius is are both radii of , . 10, AC = 2(10) or 20 units.

ANSWER: Yes; they are both radii of . Therefore, AB = 14.6 units. 17. If DA = 7.4 centimeters, what is EF? ANSWER: SOLUTION: 14.6 Here, is a diameter and is a radius. 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Therefore, EF = 3.7 cm.

ANSWER: Since circle J has a radius of 10 units, JC = 10. 3.7 cm

Circle J has a radius of 10 units, has a

radius of 8 units, and BC = 5.4 units. Find each Therefore, JK = 12.6 units. measure. ANSWER: 12.6 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The 18. CK radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before SOLUTION: finding AD, we need to find CK. Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units.

ANSWER: Therefore, AD = 30.6 units. 2.6 ANSWER: 19. AB 30.6 SOLUTION: 22. PIZZA Find the radius and circumference of the Since is a diameter of circle J and the radius is pizza shown. Round to the nearest hundredth, if 10, AC = 2(10) or 20 units. necessary.

Therefore, AB = 14.6 units.

ANSWER: 14.6 20. JK SOLUTION: SOLUTION: The diameter of the pizza is 16 inches. First find CK. Since circle K has a radius of 8 units, BK = 8.

Since circle J has a radius of 10 units, JC = 10. . So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by Therefore, JK = 12.6 units.

ANSWER: 12.6 Therefore, the circumference of the pizza is about 21. AD 50.27 inches. SOLUTION: ANSWER: We can find AD using AD = AC + CK + KD. The 8 in.; 50.27 in. radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before 23. BICYCLES A bicycle has tires with a diameter of finding AD, we need to find CK. 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter.

Therefore, AD = 30.6 units.

ANSWER: 30.6 So, the radius of the tires is 13 inches. 22. PIZZA Find the radius and circumference of the The circumference C of a circle with diameter d is pizza shown. Round to the nearest hundredth, if given by necessary.

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with SOLUTION: the given circumference. Round to the nearest The diameter of the pizza is 16 inches. hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is given by . Here, C = 18 in. Use the formula to find the So, the radius of the pizza is 8 inches. diameter. Then find the radius. The circumference C of a circle of diameter d is given by

Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of Therefore, the diameter is about 5.73 inches and the 26 inches. Find the radius and circumference of a radius is about 2.86 inches. tire. Round to the nearest hundredth, if necessary. ANSWER: SOLUTION: 5.73 in.; 2.86 in. The diameter of a tire is 26 inches. The radius is half the diameter. 25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the So, the radius of the tires is 13 inches. diameter. Then find the radius. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in.

Find the diameter and radius of a circle by with Therefore, the diameter is about 39.47 feet and the the given circumference. Round to the nearest radius is about 19.74 feet. hundredth. 24. C = 18 in. ANSWER: 39.47 ft; 19.74 ft SOLUTION: The circumference C of a circle with diameter d is 26. C = 375.3 cm

given by SOLUTION: Here, C = 18 in. Use the formula to find the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: Therefore, the diameter is about 119.46 centimeters 5.73 in.; 2.86 in. and the radius is about 59.73 centimeters. ANSWER: 25. C = 124 ft 119.46 cm; 59.73 cm SOLUTION: 27. C = 2608.25 m The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 124 ft. Use the formula to find the The circumference C of a circle with diameter d is diameter. Then find the radius. given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. Therefore, the diameter is about 830.23 meters and ANSWER: the radius is about 415.12 meters. 39.47 ft; 19.74 ft ANSWER: 26. C = 375.3 cm 830.23 m; 415.12 m SOLUTION: CCSS SENSE-MAKING Find the exact The circumference C of a circle with diameter d is circumference of each circle by using the given inscribed or given by circumscribed . Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the to find the diameter.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. ANSWER: 119.46 cm; 59.73 cm The diameter of the circle is 17 cm. The circumference C of a circle is given by 27. C = 2608.25 m SOLUTION: The circumference C of a circle with diameter d is given by Therefore, the circumference of the circle is 17 Here, C = 2608.25 meters. Use the formula to find centimeters. the diameter. Then find the radius. ANSWER:

29. Therefore, the diameter is about 830.23 meters and SOLUTION: the radius is about 415.12 meters. The hypotenuse of the right triangle is a diameter of ANSWER: the circle. Use the Pythagorean Theorem to find the diameter. 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon. The diameter of the circle is 12 feet. The circumference C of a circle is given by

Therefore, the circumference of the circle is 12 28. feet.

SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

30. The diameter of the circle is 17 cm. The circumference C of a circle is given by SOLUTION: Each of the inscribed will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. Therefore, the circumference of the circle is 17 centimeters.

ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by

29. Therefore, the circumference of the circle is inches. SOLUTION: The hypotenuse of the right triangle is a diameter of ANSWER: the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 12 feet. The circumference C of a circle is given by 31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Therefore, the circumference of the circle is 12 Use the Pythagorean Theorem to find the diameter feet. of the circle.

ANSWER:

The diameter of the circle is 10 inches. The circumference C of a circle is given by

30.

SOLUTION: Therefore, the circumference of the circle is 10 Each diagonal of the inscribed rectangle will pass inches. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter ANSWER: of the circle.

The diameter of the circle is inches. The

circumference C of a circle is given by 32. SOLUTION: A diameter to a side of the and the length of each side will measure the same. So, d = 25 millimeters. Therefore, the circumference of the circle is The circumference C of a circle is given by inches.

ANSWER:

Therefore, the circumference of the circle is 25 millimeters.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter

of the circle. 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The diameter of the circle is 10 inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 10 Therefore, the circumference of the circle is 14 inches. yards. ANSWER: ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight 32. for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: A diameter perpendicular to a side of the square and SOLUTION: the length of each side will measure the same. So, d The circumference C of a circle with diameter d is = 25 millimeters. given by The circumference C of a circle is given by Here, C = 66.92 cm. Use the formula to find the diameter.

Therefore, the circumference of the circle is 25 millimeters.

ANSWER: The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio 33. shown. a. What is the patio’s approximate circumference? SOLUTION: b. If Mr. Martinez changes the plans so that the A diameter perpendicular to a side of the square and inner circle has a circumference of approximately 25 the length of each side will measure the same. So, d feet, what should the radius of the circle be to the = 14 yards. nearest foot? The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 yards. ANSWER: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and Use the formula to find the circumference. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. Therefore, the circumference is about 31.42 feet. b. Here, C = 25 ft. Use the formula to find the SOLUTION: inner radius of the inner circle. The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter.

So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft The diameter of the disc is about 21.3 centimeters b. 4 ft and the maximum weight allowed is 8.3 times the diameter in centimeters. The radius, diameter, or circumference of a Therefore, the maximum weight is 8.3 × 21.3 or circle is given. Find each missing measure to about 176.8 grams. the nearest hundredth.

ANSWER: 36. 176.8 g SOLUTION: PATIOS 35. Mr. Martinez is going to build the patio The radius is half the diameter and the

shown. circumference C of a circle with diameter d is given a. What is the patio’s approximate circumference? by b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. SOLUTION: ANSWER: a. The radius of the patio is 3 + 2 = 5 ft. The 4.25 in.; 26.70 in. circumference C of a circle with radius r is given by

Use the formula to find the circumference. 37.

SOLUTION: The diameter is twice the radius and the Therefore, the circumference is about 31.42 feet. circumference C of a circle with diameter d is given b. Here, Cinner = 25 ft. Use the formula to find the by radius of the inner circle.

So, the radius of the inner circle is about 4 feet. ANSWER: So, the diameter is 22.80 feet and the circumference is about 71.63 feet. a. 31.42 ft b. 4 ft ANSWER: 22.80 ft; 71.63 ft The radius, diameter, or circumference of a circle is given. Find each missing measure to 38. the nearest hundredth. SOLUTION: 36. The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 35x cm. Use the formula to find the The radius is half the diameter and the diameter.Then find the radius. circumference C of a circle with diameter d is given by

Therefore, the radius is 4.25 inches and the Therefore, the diameter is about 11.14x centimeters circumference is about 26.70 inches. and the radius is about 5.57x centimeters.

ANSWER: ANSWER: 4.25 in.; 26.70 in. 11.14x cm; 5.57x cm

37. 39.

SOLUTION: SOLUTION: The diameter is twice the radius and the The diameter is twice the radius and the circumference C of a circle with diameter d is given circumference C of a circle with diameter d is given by by

So, the diameter is 22.80 feet and the circumference So, the diameter is 0.25x units and the circumference is about 71.63 feet. is about 0.79x units.

ANSWER: ANSWER: 22.80 ft; 71.63 ft 0.25x; 0.79x 38. Determine whether the circles in the figures below appear to be congruent, concentric, or SOLUTION: neither. The circumference C of a circle with diameter d is 40. Refer to the photo on page 703. given by Here, C = 35x cm. Use the formula to find the SOLUTION: diameter.Then find the radius. The circles have the same center and they are in the same . So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

Therefore, the diameter is about 11.14x centimeters ANSWER: and the radius is about 5.57x centimeters. neither

ANSWER: 42. Refer to the photo on page 703. 11.14x cm; 5.57x cm SOLUTION: The circles appear to have the same radius. So, they 39. appear to be congruent.

SOLUTION: ANSWER: The diameter is twice the radius and the congruent circumference C of a circle with diameter d is given 43. HISTORY The Indian Shell Ring on Hilton Head by Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x

Determine whether the circles in the figures below appear to be congruent, concentric, or neither. SOLUTION: 40. Refer to the photo on page 703. The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a SOLUTION: circle with diameter d is given by The circles have the same center and they are in the So, the circumference is same plane. So, they are concentric circles. Therefore, someone have to ANSWER: walk about 471.2 feet to go completely around the concentric ring. ANSWER: 41. Refer to the photo on page 703. 471.2 ft

SOLUTION: 44. CCSS MODELING A brick path is being installed The circles neither have the same radius nor are they around a circular pond. The pond has a concentric. circumference of 68 feet. The outer of the path is going to be 4 feet from the pond all the way ANSWER: around. What is the approximate circumference of neither the path? Round to the nearest hundredth. 42. Refer to the photo on page 703. SOLUTION: The circumference C of a circle with radius r is SOLUTION: given by Here, C = 68 ft. Use the The circles appear to have the same radius. So, they pond appear to be congruent. formula to find the radius of the pond.

ANSWER: congruent

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would So, the radius of the pond is about 10.82 feet and the someone have to walk to go completely around the radius to the outer edge of the path will be 10.82 + 4 ring? Round to the nearest tenth. = 14.82 ft. Use the radius to find the circumference of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing in circles. SOLUTION: a. GEOMETRIC Use a compass to draw three The radius of the circle is 3 units, that is 75 feet. So, circles in which the scale factor from each circle to the diameter is 150 ft. The circumference C of a the next larger circle is 1 : 2. circle with diameter d is given by b. TABULAR Calculate the radius (to the nearest So, the circumference is tenth) and circumference (to the nearest hundredth) Therefore, someone have to of each circle. Record your results in a table. walk about 471.2 feet to go completely around the c. VERBAL Explain why these three circles are ring. geometrically similar. d. VERBAL Make a conjecture about the ratio ANSWER: between the of two circles when the 471.2 ft ratio between their radii is 2. e. ANALYTICAL The scale factor from to 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a is . Write an relating the circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way circumference (CA) of to the circumference around. What is the approximate circumference of (CB) of . the path? Round to the nearest hundredth. f. NUMERICAL If the scale factor from to SOLUTION: is , and the circumference of is 12 The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the inches, what is the circumference of ? formula to find the radius of the pond. SOLUTION: a. Sample answer:

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path. b. Sample answer.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: c. They all have the same –circular. 93.13 ft d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. 45. MULTIPLE REPRESENTATIONS In this So, if the radii are in the ratio b : a, so will the problem, you will explore changing dimensions in

circles. circumference. Therefore, . a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to f. Use the formula from part e. Here, the the next larger circle is 1 : 2. circumference of is 12 inches and . b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. Therefore, the circumference of is 4 inches. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ANSWER: ratio between their radii is 2. a. Sample answer: e. ANALYTICAL The scale factor from to is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to b. Sample answer. is , and the circumference of is 12 inches, what is the circumference of ? SOLUTION: a. Sample answer:

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in. b. Sample answer. 46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a of horizontal lines that are centimeters apart on a sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . f. Use the formula from part e. Here, the a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a circumference of is 12 inches and . hit. Record the of hits after 25, 50, and 100 drops.

Therefore, the circumference of is 4 inches. b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 ANSWER: drops. a. Sample answer: c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, b. Sample answer. which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. c. They all have the same shape—circular. d. The ratio of their circumferences is also 2. M APS e. 47. The concentric circles on the map below show the that are 5, 10, 15, 20, 25, and 30 f. 4 in. miles from downtown Phoenix.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center

circle? b. As the radii of the circles increases by 5 miles, by a. Drop the needle onto the paper. When the needle how much does the circumference increase? lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 SOLUTION: drops. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π?

SOLUTION: The difference of the circumferences of the a. See students’ work. outermost circle and the center circle is 60π - 10π = b. See students’ work. 50π or about 157.1 miles. c. Sample answer: The values are approaching 3.14, b. Let the radius of one circle be x miles and the which is approximately equal to π. radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

π + 47. M APS The concentric circles on the map below The difference of the two circumferences is (2x show the areas that are 5, 10, 15, 20, 25, and 30 10π) - (2xπ) = 10π or about 31.4 miles. miles from downtown Phoenix. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines?

a. How much greater is the circumference of the SOLUTION: outermost circle than the circumference of the center Sample answer: A and a circle may intersect in circle? one , two points, or may not intersect at all. A b. As the radii of the circles increases by 5 miles, by line that intersects a circle in one point can be how much does the circumference increase? described as a . A line that intersects a circle SOLUTION: in exactly two points can be described as a secant. a. The radius of the outermost circle is 30 miles and A with endpoints on a circle can be that of the center circle is 5 miles. Find the described as a chord. If the chord passes through circumference of each circle. the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius. ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that The difference of the circumferences of the intersects a circle in exactly two points can be outermost circle and the center circle is 60π - 10π = described as a secant. A line segment with 50π or about 157.1 miles. endpoints on a circle can be described as a chord. b. Let the radius of one circle be x miles and the If the chord passes through the center of the circle, radius of the next larger circle be x + 5 miles. Find it can be described as a diameter. A line segment the circumference of each circle. with endpoints at the center and on the circle can be described as a radius.

49. The difference of the two circumferences is (2xπ + REASONING In the figure, a circle with radius r is 10π) - (2xπ) = 10π or about 31.4 miles. inscribed in a and circumscribed Therefore, as the radius of each circle increases by 5 about another. miles, the circumference increases by about 31.4 a. What are the of the circumscribed and miles. inscribed in terms of r? Explain. b. Is the circumference C of the circle greater or ANSWER: less than the of the circumscribed polygon? a. 157.1 mi the inscribed polygon? Write a compound inequality b. by about 31.4 mi comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the 48. WRITING IN MATH How can we describe the diameter d of the circle and interpret its meaning. relationships that exist between d. As the number of sides of both the circumscribed circles and lines? and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part SOLUTION: c, and what does this imply? Sample answer: A line and a circle may intersect in SOLUTION: one point, two points, or may not intersect at all. A line that intersects a circle in one point can be a. The circumscribed polygon is a square with sides described as a tangent. A line that intersects a circle of length 2r. in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a The inscribed polygon is a with sides of diameter. A line segment with endpoints at the length r. (Each triangle formed using a central center and on the circle can be described as a will be equilateral.) radius.

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment So, the perimeters are 8r and 6r. with endpoints at the center and on the circle can be b. described as a radius.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 49. 8r < C < 6r REASONING In the figure, a circle with radius r is c. Since the diameter is twice the radius, 8r = 4(2r) inscribed in a regular polygon and circumscribed or 4d and 6r = 3(2r) or 3d. about another. The inequality is then 4d < C < 3d. Therefore, the a. What are the perimeters of the circumscribed and circumference of the circle is between 3 and 4 times inscribed polygons in terms of r? Explain. its diameter. b. Is the circumference C of the circle greater or d. These limits will approach a value of πd, implying less than the perimeter of the circumscribed polygon? that C = πd. the inscribed polygon? Write a compound inequality comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the ANSWER: diameter d of the circle and interpret its meaning. a. 8r and 6r ; Twice the radius of the circle, 2r is d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to the side length of the square, so the perimeter of the the upper and lower limits of the inequality from part square is 4(2r) or 8r. The c, and what does this imply? regular hexagon is made up of six equilateral with side length r, so the perimeter of the SOLUTION: hexagon is 6(r) or 6r. a. The circumscribed polygon is a square with sides b. less; greater; 6r < C < 8r of length 2r. c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a 50. CHALLENGE The sum of the circumferences of will be equilateral.) circles H, J, and K shown at the right is units. Find KJ.

SOLUTION: So, the perimeters are 8r and 6r. b. The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter.

d. These limits will approach a value of πd, implying The sum of the circumferences is 56π units. that C = πd. Substitute the three circumferences and solve for x.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the So, the radius of is 4 units, the radius of is square is 4(2r) or 8r. The 2(4) or 8 units, and the radius of is 4(4) or 16 regular hexagon is made up of six equilateral units. triangles with side length r, so the perimeter of the The measure of KJ is equal to the sum of the radii hexagon is 6(r) or 6r. of and . b. less; greater; 6r < C < 8r Therefore, KJ = 16 + 8 or 24 units. c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. ANSWER: d. These limits will approach a value of πd, implying 24 units that C = πd. 51. REASONING Is the distance from the center of a 50. CHALLENGE The sum of the circumferences of circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? circles H, J, and K shown at the right is units. Explain. Find KJ. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A SOLUTION: segment drawn from the center to a point inside the The radius of is 4x units, the radius of is x circle will always have a length less than the radius units, and the radius of is 2x units. Find the of the circle. circumference of each circle. 52. PROOF Use the definition of a circle and dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane from a given point. For any two circles and , there exists a that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make The sum of the circumferences is 56π units. up are from center B. Substitute the three circumferences and solve for x. For some value k, r2 = kr1.

So, the radius of is 4 units, the radius of is

2(4) or 8 units, and the radius of is 4(4) or 16

units.

The measure of KJ is equal to the sum of the radii Therefore, is mapped onto . Since there of and . exists a rigid motion followed by a that maps Therefore, KJ = 16 + 8 or 24 units. onto , the circles are similar. Thus, all circles are similar. ANSWER:

24 units

51. REASONING Is the distance from the center of a ANSWER: circle to a point in the interior of a circle sometimes, A circle is a locus of points in a plane equidistant always, or never less than the radius of the circle? from a given point. For any two circles and Explain. , there exists a translation that maps center A SOLUTION: onto center B, moving so that it is concentric A radius is a segment drawn between the center of with . There also exists a dilation with scale the circle and a point on the circle. A segment drawn factor k such that each point that makes up is from the center to a point inside the circle will always moved to be the same distance from center A as the have a length less than the radius of the circle. points that make up are from center B. ANSWER: Therefore, is mapped onto . Since there Always; a radius is a segment drawn between the exists a rigid motion followed by a scaling that maps center of the circle and a point on the circle. A onto , the circles are similar. Thus, all segment drawn from the center to a point inside the circles are similar. circle will always have a length less than the radius of the circle. 53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference 52. PROOF Use the locus definition of a circle and of ? dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such SOLUTION: that each point that makes up is moved to be the Draw a perpendicular to the side Also join same distance from center A as the points that make

up are from center B. For some value k, r2 = kr1.

Since is equilateral triangle, MN = 8 inches

and RN = 4 inches. Also, bisects Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps Use the definition of onto , the circles are tangent of an angle to find PR, which is the radius of similar. Thus, all circles are similar.

ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A Therefore, the radius of is onto center B, moving so that it is concentric The circumference C of a circle with radius r is with . There also exists a dilation with scale given by factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all

circles are similar. The circumference of is inches.

53. CHALLENGE In the figure, is inscribed in ANSWER: equilateral triangle LMN. What is the circumference of ? in.

54. WRITING IN MATH Research and write about the history of and its importance to the study of geometry. SOLUTION: See students’ work. SOLUTION: ANSWER: Draw a perpendicular to the side Also join See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects SOLUTION: Use the definition of Use the Pythagorean Theorem to find the diameter tangent of an angle to find PR, which is the radius of of the circle.

The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of is The circumference C of a circle with radius r is given by

ANSWER: 40.8

56. What is the radius of a table with a circumference of 10 feet? The circumference of is inches. A 1.6 ft B 2.5 ft ANSWER: C 3.2 ft D in. 5 ft SOLUTION: 54. WRITING IN MATH Research and write about The circumference C of a circle with radius r is the history of pi and its importance to the study of given by geometry. Here, C = 10 ft. Use the formula to find the radius. SOLUTION: See students’ work.

ANSWER: See students’ work. The radius of the circle is about 1.6 feet, so the 55. GRIDDED RESPONSE What is the correct choice is A. circumference of ? Round to the nearest tenth. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? SOLUTION: F 10 Use the Pythagorean Theorem to find the diameter G 9 of the circle. H 8 J 7 The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with radius r is Use the formula to find the circumference. given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

ANSWER:

40.8 Therefore, the radius of the garden must be less than 56. What is the radius of a table with a circumference of or equal to about 7.96 feet. Therefore, the only

10 feet? correct choice is J. A 1.6 ft ANSWER: B 2.5 ft J C 3.2 ft D 5 ft 58. SAT/ACT What is the radius of a circle with an

SOLUTION: of square units? The circumference C of a circle with radius r is A 0.4 units given by Here, C = 10 ft. Use the formula to find the radius. B 0.5 units C 2 units D 4 units E 16 units SOLUTION: The of radius r is given by the The radius of the circle is about 1.6 feet, so the correct choice is A. formula Here A = square units. Use the formula to find the radius. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the Therefore, the correct choice is B. garden? F 10 ANSWER: G 9 B H 8 Copy each figure and point B. Then use a ruler J 7 to draw the image of the figure under a dilation SOLUTION: with center B and the scale factor r indicated. The circumference C of a circle with radius r is 59. given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

SOLUTION: Draw a ray from point B through a of the hexagon. Use the ruler to measure the distance from Therefore, the radius of the garden must be less than B to the vertex. Mark a point on the ray that is of or equal to about 7.96 feet. Therefore, the only that distance from B. Repeat the process for rays

correct choice is J. through the other five vertices. Connect the six points ANSWER: on the rays to construct the dilated image of the hexagon. J

58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the ANSWER: formula Here A = square units. Use the formula to find the radius.

60.

Therefore, the correct choice is B.

ANSWER: B SOLUTION: Draw a ray from point B through a vertex of the Copy each figure and point B. Then use a ruler triangle. Use the ruler to measure the distance from to draw the image of the figure under a dilation with center B and the scale factor r indicated. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays 59. through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the ANSWER: hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points 61. on the rays to construct the dilated image of the hexagon.

SOLUTION: Draw a ray from point B through a vertex of the . Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

60. ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of 62. that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of

the triangle. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for ANSWER: rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the ANSWER: four points on the rays to construct the dilated image of the trapezoid.

ANSWER: State whether each figure has rotational . Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and of symmetry. 63. Refer to the image on page 705.

SOLUTION: 62. The figure cannot be mapped onto itself by any rotation. So, it does not have a .

ANSWER: No SOLUTION: Refer to the image on page 705. Draw a ray from point B through a vertex of the 64. triangle. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is 3 The figure can be mapped onto itself by a rotation of times that distance from B. Repeat the process for 90º. So, it has a rotational symmetry. So, the rays through the other two vertices. Connect the magnitude of symmetry is 90º and order of symmetry three points on the rays to construct the dilated image is 4. of the triangle. ANSWER: Yes; 4, 90

65. ANSWER: SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. State whether each figure has rotational symmetry. Write yes or no. If so, copy the ANSWER: figure, locate the center of symmetry, and state No the order and magnitude of symmetry. 63. Refer to the image on page 705. Determine the truth value of the following statement for each set of conditions. Explain SOLUTION: your reasoning. The figure cannot be mapped onto itself by any If you are over 18 years old, then you vote in all rotation. So, it does not have a rotational symmetry. elections. 67. You are 19 years old and you vote. ANSWER: No SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true 64. Refer to the image on page 705. for the conditions.

SOLUTION: ANSWER: The figure can be mapped onto itself by a rotation of True; sample answer: Since the hypothesis is true 90º. So, it has a rotational symmetry. So, the and the conclusion is true, then the statement is true magnitude of symmetry is 90º and order of symmetry for the conditions. is 4.

ANSWER: 68. You are 21 years old and do not vote. Yes; 4, 90 SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

65. Find x. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No 69. 66. Refer to the image on page 705. SOLUTION: The sum of the measures of around a point is SOLUTION: 360º. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No ANSWER: Determine the truth value of the following 90 statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION:

True; sample answer: Since the hypothesis is true 70. and the conclusion is true, then the statement is true SOLUTION: for the conditions. The angles with the measures xº and 2xº form a

ANSWER: straight line. So, the sum is 180.

True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

ANSWER: 68. You are 21 years old and do not vote. 60 SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true 71. and the conclusion is false, then the statement is false for the conditions. SOLUTION: The angles with the measures 6xº and 3xº form a Find x. straight line. So, the sum is 180.

ANSWER: 69. 20 SOLUTION: The sum of the measures of angles around a point is 360º.

72. ANSWER: SOLUTION: 90 The sum of the three angles is 180.

ANSWER: 45 70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius. For Exercises 1–4, refer to .

Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 1. Name the circle. inches, 18 inches, and 11 inches, respectively. SOLUTION: Find each measure. The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord 5. FG b. a diameter SOLUTION: c. a radius Since the diameter of B is 18 inches and A is 8 SOLUTION: inches, AG = 18 and AF = (8) or 4. a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center Therefore, FG = 14 in. and on the circle. Here, is ANSWER: radius. 14 in. ANSWER: a. 6. FB b. SOLUTION: c. Since the diameter of B is 18 inches and A is 4 3. If CN = 8 centimeters, find DN. inches, AB = (18) or 9 and AF = (8) or 4. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm. Therefore, FB = 5 inches. ANSWER: 8 cm ANSWER: 5 in. 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the Here, is a radius and the diameter is twice the radius and circumference of the ride? Round to the radius. nearest hundredth, if necessary.

SOLUTION: Therefore, the diameter is 26 ft. The radius is half the diameter. So, the radius of the circular ride is 22 feet. ANSWER: 26 ft

The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet.

5. FG What are the diameter and radius of the pool? Round to the nearest hundredth. SOLUTION: Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4.

Therefore, FG = 14 in. SOLUTION: ANSWER: 14 in.

6. FB SOLUTION:

Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

The diameter of the pool is about 17.98 feet and the Therefore, FB = 5 inches. radius of the pool is about 8.99 feet.

10-1ANSWER: Circles and Circumference ANSWER: 5 in. 17.98 ft; 8.99 ft

7. RIDES The circular ride described at the beginning 9. SHORT RESPONSE The right triangle shown is of the lesson has a diameter of 44 feet. What are the inscribed in . Find the exact circumference of radius and circumference of the ride? Round to the . nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet.

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle. Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft The diameter of the circle is 4 centimeters. 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to . SOLUTION:

10. Name the center of the circle. SOLUTION: R ANSWER:

R The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. 11. Identify a chord that is also a diameter. ANSWER: SOLUTION: 17.98 ft; 8.99 ft Two chords are shown: and . goes 9. SHORT RESPONSE The right triangle shown is through the center, R, so is a diameter. inscribed in . Find the exact circumference of eSolutions Manual. - Powered by Cognero Page 2 ANSWER:

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center SOLUTION: The diameter of the circle is the hypotenuse of the and on the circle. But has both the end points on right triangle. the circle, so it is a chord. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT?

The diameter of the circle is 4 centimeters. SOLUTION: Here, is a diameter and is a radius.

The circumference of the circle is 4π centimeters. Therefore, RT = 8.1 cm. ANSWER: ANSWER:

8.1 cm

For Exercises 10–13, refer to . For Exercises 14–17, refer to .

10. Name the center of the circle. 14. Identify a chord that is not a diameter. SOLUTION: SOLUTION: R The chords do not pass through the ANSWER: center. So, they are not diameters. R ANSWER:

11. Identify a chord that is also a diameter. SOLUTION: 15. If CF = 14 inches, what is the diameter of the circle? Two chords are shown: and . goes SOLUTION:

through the center, R, so is a diameter. Here, is a radius and the diameter is twice the radius. . ANSWER:

Therefore, the diameter of the circle is 28 inches. 12. Is a radius? Explain. SOLUTION: ANSWER: A radius is a segment with endpoints at the center 28 in. and on the circle. But has both the end points on 16. Is ? Explain. the circle, so it is a chord. SOLUTION: ANSWER: All radii of a circle are congruent. Since No; it is a chord. are both radii of , . 13. If SU = 16.2 centimeters, what is RT? ANSWER: SOLUTION: Yes; they are both radii of . Here, is a diameter and is a radius. 17. If DA = 7.4 centimeters, what is EF? SOLUTION: Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm.

ANSWER: 8.1 cm Therefore, EF = 3.7 cm. For Exercises 14–17, refer to . ANSWER: 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

14. Identify a chord that is not a diameter. SOLUTION: The chords do not pass through the center. So, they are not diameters.

ANSWER: 18. CK

SOLUTION: 15. If CF = 14 inches, what is the diameter of the circle? Since the radius of K is 8 units, BK = 8. SOLUTION: Here, is a radius and the diameter is twice the

radius. . Therefore, CK = 2.6 units. ANSWER: 2.6

Therefore, the diameter of the circle is 28 inches. 19. AB ANSWER: SOLUTION: 28 in. Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. 16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since Therefore, AB = 14.6 units. are both radii of , . ANSWER: ANSWER: 14.6 Yes; they are both radii of . 20. JK 17. If DA = 7.4 centimeters, what is EF? SOLUTION: SOLUTION: First find CK. Since circle K has a radius of 8 units, Here, is a diameter and is a radius. BK = 8.

Since circle J has a radius of 10 units, JC = 10.

Therefore, EF = 3.7 cm.

ANSWER: Therefore, JK = 12.6 units. 3.7 cm ANSWER: Circle J has a radius of 10 units, has a 12.6 radius of 8 units, and BC = 5.4 units. Find each measure. 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

18. CK

SOLUTION: Since the radius of K is 8 units, BK = 8.

Therefore, AD = 30.6 units.

ANSWER: Therefore, CK = 2.6 units. 30.6 ANSWER: PIZZA 2.6 22. Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary. 19. AB SOLUTION: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units. SOLUTION: ANSWER: The diameter of the pizza is 16 inches. 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, . BK = 8. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Since circle J has a radius of 10 units, JC = 10.

Therefore, the circumference of the pizza is about Therefore, JK = 12.6 units. 50.27 inches.

ANSWER: ANSWER: 12.6 8 in.; 50.27 in.

21. AD 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a SOLUTION: tire. Round to the nearest hundredth, if necessary. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J SOLUTION: is 10 units, so KD = 8 and AC = 2(10) or 20. Before The diameter of a tire is 26 inches. The radius is half finding AD, we need to find CK. the diameter.

So, the radius of the tires is 13 inches. Therefore, AD = 30.6 units. The circumference C of a circle with diameter d is given by ANSWER: 30.6

22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if Therefore, the circumference of the bicycle tire is necessary. about 81.68 inches. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in.

SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The diameter of the pizza is 16 inches. given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the circumference of the pizza is about 50.27 inches. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. ANSWER: 8 in.; 50.27 in. ANSWER: 5.73 in.; 2.86 in. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. 25. C = 124 ft SOLUTION: SOLUTION: The diameter of a tire is 26 inches. The radius is half The circumference C of a circle with diameter d is the diameter. given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches. Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. ANSWER: 13 in.; 81.68 in. ANSWER: 39.47 ft; 19.74 ft Find the diameter and radius of a circle by with the given circumference. Round to the nearest 26. C = 375.3 cm hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 375.3 cm. Use the formula to find the given by diameter. Then find the radius. Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. ANSWER: 119.46 cm; 59.73 cm ANSWER: 5.73 in.; 2.86 in. 27. C = 2608.25 m SOLUTION: 25. C = 124 ft The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

Therefore, the diameter is about 39.47 feet and the ANSWER: radius is about 19.74 feet. 830.23 m; 415.12 m ANSWER: CCSS SENSE-MAKING Find the exact 39.47 ft; 19.74 ft circumference of each circle by using the given inscribed or 26. C = 375.3 cm circumscribed polygon. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. 28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The Therefore, the diameter is about 119.46 centimeters circumference C of a circle is given by and the radius is about 59.73 centimeters.

ANSWER: 119.46 cm; 59.73 cm Therefore, the circumference of the circle is 17 27. C = 2608.25 m centimeters. SOLUTION: ANSWER: The circumference C of a circle with diameter d is

given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER:

830.23 m; 415.12 m The diameter of the circle is 12 feet. The circumference C of a circle is given by CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon. Therefore, the circumference of the circle is 12 feet.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter The diameter of the circle is 17 cm. The of the circle. circumference C of a circle is given by

Therefore, the circumference of the circle is 17 centimeters. The diameter of the circle is inches. The circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

31. SOLUTION: Each diagonal of the inscribed rectangle will pass The diameter of the circle is 12 feet. The through the origin, so it is a diameter of the circle. circumference C of a circle is given by Use the Pythagorean Theorem to find the diameter of the circle.

Therefore, the circumference of the circle is 12 feet. The diameter of the circle is 10 inches. The ANSWER: circumference C of a circle is given by

Therefore, the circumference of the circle is 10 inches.

30. ANSWER: SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

32. SOLUTION: A diameter perpendicular to a side of the square and The diameter of the circle is inches. The the length of each side will measure the same. So, d circumference C of a circle is given by = 25 millimeters. The circumference C of a circle is given by

Therefore, the circumference of the circle is Therefore, the circumference of the circle is 25 inches. millimeters.

ANSWER: ANSWER:

31. SOLUTION: 33. Each diagonal of the inscribed rectangle will pass SOLUTION: through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter A diameter perpendicular to a side of the square and of the circle. the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The circumference C of a circle is given by Therefore, the circumference of the circle is 14 yards.

ANSWER:

Therefore, the circumference of the circle is 10 inches. 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and ANSWER: clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is 32. given by SOLUTION: Here, C = 66.92 cm. Use the formula to find the diameter. A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the Therefore, the circumference of the circle is 25 diameter in centimeters. millimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. ANSWER: ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the 33. nearest foot? SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

SOLUTION:

Therefore, the circumference of the circle is 14 a. The radius of the patio is 3 + 2 = 5 ft. The yards. circumference C of a circle with radius r is given by

ANSWER: Use the formula to find the circumference.

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and Therefore, the circumference is about 31.42 feet. clubs. For professional competitions, the maximum b. Here, C = 25 ft. Use the formula to find the weight of a disc in grams is 8.3 times its diameter in inner centimeters. What is the maximum allowable weight radius of the inner circle. for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is

given by So, the radius of the inner circle is about 4 feet. Here, C = 66.92 cm. Use the formula to find the diameter. ANSWER: a. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to The diameter of the disc is about 21.3 centimeters the nearest hundredth. and the maximum weight allowed is 8.3 times the 36. diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. SOLUTION: The radius is half the diameter and the ANSWER: circumference C of a circle with diameter d is given 176.8 g by

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

SOLUTION: 37. a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by SOLUTION: The diameter is twice the radius and the Use the formula to find the circumference. circumference C of a circle with diameter d is given by

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

So, the radius of the inner circle is about 4 feet. ANSWER: 22.80 ft; 71.63 ft ANSWER: a. 31.42 ft 38. b. 4 ft SOLUTION: The radius, diameter, or circumference of a The circumference C of a circle with diameter d is circle is given. Find each missing measure to given by the nearest hundredth. Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. 36.

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters.

ANSWER: Therefore, the radius is 4.25 inches and the 11.14x cm; 5.57x cm circumference is about 26.70 inches.

39. ANSWER: 4.25 in.; 26.70 in. SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given 37. by SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: So, the diameter is 22.80 feet and the circumference 0.25x; 0.79x is about 71.63 feet. Determine whether the circles in the figures ANSWER: below appear to be congruent, concentric, or 22.80 ft; 71.63 ft neither. 40. Refer to the photo on page 703. 38. SOLUTION: SOLUTION: The circles have the same center and they are in the The circumference C of a circle with diameter d is same plane. So, they are concentric circles. given by ANSWER: Here, C = 35x cm. Use the formula to find the concentric diameter.Then find the radius.

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. Therefore, the diameter is about 11.14x centimeters SOLUTION: and the radius is about 5.57x centimeters. The circles appear to have the same radius. So, they appear to be congruent. ANSWER: 11.14x cm; 5.57x cm ANSWER: congruent

39. 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the SOLUTION: coordinate grid represents 25 feet, how far would The diameter is twice the radius and the someone have to walk to go completely around the circumference C of a circle with diameter d is given ring? Round to the nearest tenth. by

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: SOLUTION: 0.25x; 0.79x The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a Determine whether the circles in the figures circle with diameter d is given by below appear to be congruent, concentric, or So, the circumference is neither. Therefore, someone have to 40. Refer to the photo on page 703. walk about 471.2 feet to go completely around the SOLUTION: ring. The circles have the same center and they are in the same plane. So, they are concentric circles. ANSWER: 471.2 ft ANSWER: concentric 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path 41. Refer to the photo on page 703. is going to be 4 feet from the pond all the way SOLUTION: around. What is the approximate circumference of The circles neither have the same radius nor are they the path? Round to the nearest hundredth. concentric. SOLUTION: ANSWER: The circumference C of a circle with radius r is neither given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. 42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they appear to be congruent.

ANSWER: So, the radius of the pond is about 10.82 feet and the congruent radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference 43. HISTORY The Indian Shell Ring on Hilton Head of the brick path. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth. Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest SOLUTION: tenth) and circumference (to the nearest hundredth) The radius of the circle is 3 units, that is 75 feet. So, of each circle. Record your results in a table. the diameter is 150 ft. The circumference C of a c. VERBAL Explain why these three circles are circle with diameter d is given by geometrically similar. So, the circumference is d. VERBAL Make a conjecture about the ratio Therefore, someone have to between the circumferences of two circles when the ratio between their radii is 2. walk about 471.2 feet to go completely around the e. ANALYTICAL The scale factor from to ring. is . Write an equation relating the ANSWER:

471.2 ft circumference (CA) of to the circumference

44. CCSS MODELING A brick path is being installed (CB) of . around a circular pond. The pond has a f. NUMERICAL If the scale factor from to circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way is , and the circumference of is 12 around. What is the approximate circumference of inches, what is the circumference of ? the path? Round to the nearest hundredth. SOLUTION: SOLUTION: a. The circumference C of a circle with radius r is Sample answer: given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

b. Sample answer.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

c. They all have the same shape–circular. Therefore, the circumference of the path is about d. The ratio of their circumferences is also 2. 93.13 feet. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the ANSWER: 93.13 ft circumference. Therefore, . f. Use the formula from part e. Here, the MULTIPLE REPRESENTATIONS 45. In this problem, you will explore changing dimensions in circumference of is 12 inches and . circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest Therefore, the circumference of is 4 inches. tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. ANSWER: c. VERBAL Explain why these three circles are a. Sample answer: geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to is . Write an equation relating the b. Sample answer. circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to is , and the circumference of is 12 inches, what is the circumference of ? c. They all have the same shape—circular. SOLUTION: d. The ratio of their circumferences is also 2. a. Sample answer: e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a b. Sample answer. sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. a. Drop the needle onto the paper. When the needle So, if the radii are in the ratio b : a, so will the lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 circumference. Therefore, . drops.

f. Use the formula from part e. Here, the circumference of is 12 inches and . b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to Therefore, the circumference of is 4 inches. π? ANSWER: SOLUTION: a. Sample answer: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER:

b. Sample answer. a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 c. They all have the same shape—circular. miles from downtown Phoenix. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase?

SOLUTION: a. The radius of the outermost circle is 30 miles and a. Drop the needle onto the paper. When the needle that of the center circle is 5 miles. Find the lands, record whether it touches one of the lines as a circumference of each circle. hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops. The difference of the circumferences of the outermost circle and the center circle is 60π - 10π = c. How are the values you found in part b related to 50π or about 157.1 miles. π? b. Let the radius of one circle be x miles and the SOLUTION: radius of the next larger circle be x + 5 miles. Find a. See students’ work. the circumference of each circle. b. See students’ work. c. Sample answer: The values are approaching 3.14,

which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. The difference of the two circumferences is (2xπ + c. Sample answer: The values are approaching 3.14, 10π) - (2xπ) = 10π or about 31.4 miles. which is approximately equal to π. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 ANSWER: miles from downtown Phoenix. a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be a. How much greater is the circumference of the described as a tangent. A line that intersects a circle outermost circle than the circumference of the center in exactly two points can be described as a secant. circle? A line segment with endpoints on a circle can be b. As the radii of the circles increases by 5 miles, by described as a chord. If the chord passes through how much does the circumference increase? the center of the circle, it can be described as a SOLUTION: diameter. A line segment with endpoints at the a. The radius of the outermost circle is 30 miles and center and on the circle can be described as a that of the center circle is 5 miles. Find the radius. circumference of each circle. ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, The difference of the circumferences of the it can be described as a diameter. A line segment outermost circle and the center circle is 60π - 10π = with endpoints at the center and on the circle can be 50π or about 157.1 miles. described as a radius. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. The difference of the two circumferences is (2xπ + b. Is the circumference C of the circle greater or 10π) - (2xπ) = 10π or about 31.4 miles. less than the perimeter of the circumscribed polygon? Therefore, as the radius of each circle increases by 5 the inscribed polygon? Write a compound inequality miles, the circumference increases by about 31.4 comparing C to these perimeters. miles. c. Rewrite the inequality from part b in terms of the ANSWER: diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed a. 157.1 mi and inscribed polygons increase, what will happen to b. by about 31.4 mi the upper and lower limits of the inequality from part c, and what does this imply? 48. WRITING IN MATH How can we describe the relationships that exist between SOLUTION: circles and lines? a. The circumscribed polygon is a square with sides SOLUTION: of length 2r. Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle The inscribed polygon is a hexagon with sides of in exactly two points can be described as a secant. length r. (Each triangle formed using a central angle A line segment with endpoints on a circle can be will be equilateral.) described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER:

Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be So, the perimeters are 8r and 6r. described as a secant. A line segment with b. endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment The circumference is less than the perimeter of the with endpoints at the center and on the circle can be circumscribed polygon and greater than the described as a radius. perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the 49. circumference of the circle is between 3 and 4 times REASONING In the figure, a circle with radius r is its diameter. inscribed in a regular polygon and circumscribed d. These limits will approach a value of πd, implying about another. that C = πd. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or ANSWER: less than the perimeter of the circumscribed polygon? a. 8r and 6r ; Twice the radius of the circle, 2r is the inscribed polygon? Write a compound inequality the side length of the square, so the perimeter of the comparing C to these perimeters. square is 4(2r) or 8r. The c. Rewrite the inequality from part b in terms of the regular hexagon is made up of six equilateral diameter d of the circle and interpret its meaning. triangles with side length r, so the perimeter of the d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to hexagon is 6(r) or 6r. the upper and lower limits of the inequality from part b. less; greater; 6r < C < 8r c, and what does this imply? c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. SOLUTION: d. These limits will approach a value of πd, implying a. The circumscribed polygon is a square with sides that C = πd. of length 2r. 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units.

Find KJ. The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The sum of the circumferences is 56π units. . Therefore, the The inequality is then 4d < C < 3d Substitute the three circumferences and solve for x. circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 ANSWER: units. a. 8r and 6r ; Twice the radius of the circle, 2r is The measure of KJ is equal to the sum of the radii the side length of the square, so the perimeter of the of and . square is 4(2r) or 8r. The Therefore, KJ = 16 + 8 or 24 units. regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the ANSWER: hexagon is 6(r) or 6r. 24 units b. less; greater; 6r < C < 8r 51. REASONING Is the distance from the center of a c. 3d < C < 4d; The circumference of the circle is circle to a point in the interior of a circle sometimes, between 3 and 4 times its diameter. always, or never less than the radius of the circle? d. These limits will approach a value of πd, implying Explain. that C = πd. SOLUTION: 50. CHALLENGE The sum of the circumferences of A radius is a segment drawn between the center of circles H, J, and K shown at the right is units. the circle and a point on the circle. A segment drawn Find KJ. from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

SOLUTION: 52. PROOF Use the locus definition of a circle and The radius of is 4x units, the radius of is x dilations to prove that all circles are similar. units, and the radius of is 2x units. Find the SOLUTION: circumference of each circle. A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr . 1

The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

Therefore, is mapped onto . Since there So, the radius of is 4 units, the radius of is exists a rigid motion followed by a scaling that maps 2(4) or 8 units, and the radius of is 4(4) or 16 onto , the circles are units. similar. Thus, all circles are similar. The measure of KJ is equal to the sum of the radii of and . Therefore, KJ = 16 + 8 or 24 units. ANSWER: ANSWER: A circle is a locus of points in a plane equidistant 24 units from a given point. For any two circles and , there exists a translation that maps center A 51. REASONING Is the distance from the center of a onto center B, moving so that it is concentric circle to a point in the interior of a circle sometimes, with . There also exists a dilation with scale always, or never less than the radius of the circle? factor k such that each point that makes up is Explain. moved to be the same distance from center A as the SOLUTION: points that make up are from center B. A radius is a segment drawn between the center of Therefore, is mapped onto . Since there the circle and a point on the circle. A segment drawn exists a rigid motion followed by a scaling that maps from the center to a point inside the circle will always onto , the circles are similar. Thus, all have a length less than the radius of the circle. circles are similar. ANSWER: 53. CHALLENGE In the figure, is inscribed in Always; a radius is a segment drawn between the equilateral triangle LMN. What is the circumference center of the circle and a point on the circle. A

segment drawn from the center to a point inside the of ? circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant SOLUTION: from a given point. For any two circles and , there exists a translation that Draw a perpendicular to the side Also join maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1. Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

Therefore, the radius of is ANSWER: The circumference C of a circle with radius r is A circle is a locus of points in a plane equidistant given by from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the

points that make up are from center B. The circumference of is inches. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps ANSWER: onto , the circles are similar. Thus, all circles are similar. in.

53. CHALLENGE In the figure, is inscribed in 54. WRITING IN MATH Research and write about equilateral triangle LMN. What is the circumference the history of pi and its importance to the study of of ? geometry. SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the SOLUTION: circumference of ? Round to the nearest tenth. Draw a perpendicular to the side Also join

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle. Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects The circumference C of a circle with diameter d is Use the definition of given by tangent of an angle to find PR, which is the radius of Use the formula to find the circumference.

Therefore, the radius of is ANSWER: The circumference C of a circle with radius r is 40.8 given by 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft

D 5 ft The circumference of is inches. SOLUTION: ANSWER: The circumference C of a circle with radius r is given by in. Here, C = 10 ft. Use the formula to find the radius.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry.

SOLUTION: The radius of the circle is about 1.6 feet, so the See students’ work. correct choice is A.

ANSWER: ANSWER: See students’ work. A

55. GRIDDED RESPONSE What is the 57. ALGEBRA Bill is planning a circular vegetable circumference of ? Round to the nearest tenth. garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? F 10 G 9 H 8 J 7 SOLUTION: SOLUTION: Use the Pythagorean Theorem to find the diameter The circumference C of a circle with radius r is of the circle. given by Because 50 feet of fence can be used to border the The circumference C of a circle with diameter d is garden, Cmax = 50. Use the circumference formula to given by find the maximum radius. Use the formula to find the circumference.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J. ANSWER: 40.8 ANSWER: J 56. What is the radius of a table with a circumference of 10 feet? 58. SAT/ACT What is the radius of a circle with an A 1.6 ft area of square units? B 2.5 ft C 3.2 ft A 0.4 units D 5 ft B 0.5 units C 2 units SOLUTION: D 4 units The circumference C of a circle with radius r is E 16 units given by Here, C = 10 ft. Use the formula to find the radius. SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER:

A Therefore, the correct choice is B. ANSWER: 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use B up to 50 feet of fence, what radius can he use for the Copy each figure and point B. Then use a ruler garden? to draw the image of the figure under a dilation F 10 with center B and the scale factor r indicated. G 9 H 8 59. J 7 SOLUTION: The circumference C of a circle with radius r is given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to SOLUTION: find the maximum radius. Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the Therefore, the radius of the garden must be less than hexagon. or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units ANSWER: D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the

formula Here A = square units. Use the 60. formula to find the radius.

SOLUTION:

Therefore, the correct choice is B. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is of B that distance from B. Repeat the process for rays Copy each figure and point B. Then use a ruler through the other two vertices. Connect the three to draw the image of the figure under a dilation points on the rays to construct the dilated image of with center B and the scale factor r indicated. the triangle.

59.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the 61. hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the SOLUTION: hexagon. Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

ANSWER:

60.

62. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from

B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other two vertices. Connect the three Draw a ray from point B through a vertex of the points on the rays to construct the dilated image of triangle. Use the ruler to measure the distance from the triangle. B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

61.

SOLUTION: ANSWER: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state ANSWER: the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No 62.

64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of SOLUTION: 90º. So, it has a rotational symmetry. So, the Draw a ray from point B through a vertex of the magnitude of symmetry is 90º and order of symmetry triangle. Use the ruler to measure the distance from is 4. B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for ANSWER: rays through the other two vertices. Connect the Yes; 4, 90 three points on the rays to construct the dilated image of the triangle.

65. SOLUTION: The given figure cannot be mapped onto itself by any

rotation. So, it does not have a rotational symmetry.

ANSWER: ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

Determine the truth value of the following State whether each figure has rotational statement for each set of conditions. Explain symmetry. Write yes or no. If so, copy the your reasoning. figure, locate the center of symmetry, and state If you are over 18 years old, then you vote in all the order and magnitude of symmetry. elections. 63. Refer to the image on page 705. 67. You are 19 years old and you vote. SOLUTION: SOLUTION: The figure cannot be mapped onto itself by any True; sample answer: Since the hypothesis is true rotation. So, it does not have a rotational symmetry. and the conclusion is true, then the statement is true for the conditions. ANSWER: No ANSWER: True; sample answer: Since the hypothesis is true 64. Refer to the image on page 705. and the conclusion is true, then the statement is true for the conditions. SOLUTION: The figure can be mapped onto itself by a rotation of 68. You are 21 years old and do not vote. 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false ANSWER: for the conditions. Yes; 4, 90 ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. SOLUTION:

The given figure cannot be mapped onto itself by any 69. rotation. So, it does not have a rotational symmetry. SOLUTION: ANSWER: The sum of the measures of angles around a point is 360º. No 66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any ANSWER: rotation. So, it does not have a rotational symmetry. 90 ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all 70. elections. SOLUTION: 67. You are 19 years old and you vote. The angles with the measures xº and 2xº form a SOLUTION: straight line. So, the sum is 180. True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

ANSWER: ANSWER: True; sample answer: Since the hypothesis is true 60 and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false 71. for the conditions. SOLUTION: ANSWER: The angles with the measures 6xº and 3xº form a False; sample answer: Since the hypothesis is true straight line. So, the sum is 180. and the conclusion is false, then the statement is false for the conditions.

Find x. ANSWER: 20

69. SOLUTION: The sum of the measures of angles around a point is 72. 360º. SOLUTION: The sum of the three angles is 180.

ANSWER: 90 ANSWER: 45

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

For Exercises 1–4, refer to . Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

5. FG 2. Identify each. a. a chord SOLUTION: b. a diameter Since the diameter of B is 18 inches and A is 8 c. a radius inches, AG = 18 and AF = (8) or 4. SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords.

b. A diameter of a circle is a chord that passes Therefore, FG = 14 in. through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center ANSWER: and on the circle. Here, is 14 in. radius. 6. FB ANSWER: a. SOLUTION: b. Since the diameter of B is 18 inches and A is 4 c. inches, AB = (18) or 9 and AF = (8) or 4. 3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, Therefore, FB = 5 inches. they are equal in length. Therefore, DN = 8 cm. ANSWER: ANSWER: 5 in. 8 cm RIDES 4. If EN = 13 feet, what is the diameter of the circle? 7. The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the SOLUTION: radius and circumference of the ride? Round to the nearest hundredth, if necessary. Here, is a radius and the diameter is twice the radius. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 Therefore, the circumference of the ride is about inches, 18 inches, and 11 inches, respectively. 138.23 feet. Find each measure. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. 5. FG SOLUTION: Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4.

SOLUTION: Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB

SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

Therefore, FB = 5 inches. ANSWER: 17.98 ft; 8.99 ft ANSWER: 5 in. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of 7. RIDES The circular ride described at the beginning

of the lesson has a diameter of 44 feet. What are the . radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER:

22 ft; 138.23 ft The diameter of the circle is 4 centimeters.

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to .

SOLUTION:

10. Name the center of the circle.

SOLUTION: R ANSWER: R

The diameter of the pool is about 17.98 feet and the 11. Identify a chord that is also a diameter. radius of the pool is about 8.99 feet. SOLUTION: ANSWER: Two chords are shown: and . goes 17.98 ft; 8.99 ft through the center, R, so is a diameter. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of ANSWER: .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center

and on the circle. But has both the end points on SOLUTION: the circle, so it is a chord. The diameter of the circle is the hypotenuse of the right triangle. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION:

The diameter of the circle is 4 centimeters. Here, is a diameter and is a radius.

The circumference of the circle is 4π Therefore, RT = 8.1 cm. centimeters. ANSWER: ANSWER: 8.1 cm

For Exercises 14–17, refer to . For Exercises 10–13, refer to .

14. Identify a chord that is not a diameter.

10. Name the center of the circle. SOLUTION: SOLUTION: The chords do not pass through the R center. So, they are not diameters. ANSWER: ANSWER: 10-1R Circles and Circumference

11. Identify a chord that is also a diameter. 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Two chords are shown: and . goes Here, is a radius and the diameter is twice the through the center, R, so is a diameter. radius. .

ANSWER:

Therefore, the diameter of the circle is 28 inches.

ANSWER: 12. Is a radius? Explain. 28 in. SOLUTION: A radius is a segment with endpoints at the center 16. Is ? Explain. and on the circle. But has both the end points on SOLUTION: the circle, so it is a chord. All radii of a circle are congruent. Since ANSWER: are both radii of , . No; it is a chord. ANSWER: 13. If SU = 16.2 centimeters, what is RT? Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? Here, is a diameter and is a radius. SOLUTION: Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm.

ANSWER: Therefore, EF = 3.7 cm. 8.1 cm ANSWER: For Exercises 14–17, refer to . 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

14. Identify a chord that is not a diameter. SOLUTION: The chords do not pass through the center. So, they are not diameters. 18. CK ANSWER: SOLUTION: Since the radius of K is 8 units, BK = 8.

15. If CF = 14 inches, what is the diameter of the circle? eSolutions Manual - Powered by Cognero Page 3 SOLUTION: Therefore, CK = 2.6 units. Here, is a radius and the diameter is twice the radius. . ANSWER: 2.6

19. AB Therefore, the diameter of the circle is 28 inches. SOLUTION: ANSWER: Since is a diameter of circle J and the radius is 28 in. 10, AC = 2(10) or 20 units.

16. Is ? Explain.

SOLUTION: Therefore, AB = 14.6 units. All radii of a circle are congruent. Since ANSWER: are both radii of , . 14.6 ANSWER: 20. JK Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? First find CK. Since circle K has a radius of 8 units, BK = 8. SOLUTION: Here, is a diameter and is a radius.

Since circle J has a radius of 10 units, JC = 10.

Therefore, EF = 3.7 cm. Therefore, JK = 12.6 units.

ANSWER: ANSWER: 3.7 cm 12.6 Circle J has a radius of 10 units, has a 21. AD radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

18. CK SOLUTION: Since the radius of K is 8 units, BK = 8. Therefore, AD = 30.6 units.

ANSWER: 30.6 Therefore, CK = 2.6 units. 22. PIZZA Find the radius and circumference of the ANSWER: pizza shown. Round to the nearest hundredth, if 2.6 necessary.

19. AB SOLUTION: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units. SOLUTION: The diameter of the pizza is 16 inches. ANSWER: 14.6 20. JK SOLUTION: . First find CK. Since circle K has a radius of 8 units, So, the radius of the pizza is 8 inches. BK = 8. The circumference C of a circle of diameter d is given by

Since circle J has a radius of 10 units, JC = 10.

Therefore, the circumference of the pizza is about 50.27 inches.

Therefore, JK = 12.6 units. ANSWER: 8 in.; 50.27 in. ANSWER: 12.6 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a 21. AD tire. Round to the nearest hundredth, if necessary. SOLUTION: SOLUTION: We can find AD using AD = AC + CK + KD. The The diameter of a tire is 26 inches. The radius is half radius of circle K is 8 units and the radius of circle J the diameter. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is

given by Therefore, AD = 30.6 units.

ANSWER: 30.6 Therefore, the circumference of the bicycle tire is 22. PIZZA Find the radius and circumference of the about 81.68 inches. pizza shown. Round to the nearest hundredth, if necessary. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The diameter of the pizza is 16 inches. Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the diameter is about 5.73 inches and the Therefore, the circumference of the pizza is about radius is about 2.86 inches. 50.27 inches. ANSWER: ANSWER: 5.73 in.; 2.86 in. 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 25. C = 124 ft 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The diameter of a tire is 26 inches. The radius is half Here, C = 124 ft. Use the formula to find the the diameter. diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the diameter is about 39.47 feet and the Therefore, the circumference of the bicycle tire is radius is about 19.74 feet. about 81.68 inches. ANSWER: ANSWER: 39.47 ft; 19.74 ft 13 in.; 81.68 in.

Find the diameter and radius of a circle by with 26. C = 375.3 cm the given circumference. Round to the nearest SOLUTION: hundredth. The circumference C of a circle with diameter d is 24. C = 18 in. given by SOLUTION: Here, C = 375.3 cm. Use the formula to find the

The circumference C of a circle with diameter d is diameter. Then find the radius. given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the diameter is about 5.73 inches and the 119.46 cm; 59.73 cm radius is about 2.86 inches. 27. C = 2608.25 m ANSWER: 5.73 in.; 2.86 in. SOLUTION: The circumference C of a circle with diameter d is 25. C = 124 ft given by Here, C = 2608.25 meters. Use the formula to find SOLUTION: the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given ANSWER: inscribed or 39.47 ft; 19.74 ft circumscribed polygon. 26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the 28. diameter. Then find the radius. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the circumference of the circle is 17 119.46 cm; 59.73 cm centimeters.

27. C = 2608.25 m ANSWER: SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29.

SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: The diameter of the circle is 12 feet. The

830.23 m; 415.12 m circumference C of a circle is given by CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or Therefore, the circumference of the circle is 12 circumscribed polygon. feet.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of

the circle. Use the Pythagorean Theorem to find the 30. diameter. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. The diameter of the circle is 17 cm. The circumference C of a circle is given by

The diameter of the circle is inches. The Therefore, the circumference of the circle is 17 circumference C of a circle is given by centimeters.

ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 31. SOLUTION: Each diagonal of the inscribed rectangle will pass

through the origin, so it is a diameter of the circle. The diameter of the circle is 12 feet. The Use the Pythagorean Theorem to find the diameter

circumference C of a circle is given by of the circle.

Therefore, the circumference of the circle is 12 The diameter of the circle is 10 inches. The feet. circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER: 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The diameter of the circle is inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 25 millimeters. Therefore, the circumference of the circle is inches. ANSWER: ANSWER:

31. 33. SOLUTION: SOLUTION: Each diagonal of the inscribed rectangle will pass A diameter perpendicular to a side of the square and through the origin, so it is a diameter of the circle. the length of each side will measure the same. So, d Use the Pythagorean Theorem to find the diameter = 14 yards. of the circle. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 The diameter of the circle is 10 inches. The yards. circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 10 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and inches. clubs. For professional competitions, the maximum ANSWER: weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the 32. diameter. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters.

Therefore, the circumference of the circle is 25 Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. millimeters. ANSWER: ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by Therefore, the circumference of the circle is 14 yards. Use the formula to find the circumference.

ANSWER:

Therefore, the circumference is about 31.42 feet. 34. DISC GOLF Disc golf is similar to regular golf, b. except that a flying disc is used instead of a ball and Here, Cinner = 25 ft. Use the formula to find the clubs. For professional competitions, the maximum radius of the inner circle. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is So, the radius of the inner circle is about 4 feet.

given by ANSWER: Here, C = 66.92 cm. Use the formula to find the diameter. a. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

The diameter of the disc is about 21.3 centimeters 36. and the maximum weight allowed is 8.3 times the diameter in centimeters. SOLUTION: Therefore, the maximum weight is 8.3 × 21.3 or The radius is half the diameter and the about 176.8 grams. circumference C of a circle with diameter d is given by ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown.

a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

37. SOLUTION: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by The diameter is twice the radius and the circumference C of a circle with diameter d is given Use the formula to find the circumference. by

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft So, the radius of the inner circle is about 4 feet. 38. ANSWER: a. 31.42 ft SOLUTION: b. 4 ft The circumference C of a circle with diameter d is given by The radius, diameter, or circumference of a Here, C = 35x cm. Use the formula to find the circle is given. Find each missing measure to diameter.Then find the radius. the nearest hundredth.

36.

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 11.14x cm; 5.57x cm

Therefore, the radius is 4.25 inches and the 39. circumference is about 26.70 inches.

ANSWER: SOLUTION: 4.25 in.; 26.70 in. The diameter is twice the radius and the circumference C of a circle with diameter d is given by 37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. ANSWER: 0.25x; 0.79x

So, the diameter is 22.80 feet and the circumference Determine whether the circles in the figures is about 71.63 feet. below appear to be congruent, concentric, or neither. ANSWER: 40. Refer to the photo on page 703. 22.80 ft; 71.63 ft SOLUTION: 38. The circles have the same center and they are in the same plane. So, they are concentric circles. SOLUTION: The circumference C of a circle with diameter d is ANSWER: given by concentric Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. 41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION:

Therefore, the diameter is about 11.14x centimeters The circles appear to have the same radius. So, they appear to be congruent. and the radius is about 5.57x centimeters. ANSWER: ANSWER: congruent 11.14x cm; 5.57x cm 43. HISTORY The Indian Shell Ring on Hilton Head

39. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would SOLUTION: someone have to walk to go completely around the ring? Round to the nearest tenth. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. SOLUTION: ANSWER: The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a 0.25x; 0.79x circle with diameter d is given by Determine whether the circles in the figures So, the circumference is below appear to be congruent, concentric, or Therefore, someone have to neither. walk about 471.2 feet to go completely around the 40. Refer to the photo on page 703. ring.

SOLUTION: ANSWER: The circles have the same center and they are in the 471.2 ft same plane. So, they are concentric circles. 44. CCSS MODELING A brick path is being installed ANSWER: around a circular pond. The pond has a concentric circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way 41. Refer to the photo on page 703. around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: The circles neither have the same radius nor are they SOLUTION: concentric. The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the ANSWER: formula to find the radius of the pond. neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they

appear to be congruent. So, the radius of the pond is about 10.82 feet and the ANSWER: radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference congruent of the brick path. 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the Therefore, the circumference of the path is about ring? Round to the nearest tenth. 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. SOLUTION: c. VERBAL Explain why these three circles are The radius of the circle is 3 units, that is 75 feet. So, geometrically similar. the diameter is 150 ft. The circumference C of a d. VERBAL Make a conjecture about the ratio circle with diameter d is given by between the circumferences of two circles when the So, the circumference is ratio between their radii is 2. Therefore, someone have to e. ANALYTICAL The scale factor from to walk about 471.2 feet to go completely around the ring. is . Write an equation relating the circumference (C ) of to the circumference ANSWER: A 471.2 ft (CB) of . f. NUMERICAL If the scale factor from to 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a is , and the circumference of is 12 circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way inches, what is the circumference of ? around. What is the approximate circumference of SOLUTION: the path? Round to the nearest hundredth. a. Sample answer: SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

b. Sample answer.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. Therefore, the circumference of the path is about So, if the radii are in the ratio b : a, so will the 93.13 feet. circumference. Therefore, . ANSWER: 93.13 ft f. Use the formula from part e. Here, the circumference of is 12 inches and . 45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. Therefore, the circumference of is 4 inches. b. TABULAR Calculate the radius (to the nearest ANSWER: tenth) and circumference (to the nearest hundredth) a. Sample answer: of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to b. Sample answer. is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to is , and the circumference of is 12 c. They all have the same shape—circular. inches, what is the circumference of ? d. The ratio of their circumferences is also 2.

SOLUTION: e. a. Sample answer: f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

b. Sample answer.

c. They all have the same shape–circular. a. Drop the needle onto the paper. When the needle d. The ratio of their circumferences is also 2. lands, record whether it touches one of the lines as a e. The circumference is directly proportional to radii. hit. Record the number of hits after 25, 50, and 100 So, if the radii are in the ratio b : a, so will the drops.

circumference. Therefore, . f. Use the formula from part e. Here, the b. Calculate the ratio of two times the total number circumference of is 12 inches and . of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? Therefore, the circumference of is 4 inches. SOLUTION: ANSWER: a. See students’ work. a. Sample answer: b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. b. Sample answer. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper. a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle. a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 The difference of the circumferences of the drops. outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. c. How are the values you found in part b related to b. Let the radius of one circle be x miles and the π? radius of the next larger circle be x + 5 miles. Find the circumference of each circle. SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: The difference of the two circumferences is (2xπ + a. See students’ work. 10π) - (2xπ) = 10π or about 31.4 miles. b. See students’ work. Therefore, as the radius of each circle increases by 5 c. Sample answer: The values are approaching 3.14, miles, the circumference increases by about 31.4 which is approximately equal to π. miles.

ANSWER: 47. M APS The concentric circles on the map below a. 157.1 mi show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. b. by about 31.4 mi 48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. a. How much greater is the circumference of the A line segment with endpoints on a circle can be outermost circle than the circumference of the center circle? described as a chord. If the chord passes through b. As the radii of the circles increases by 5 miles, by the center of the circle, it can be described as a how much does the circumference increase? diameter. A line segment with endpoints at the center and on the circle can be described as a SOLUTION: radius. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the ANSWER: circumference of each circle. Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be The difference of the circumferences of the described as a radius. outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. 49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or The difference of the two circumferences is (2xπ + less than the perimeter of the circumscribed polygon? 10π) - (2xπ) = 10π or about 31.4 miles. the inscribed polygon? Write a compound inequality Therefore, as the radius of each circle increases by 5 comparing C to these perimeters. miles, the circumference increases by about 31.4 c. Rewrite the inequality from part b in terms of the miles. diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed ANSWER: and inscribed polygons increase, what will happen to a. 157.1 mi the upper and lower limits of the inequality from part b. by about 31.4 mi c, and what does this imply? SOLUTION: 48. WRITING IN MATH How can we describe the relationships that exist between a. The circumscribed polygon is a square with sides circles and lines? of length 2r. SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A The inscribed polygon is a hexagon with sides of line that intersects a circle in one point can be length r. (Each triangle formed using a central angle described as a tangent. A line that intersects a circle will be equilateral.) in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one So, the perimeters are 8r and 6r. point can be described as a tangent. A line that b. intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, The circumference is less than the perimeter of the it can be described as a diameter. A line segment circumscribed polygon and greater than the with endpoints at the center and on the circle can be perimeter of the inscribed polygon. described as a radius. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. 49. d. These limits will approach a value of πd, implying REASONING In the figure, a circle with radius r is that C = πd. inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and ANSWER: inscribed polygons in terms of r? Explain. a. 8r and 6r ; Twice the radius of the circle, 2r is b. Is the circumference C of the circle greater or the side length of the square, so the perimeter of the less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality square is 4(2r) or 8r. The comparing C to these perimeters. regular hexagon is made up of six equilateral c. Rewrite the inequality from part b in terms of the triangles with side length r, so the perimeter of the diameter d of the circle and interpret its meaning. hexagon is 6(r) or 6r. d. As the number of sides of both the circumscribed b. less; greater; 6r < C < 8r and inscribed polygons increase, what will happen to c. 3d < C < 4d; The circumference of the circle is the upper and lower limits of the inequality from part between 3 and 4 times its diameter. c, and what does this imply? d. These limits will approach a value of πd, implying SOLUTION: that C = πd. a. The circumscribed polygon is a square with sides 50. CHALLENGE The sum of the circumferences of of length 2r. circles H, J, and K shown at the right is units. Find KJ.

The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) The sum of the circumferences is 56π units. or 4d and 6r = 3(2r) or 3d. Substitute the three circumferences and solve for x. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. ANSWER: The measure of KJ is equal to the sum of the radii a. 8r and 6r ; Twice the radius of the circle, 2r is of and . the side length of the square, so the perimeter of the Therefore, KJ = 16 + 8 or 24 units. square is 4(2r) or 8r. The ANSWER: regular hexagon is made up of six equilateral 24 units triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. 51. REASONING Is the distance from the center of a b. less; greater; 6r < C < 8r circle to a point in the interior of a circle sometimes, c. 3d < C < 4d; The circumference of the circle is always, or never less than the radius of the circle? between 3 and 4 times its diameter. Explain. d. These limits will approach a value of πd, implying SOLUTION: that C = πd. A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn 50. CHALLENGE The sum of the circumferences of from the center to a point inside the circle will always circles H, J, and K shown at the right is units. have a length less than the radius of the circle. Find KJ. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: The radius of is 4x units, the radius of is x SOLUTION: units, and the radius of is 2x units. Find the A circle is a locus of points in a plane equidistant circumference of each circle. from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are So, the radius of is 4 units, the radius of is similar. Thus, all circles are similar. 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii of and . ANSWER: Therefore, KJ = 16 + 8 or 24 units. A circle is a locus of points in a plane equidistant from a given point. For any two circles and ANSWER: , there exists a translation that maps center A 24 units onto center B, moving so that it is concentric 51. REASONING Is the distance from the center of a with . There also exists a dilation with scale circle to a point in the interior of a circle sometimes, factor k such that each point that makes up is always, or never less than the radius of the circle? moved to be the same distance from center A as the Explain. points that make up are from center B. Therefore, is mapped onto . Since there SOLUTION: exists a rigid motion followed by a scaling that maps A radius is a segment drawn between the center of onto , the circles are similar. Thus, all the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always circles are similar. have a length less than the radius of the circle. 53. CHALLENGE In the figure, is inscribed in ANSWER: equilateral triangle LMN. What is the circumference Always; a radius is a segment drawn between the of ? center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: SOLUTION: A circle is a locus of points in a plane equidistant Draw a perpendicular to the side Also join from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r = 2 Since is equilateral triangle, MN = 8 inches kr . 1 and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. Therefore, the radius of is The circumference C of a circle with radius r is ANSWER: given by A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is The circumference of is inches. moved to be the same distance from center A as the points that make up are from center B. ANSWER: Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps in. onto , the circles are similar. Thus, all circles are similar. 54. WRITING IN MATH Research and write about 53. CHALLENGE In the figure, is inscribed in the history of pi and its importance to the study of geometry. equilateral triangle LMN. What is the circumference of ? SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. SOLUTION: Draw a perpendicular to the side Also join

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

Since is equilateral triangle, MN = 8 inches The circumference C of a circle with diameter d is and RN = 4 inches. Also, bisects given by Use the definition of Use the formula to find the circumference. tangent of an angle to find PR, which is the radius of

ANSWER: Therefore, the radius of is 40.8 The circumference C of a circle with radius r is given by 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft

SOLUTION: The circumference of is inches. The circumference C of a circle with radius r is given by ANSWER: Here, C = 10 ft. Use the formula to find the radius.

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. The radius of the circle is about 1.6 feet, so the correct choice is A. SOLUTION: See students’ work. ANSWER: A ANSWER: See students’ work. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use 55. GRIDDED RESPONSE What is the up to 50 feet of fence, what radius can he use for the circumference of ? Round to the nearest tenth. garden? F 10 G 9 H 8 J 7 SOLUTION: SOLUTION: The circumference C of a circle with radius r is Use the Pythagorean Theorem to find the diameter given by of the circle. Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to The circumference C of a circle with diameter d is find the maximum radius. given by Use the formula to find the circumference.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: ANSWER: J 40.8 58. SAT/ACT What is the radius of a circle with an 56. What is the radius of a table with a circumference of 10 feet? area of square units? A 1.6 ft A 0.4 units B 2.5 ft B 0.5 units C 3.2 ft C 2 units D 5 ft D 4 units SOLUTION: E 16 units The circumference C of a circle with radius r is SOLUTION: given by The area of a circle of radius r is given by the Here, C = 10 ft. Use the formula to find the radius. formula Here A = square units. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the correct choice is A. Therefore, the correct choice is B. ANSWER: A ANSWER: B 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use Copy each figure and point B. Then use a ruler up to 50 feet of fence, what radius can he use for the to draw the image of the figure under a dilation garden? with center B and the scale factor r indicated. F 10 59. G 9 H 8 J 7 SOLUTION: The circumference C of a circle with radius r is given by Because 50 feet of fence can be used to border the SOLUTION: garden, C = 50. Use the circumference formula to Draw a ray from point B through a vertex of the max hexagon. Use the ruler to measure the distance from find the maximum radius. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units?

A 0.4 units B 0.5 units ANSWER: C 2 units D 4 units E 16 units

SOLUTION: The area of a circle of radius r is given by the 60. formula Here A = square units. Use the formula to find the radius.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from Therefore, the correct choice is B. B to the vertex. Mark a point on the ray that is of ANSWER: that distance from B. Repeat the process for rays B through the other two vertices. Connect the three points on the rays to construct the dilated image of Copy each figure and point B. Then use a ruler the triangle. to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

SOLUTION: 61. Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other five vertices. Connect the six points on the rays to construct the dilated image of the Draw a ray from point B through a vertex of the hexagon. trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER: ANSWER:

60.

62.

SOLUTION: Draw a ray from point B through a vertex of the

triangle. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is of Draw a ray from point B through a vertex of the that distance from B. Repeat the process for rays triangle. Use the ruler to measure the distance from through the other two vertices. Connect the three B to the vertex. Mark a point on the ray that is 3 points on the rays to construct the dilated image of times that distance from B. Repeat the process for the triangle. rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

61.

ANSWER: SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. ANSWER: SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

62. 64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is 3 Yes; 4, 90 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain State whether each figure has rotational your reasoning. symmetry. Write yes or no. If so, copy the If you are over 18 years old, then you vote in all figure, locate the center of symmetry, and state elections. the order and magnitude of symmetry. 67. You are 19 years old and you vote. 63. Refer to the image on page 705. SOLUTION: SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true The figure cannot be mapped onto itself by any for the conditions. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: True; sample answer: Since the hypothesis is true No and the conclusion is true, then the statement is true for the conditions. 64. Refer to the image on page 705. SOLUTION: 68. You are 21 years old and do not vote. The figure can be mapped onto itself by a rotation of SOLUTION: 90º. So, it has a rotational symmetry. So, the False; sample answer: Since the hypothesis is true magnitude of symmetry is 90º and order of symmetry and the conclusion is false, then the statement is false is 4. for the conditions.

ANSWER: ANSWER: Yes; 4, 90 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. 69. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. The sum of the measures of angles around a point is 360º. ANSWER: No

66. Refer to the image on page 705. ANSWER: SOLUTION: 90 The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain 70. your reasoning. If you are over 18 years old, then you vote in all SOLUTION: elections. The angles with the measures xº and 2xº form a 67. You are 19 years old and you vote. straight line. So, the sum is 180.

SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. ANSWER: ANSWER: 60 True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote.

SOLUTION: 71. False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false SOLUTION: for the conditions. The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180. ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. ANSWER: Find x. 20

69. SOLUTION: 72. The sum of the measures of angles around a point is SOLUTION: 360º. The sum of the three angles is 180.

ANSWER: ANSWER: 90 45

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

For Exercises 1–4, refer to . Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

5. FG 2. Identify each. a. a chord SOLUTION: b. a diameter Since the diameter of B is 18 inches and A is 8 c. a radius inches, AG = 18 and AF = (8) or 4. SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords.

b. A diameter of a circle is a chord that passes Therefore, FG = 14 in. through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center ANSWER: and on the circle. Here, is 14 in. radius. 6. FB ANSWER: a. SOLUTION: b. Since the diameter of B is 18 inches and A is 4 c. inches, AB = (18) or 9 and AF = (8) or 4.

3. If CN = 8 centimeters, find DN. SOLUTION:

Here, are radii of the same circle. So, Therefore, FB = 5 inches. they are equal in length. Therefore, DN = 8 cm. ANSWER: ANSWER: 5 in. 8 cm 7. RIDES The circular ride described at the beginning 4. If EN = 13 feet, what is the diameter of the circle? of the lesson has a diameter of 44 feet. What are the SOLUTION: radius and circumference of the ride? Round to the nearest hundredth, if necessary. Here, is a radius and the diameter is twice the radius. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 Therefore, the circumference of the ride is about inches, 18 inches, and 11 inches, respectively. 138.23 feet. Find each measure. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. 5. FG SOLUTION: Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4.

SOLUTION: Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB

SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

Therefore, FB = 5 inches. ANSWER: 17.98 ft; 8.99 ft ANSWER: 5 in. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of 7. RIDES The circular ride described at the beginning . of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: The diameter of the circle is 4 centimeters. 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round

to the nearest hundredth. The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to .

SOLUTION:

10. Name the center of the circle.

SOLUTION: R ANSWER: R

The diameter of the pool is about 17.98 feet and the 11. Identify a chord that is also a diameter. radius of the pool is about 8.99 feet. SOLUTION: ANSWER: Two chords are shown: and . goes 17.98 ft; 8.99 ft through the center, R, so is a diameter.

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of ANSWER: .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on SOLUTION: the circle, so it is a chord. The diameter of the circle is the hypotenuse of the right triangle. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION:

Here, is a diameter and is a radius. The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π Therefore, RT = 8.1 cm. centimeters. ANSWER: ANSWER: 8.1 cm

For Exercises 14–17, refer to . For Exercises 10–13, refer to .

14. Identify a chord that is not a diameter.

10. Name the center of the circle. SOLUTION: SOLUTION: The chords do not pass through the R center. So, they are not diameters. ANSWER: ANSWER: R

11. Identify a chord that is also a diameter. 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Two chords are shown: and . goes Here, is a radius and the diameter is twice the

through the center, R, so is a diameter. radius. .

ANSWER: Therefore, the diameter of the circle is 28 inches. ANSWER: 12. Is a radius? Explain. 28 in. SOLUTION: A radius is a segment with endpoints at the center 16. Is ? Explain. and on the circle. But has both the end points on SOLUTION:

the circle, so it is a chord. All radii of a circle are congruent. Since ANSWER: are both radii of , . No; it is a chord. ANSWER: 13. If SU = 16.2 centimeters, what is RT? Yes; they are both radii of .

SOLUTION: 17. If DA = 7.4 centimeters, what is EF? Here, is a diameter and is a radius. SOLUTION: Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm.

ANSWER: Therefore, EF = 3.7 cm. 8.1 cm ANSWER: For Exercises 14–17, refer to . 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

14. Identify a chord that is not a diameter. SOLUTION: The chords do not pass through the center. So, they are not diameters. 18. CK ANSWER: SOLUTION: Since the radius of K is 8 units, BK = 8.

15. If CF = 14 inches, what is the diameter of the circle?

SOLUTION: Therefore, CK = 2.6 units. Here, is a radius and the diameter is twice the ANSWER: radius. . 2.6

19. AB

Therefore, the diameter of the circle is 28 inches. SOLUTION: ANSWER: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. 28 in.

16. Is ? Explain.

SOLUTION: Therefore, AB = 14.6 units. All radii of a circle are congruent. Since ANSWER:

are both radii of , . 14.6 ANSWER: 20. JK Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? First find CK. Since circle K has a radius of 8 units, BK = 8. SOLUTION: Here, is a diameter and is a radius.

Since circle J has a radius of 10 units, JC = 10.

Therefore, EF = 3.7 cm. Therefore, JK = 12.6 units.

10-1ANSWER: Circles and Circumference ANSWER: 3.7 cm 12.6

Circle J has a radius of 10 units, has a 21. AD radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

18. CK SOLUTION: Since the radius of K is 8 units, BK = 8. Therefore, AD = 30.6 units. ANSWER: 30.6 Therefore, CK = 2.6 units. 22. PIZZA Find the radius and circumference of the ANSWER: pizza shown. Round to the nearest hundredth, if 2.6 necessary.

19. AB SOLUTION: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units. SOLUTION: The diameter of the pizza is 16 inches. ANSWER: 14.6 20. JK SOLUTION: . First find CK. Since circle K has a radius of 8 units, So, the radius of the pizza is 8 inches. BK = 8. The circumference C of a circle of diameter d is given by

Since circle J has a radius of 10 units, JC = 10. Therefore, the circumference of the pizza is about 50.27 inches.

Therefore, JK = 12.6 units. ANSWER: 8 in.; 50.27 in. ANSWER: 12.6 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a 21. AD tire. Round to the nearest hundredth, if necessary. eSolutions Manual - Powered by Cognero Page 4 SOLUTION: SOLUTION: We can find AD using AD = AC + CK + KD. The The diameter of a tire is 26 inches. The radius is half radius of circle K is 8 units and the radius of circle J the diameter. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by Therefore, AD = 30.6 units.

ANSWER: 30.6 Therefore, the circumference of the bicycle tire is 22. PIZZA Find the radius and circumference of the about 81.68 inches. pizza shown. Round to the nearest hundredth, if necessary. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by Here, C = 18 in. Use the formula to find the The diameter of the pizza is 16 inches. diameter. Then find the radius.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the diameter is about 5.73 inches and the Therefore, the circumference of the pizza is about radius is about 2.86 inches. 50.27 inches. ANSWER: ANSWER: 5.73 in.; 2.86 in. 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 25. C = 124 ft 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The diameter of a tire is 26 inches. The radius is half Here, C = 124 ft. Use the formula to find the the diameter. diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the diameter is about 39.47 feet and the Therefore, the circumference of the bicycle tire is radius is about 19.74 feet. about 81.68 inches. ANSWER: ANSWER: 39.47 ft; 19.74 ft 13 in.; 81.68 in. 26. C = 375.3 cm Find the diameter and radius of a circle by with the given circumference. Round to the nearest SOLUTION: hundredth. The circumference C of a circle with diameter d is 24. C = 18 in. given by SOLUTION: Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the diameter is about 5.73 inches and the 119.46 cm; 59.73 cm radius is about 2.86 inches. 27. C = 2608.25 m ANSWER: 5.73 in.; 2.86 in. SOLUTION: The circumference C of a circle with diameter d is

25. C = 124 ft given by Here, C = 2608.25 meters. Use the formula to find SOLUTION: the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given ANSWER: inscribed or 39.47 ft; 19.74 ft circumscribed polygon.

26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the 28. diameter. Then find the radius. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the circumference of the circle is 17 119.46 cm; 59.73 cm centimeters.

27. C = 2608.25 m ANSWER: SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and

the radius is about 415.12 meters. The diameter of the circle is 12 feet. The ANSWER: circumference C of a circle is given by 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given

inscribed or Therefore, the circumference of the circle is 12 circumscribed polygon. feet.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of 30. the circle. Use the Pythagorean Theorem to find the diameter. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

The diameter of the circle is inches. The Therefore, the circumference of the circle is 17 circumference C of a circle is given by centimeters.

ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle.

The diameter of the circle is 12 feet. The Use the Pythagorean Theorem to find the diameter of the circle. circumference C of a circle is given by

Therefore, the circumference of the circle is 12 The diameter of the circle is 10 inches. The feet. circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER: 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d

= 25 millimeters. The diameter of the circle is inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 25 millimeters. Therefore, the circumference of the circle is inches. ANSWER:

ANSWER:

31. 33. SOLUTION: SOLUTION: Each diagonal of the inscribed rectangle will pass A diameter perpendicular to a side of the square and through the origin, so it is a diameter of the circle. the length of each side will measure the same. So, d Use the Pythagorean Theorem to find the diameter = 14 yards. of the circle. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 The diameter of the circle is 10 inches. The yards. circumference C of a circle is given by ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, Therefore, the circumference of the circle is 10 except that a flying disc is used instead of a ball and inches. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in ANSWER: centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the 32. diameter. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or Therefore, the circumference of the circle is 25 about 176.8 grams. millimeters. ANSWER: ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by Therefore, the circumference of the circle is 14 yards. Use the formula to find the circumference.

ANSWER:

Therefore, the circumference is about 31.42 feet. 34. DISC GOLF Disc golf is similar to regular golf, b. Here, C = 25 ft. Use the formula to find the except that a flying disc is used instead of a ball and inner clubs. For professional competitions, the maximum radius of the inner circle. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is So, the radius of the inner circle is about 4 feet. given by ANSWER: Here, C = 66.92 cm. Use the formula to find the a. diameter. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

The diameter of the disc is about 21.3 centimeters 36. and the maximum weight allowed is 8.3 times the diameter in centimeters. SOLUTION: Therefore, the maximum weight is 8.3 × 21.3 or The radius is half the diameter and the about 176.8 grams. circumference C of a circle with diameter d is given by ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown.

a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

37. SOLUTION: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by The diameter is twice the radius and the circumference C of a circle with diameter d is given

Use the formula to find the circumference. by

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft So, the radius of the inner circle is about 4 feet. 38. ANSWER: a. 31.42 ft SOLUTION: b. 4 ft The circumference C of a circle with diameter d is given by The radius, diameter, or circumference of a Here, C = 35x cm. Use the formula to find the circle is given. Find each missing measure to diameter.Then find the radius. the nearest hundredth.

36.

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 11.14x cm; 5.57x cm

Therefore, the radius is 4.25 inches and the 39. circumference is about 26.70 inches.

ANSWER: SOLUTION: 4.25 in.; 26.70 in. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. ANSWER: 0.25x; 0.79x

So, the diameter is 22.80 feet and the circumference Determine whether the circles in the figures is about 71.63 feet. below appear to be congruent, concentric, or neither. ANSWER: 40. Refer to the photo on page 703. 22.80 ft; 71.63 ft SOLUTION: 38. The circles have the same center and they are in the same plane. So, they are concentric circles. SOLUTION: The circumference C of a circle with diameter d is ANSWER: given by concentric Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. 41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION:

The circles appear to have the same radius. So, they Therefore, the diameter is about 11.14x centimeters appear to be congruent. and the radius is about 5.57x centimeters. ANSWER: ANSWER: congruent 11.14x cm; 5.57x cm 43. HISTORY The Indian Shell Ring on Hilton Head

39. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the SOLUTION: ring? Round to the nearest tenth. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. SOLUTION: ANSWER: The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a 0.25x; 0.79x circle with diameter d is given by Determine whether the circles in the figures So, the circumference is below appear to be congruent, concentric, or Therefore, someone have to neither. walk about 471.2 feet to go completely around the 40. Refer to the photo on page 703. ring.

SOLUTION: ANSWER: The circles have the same center and they are in the 471.2 ft same plane. So, they are concentric circles. 44. CCSS MODELING A brick path is being installed ANSWER: around a circular pond. The pond has a concentric circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way 41. Refer to the photo on page 703. around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: SOLUTION: The circles neither have the same radius nor are they concentric. The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the ANSWER: formula to find the radius of the pond. neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they appear to be congruent. So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 ANSWER: = 14.82 ft. Use the radius to find the circumference congruent of the brick path. 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the Therefore, the circumference of the path is about ring? Round to the nearest tenth. 93.13 feet. ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. SOLUTION: c. VERBAL Explain why these three circles are The radius of the circle is 3 units, that is 75 feet. So, geometrically similar. the diameter is 150 ft. The circumference C of a d. VERBAL Make a conjecture about the ratio circle with diameter d is given by between the circumferences of two circles when the So, the circumference is ratio between their radii is 2. Therefore, someone have to e. ANALYTICAL The scale factor from to walk about 471.2 feet to go completely around the is . Write an equation relating the ring. circumference (C ) of to the circumference ANSWER: A 471.2 ft (CB) of . f. NUMERICAL If the scale factor from to 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a is , and the circumference of is 12 circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way inches, what is the circumference of ? around. What is the approximate circumference of SOLUTION: the path? Round to the nearest hundredth. a. Sample answer: SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

b. Sample answer.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference

of the brick path. c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. Therefore, the circumference of the path is about So, if the radii are in the ratio b : a, so will the 93.13 feet. circumference. Therefore, . ANSWER: f. Use the formula from part e. Here, the 93.13 ft circumference of is 12 inches and . 45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to Therefore, the circumference of is 4 inches. the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest ANSWER: tenth) and circumference (to the nearest hundredth) a. Sample answer: of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to b. Sample answer. is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to is , and the circumference of is 12 c. They all have the same shape—circular. inches, what is the circumference of ? d. The ratio of their circumferences is also 2.

SOLUTION: e. a. Sample answer: f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

b. Sample answer.

c. They all have the same shape–circular. a. Drop the needle onto the paper. When the needle d. The ratio of their circumferences is also 2. lands, record whether it touches one of the lines as a e. The circumference is directly proportional to radii. hit. Record the number of hits after 25, 50, and 100 So, if the radii are in the ratio b : a, so will the drops.

circumference. Therefore, . f. Use the formula from part e. Here, the b. Calculate the ratio of two times the total number circumference of is 12 inches and . of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? Therefore, the circumference of is 4 inches. SOLUTION: ANSWER: a. See students’ work. b. See students’ work. a. Sample answer: c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. b. Sample answer. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper. a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle. a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number The difference of the circumferences of the of drops to the number of hits after 25, 50, and 100 π - 10π = drops. outermost circle and the center circle is 60 50π or about 157.1 miles. c. How are the values you found in part b related to b. Let the radius of one circle be x miles and the π? radius of the next larger circle be x + 5 miles. Find the circumference of each circle. SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: The difference of the two circumferences is (2xπ + a. See students’ work. 10π) - (2xπ) = 10π or about 31.4 miles. b. See students’ work. Therefore, as the radius of each circle increases by 5 c. Sample answer: The values are approaching 3.14, miles, the circumference increases by about 31.4 which is approximately equal to π. miles.

ANSWER: M APS 47. The concentric circles on the map below a. 157.1 mi show the areas that are 5, 10, 15, 20, 25, and 30 b. by about 31.4 mi miles from downtown Phoenix. 48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. a. How much greater is the circumference of the A line segment with endpoints on a circle can be outermost circle than the circumference of the center described as a chord. If the chord passes through circle? the center of the circle, it can be described as a b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? diameter. A line segment with endpoints at the center and on the circle can be described as a SOLUTION: radius. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the ANSWER: circumference of each circle. Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be The difference of the circumferences of the described as a radius. outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. 49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed

about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or The difference of the two circumferences is (2xπ + less than the perimeter of the circumscribed polygon? 10π) - (2xπ) = 10π or about 31.4 miles. the inscribed polygon? Write a compound inequality Therefore, as the radius of each circle increases by 5 comparing C to these perimeters. miles, the circumference increases by about 31.4 c. Rewrite the inequality from part b in terms of the miles. diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed ANSWER: and inscribed polygons increase, what will happen to a. 157.1 mi the upper and lower limits of the inequality from part c b. by about 31.4 mi , and what does this imply? SOLUTION: WRITING IN MATH How can we describe the 48. a. The circumscribed polygon is a square with sides relationships that exist between of length 2r. circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A The inscribed polygon is a hexagon with sides of line that intersects a circle in one point can be length r. (Each triangle formed using a central angle described as a tangent. A line that intersects a circle will be equilateral.) in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER:

Sample answer: A line that intersects a circle in one So, the perimeters are 8r and 6r. point can be described as a tangent. A line that b. intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, The circumference is less than the perimeter of the it can be described as a diameter. A line segment circumscribed polygon and greater than the with endpoints at the center and on the circle can be perimeter of the inscribed polygon. described as a radius. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. 49. d. These limits will approach a value of πd, implying REASONING In the figure, a circle with radius r is that C = πd. inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and ANSWER: inscribed polygons in terms of r? Explain. a. 8r and 6r ; Twice the radius of the circle, 2r is b. Is the circumference C of the circle greater or the side length of the square, so the perimeter of the less than the perimeter of the circumscribed polygon? square is 4(2r) or 8r. The the inscribed polygon? Write a compound inequality comparing C to these perimeters. regular hexagon is made up of six equilateral c. Rewrite the inequality from part b in terms of the triangles with side length r, so the perimeter of the diameter d of the circle and interpret its meaning. hexagon is 6(r) or 6r. d. As the number of sides of both the circumscribed b. less; greater; 6r < C < 8r and inscribed polygons increase, what will happen to c. 3d < C < 4d; The circumference of the circle is the upper and lower limits of the inequality from part between 3 and 4 times its diameter. c, and what does this imply? d. These limits will approach a value of πd, implying SOLUTION: that C = πd. a. The circumscribed polygon is a square with sides 50. CHALLENGE The sum of the circumferences of of length 2r. circles H, J, and K shown at the right is units. Find KJ.

The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) The sum of the circumferences is 56π units. or 4d and 6r = 3(2r) or 3d. Substitute the three circumferences and solve for x. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying So, the radius of that C = πd. is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. ANSWER: The measure of KJ is equal to the sum of the radii a. 8r and 6r ; Twice the radius of the circle, 2r is of and . Therefore, KJ = 16 + 8 or 24 units. the side length of the square, so the perimeter of the square is 4(2r) or 8r. The ANSWER: regular hexagon is made up of six equilateral 24 units triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. 51. REASONING Is the distance from the center of a b. less; greater; 6r < C < 8r circle to a point in the interior of a circle sometimes, c. 3d < C < 4d; The circumference of the circle is always, or never less than the radius of the circle?

between 3 and 4 times its diameter. Explain. d. These limits will approach a value of πd, implying SOLUTION: that C = πd. A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn 50. CHALLENGE The sum of the circumferences of from the center to a point inside the circle will always circles H, J, and K shown at the right is units. have a length less than the radius of the circle. Find KJ. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and

dilations to prove that all circles are similar. SOLUTION: The radius of is 4x units, the radius of is x SOLUTION: units, and the radius of is 2x units. Find the A circle is a locus of points in a plane equidistant circumference of each circle. from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

The sum of the circumferences is 56π units.

Substitute the three circumferences and solve for x.

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are So, the radius of is 4 units, the radius of is similar. Thus, all circles are similar. 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii of and . ANSWER: Therefore, KJ = 16 + 8 or 24 units. A circle is a locus of points in a plane equidistant from a given point. For any two circles and ANSWER: , there exists a translation that maps center A 24 units onto center B, moving so that it is concentric with . There also exists a dilation with scale 51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, factor k such that each point that makes up is always, or never less than the radius of the circle? moved to be the same distance from center A as the Explain. points that make up are from center B. Therefore, is mapped onto . Since there SOLUTION: exists a rigid motion followed by a scaling that maps A radius is a segment drawn between the center of onto , the circles are similar. Thus, all the circle and a point on the circle. A segment drawn circles are similar. from the center to a point inside the circle will always have a length less than the radius of the circle. 53. CHALLENGE In the figure, is inscribed in ANSWER: equilateral triangle LMN. What is the circumference Always; a radius is a segment drawn between the of ? center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: SOLUTION: A circle is a locus of points in a plane equidistant Draw a perpendicular to the side Also join from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B.

For some value k, r = 2 Since is equilateral triangle, MN = 8 inches kr1. and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. Therefore, the radius of is The circumference C of a circle with radius r is

ANSWER: given by A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale

factor k such that each point that makes up is The circumference of is inches. moved to be the same distance from center A as the points that make up are from center B. ANSWER: Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps in. onto , the circles are similar. Thus, all circles are similar. 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of 53. CHALLENGE In the figure, is inscribed in geometry. equilateral triangle LMN. What is the circumference of ? SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. SOLUTION: Draw a perpendicular to the side Also join

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

Since is equilateral triangle, MN = 8 inches The circumference C of a circle with diameter d is and RN = 4 inches. Also, bisects given by Use the definition of Use the formula to find the circumference. tangent of an angle to find PR, which is the radius of

ANSWER: Therefore, the radius of is 40.8 The circumference C of a circle with radius r is 56. What is the radius of a table with a circumference of given by 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft SOLUTION:

The circumference of is inches. The circumference C of a circle with radius r is given by ANSWER: Here, C = 10 ft. Use the formula to find the radius.

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. The radius of the circle is about 1.6 feet, so the correct choice is A. SOLUTION: See students’ work. ANSWER: A ANSWER: See students’ work. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use 55. GRIDDED RESPONSE What is the up to 50 feet of fence, what radius can he use for the circumference of ? Round to the nearest tenth. garden? F 10 G 9 H 8 J 7 SOLUTION: SOLUTION: The circumference C of a circle with radius r is given by Use the Pythagorean Theorem to find the diameter Because 50 feet of fence can be used to border the of the circle. garden, Cmax = 50. Use the circumference formula to find the maximum radius. The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: ANSWER: J 40.8 58. SAT/ACT What is the radius of a circle with an 56. What is the radius of a table with a circumference of 10 feet? area of square units? A 1.6 ft A 0.4 units B 2.5 ft B 0.5 units C 3.2 ft C 2 units D 5 ft D 4 units SOLUTION: E 16 units The circumference C of a circle with radius r is SOLUTION:

given by The area of a circle of radius r is given by the Here, C = 10 ft. Use the formula to find the radius. formula Here A = square units. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the correct choice is A. Therefore, the correct choice is B. ANSWER: ANSWER: A B 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use Copy each figure and point B. Then use a ruler up to 50 feet of fence, what radius can he use for the to draw the image of the figure under a dilation garden? with center B and the scale factor r indicated.

F 10 59. G 9 H 8 J 7 SOLUTION: The circumference C of a circle with radius r is given by SOLUTION: Because 50 feet of fence can be used to border the garden, C = 50. Use the circumference formula to Draw a ray from point B through a vertex of the max hexagon. Use the ruler to measure the distance from find the maximum radius. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units?

A 0.4 units B 0.5 units ANSWER: C 2 units D 4 units

E 16 units SOLUTION: The area of a circle of radius r is given by the 60. formula Here A = square units. Use the formula to find the radius.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from Therefore, the correct choice is B. B to the vertex. Mark a point on the ray that is of ANSWER: that distance from B. Repeat the process for rays B through the other two vertices. Connect the three points on the rays to construct the dilated image of Copy each figure and point B. Then use a ruler the triangle. to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

SOLUTION: 61. Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other five vertices. Connect the six points Draw a ray from point B through a vertex of the on the rays to construct the dilated image of the trapezoid. Use the ruler to measure the distance from hexagon. B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER: ANSWER:

60.

62.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is of Draw a ray from point B through a vertex of the that distance from B. Repeat the process for rays triangle. Use the ruler to measure the distance from through the other two vertices. Connect the three B to the vertex. Mark a point on the ray that is 3 points on the rays to construct the dilated image of times that distance from B. Repeat the process for the triangle. rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

61.

ANSWER: SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

63. Refer to the image on page 705. ANSWER: SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

62. 64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is 3 Yes; 4, 90 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

65. SOLUTION: The given figure cannot be mapped onto itself by any

rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: No 66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. State whether each figure has rotational If you are over 18 years old, then you vote in all symmetry. Write yes or no. If so, copy the elections. figure, locate the center of symmetry, and state 67. You are 19 years old and you vote. the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true The figure cannot be mapped onto itself by any for the conditions. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: True; sample answer: Since the hypothesis is true No and the conclusion is true, then the statement is true for the conditions. 64. Refer to the image on page 705. SOLUTION: 68. You are 21 years old and do not vote. The figure can be mapped onto itself by a rotation of SOLUTION: 90º. So, it has a rotational symmetry. So, the False; sample answer: Since the hypothesis is true magnitude of symmetry is 90º and order of symmetry and the conclusion is false, then the statement is false is 4. for the conditions.

ANSWER: ANSWER: Yes; 4, 90 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. 69. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any The sum of the measures of angles around a point is rotation. So, it does not have a rotational symmetry. 360º. ANSWER: No

66. Refer to the image on page 705. ANSWER: SOLUTION: 90 The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain 70. your reasoning. If you are over 18 years old, then you vote in all SOLUTION: elections. The angles with the measures xº and 2xº form a 67. You are 19 years old and you vote. straight line. So, the sum is 180.

SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. ANSWER: ANSWER: 60 True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: 71. False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false SOLUTION: for the conditions. The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180. ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. ANSWER: Find x. 20

69. SOLUTION: 72. The sum of the measures of angles around a point is SOLUTION: 360º. The sum of the three angles is 180.

ANSWER: ANSWER: 90 45

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

For Exercises 1–4, refer to . Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

5. FG 2. Identify each. a. a chord SOLUTION: b. a diameter Since the diameter of B is 18 inches and A is 8 c. a radius inches, AG = 18 and AF = (8) or 4. SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords.

b. A diameter of a circle is a chord that passes Therefore, FG = 14 in. through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center ANSWER: and on the circle. Here, is 14 in. radius. 6. FB ANSWER: a. SOLUTION: b. Since the diameter of B is 18 inches and A is 4 c. inches, AB = (18) or 9 and AF = (8) or 4.

3. If CN = 8 centimeters, find DN. SOLUTION:

Here, are radii of the same circle. So, Therefore, FB = 5 inches. they are equal in length. Therefore, DN = 8 cm. ANSWER: ANSWER: 5 in. 8 cm 7. RIDES The circular ride described at the beginning 4. If EN = 13 feet, what is the diameter of the circle? of the lesson has a diameter of 44 feet. What are the SOLUTION: radius and circumference of the ride? Round to the nearest hundredth, if necessary. Here, is a radius and the diameter is twice the radius. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 Therefore, the circumference of the ride is about inches, 18 inches, and 11 inches, respectively. 138.23 feet. Find each measure. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. 5. FG SOLUTION: Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4.

SOLUTION: Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB

SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

Therefore, FB = 5 inches. ANSWER: 17.98 ft; 8.99 ft ANSWER: 5 in. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of 7. RIDES The circular ride described at the beginning . of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: The diameter of the circle is 4 centimeters. 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round

to the nearest hundredth. The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to .

SOLUTION:

10. Name the center of the circle.

SOLUTION: R ANSWER: R

The diameter of the pool is about 17.98 feet and the 11. Identify a chord that is also a diameter. radius of the pool is about 8.99 feet. SOLUTION: ANSWER: Two chords are shown: and . goes 17.98 ft; 8.99 ft through the center, R, so is a diameter.

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of ANSWER: .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on SOLUTION: the circle, so it is a chord. The diameter of the circle is the hypotenuse of the right triangle. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION:

Here, is a diameter and is a radius. The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π Therefore, RT = 8.1 cm. centimeters. ANSWER: ANSWER: 8.1 cm

For Exercises 14–17, refer to . For Exercises 10–13, refer to .

14. Identify a chord that is not a diameter.

10. Name the center of the circle. SOLUTION: SOLUTION: The chords do not pass through the R center. So, they are not diameters. ANSWER: ANSWER: R

11. Identify a chord that is also a diameter. 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Two chords are shown: and . goes Here, is a radius and the diameter is twice the

through the center, R, so is a diameter. radius. .

ANSWER: Therefore, the diameter of the circle is 28 inches. ANSWER: 12. Is a radius? Explain. 28 in. SOLUTION: A radius is a segment with endpoints at the center 16. Is ? Explain. and on the circle. But has both the end points on SOLUTION:

the circle, so it is a chord. All radii of a circle are congruent. Since ANSWER: are both radii of , . No; it is a chord. ANSWER: 13. If SU = 16.2 centimeters, what is RT? Yes; they are both radii of .

SOLUTION: 17. If DA = 7.4 centimeters, what is EF? Here, is a diameter and is a radius. SOLUTION: Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm.

ANSWER: Therefore, EF = 3.7 cm. 8.1 cm ANSWER: For Exercises 14–17, refer to . 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

14. Identify a chord that is not a diameter. SOLUTION: The chords do not pass through the center. So, they are not diameters. 18. CK ANSWER: SOLUTION: Since the radius of K is 8 units, BK = 8.

15. If CF = 14 inches, what is the diameter of the circle?

SOLUTION: Therefore, CK = 2.6 units. Here, is a radius and the diameter is twice the ANSWER: radius. . 2.6

19. AB

Therefore, the diameter of the circle is 28 inches. SOLUTION: ANSWER: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. 28 in.

16. Is ? Explain.

SOLUTION: Therefore, AB = 14.6 units. All radii of a circle are congruent. Since ANSWER:

are both radii of , . 14.6 ANSWER: 20. JK Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? First find CK. Since circle K has a radius of 8 units, BK = 8. SOLUTION: Here, is a diameter and is a radius.

Since circle J has a radius of 10 units, JC = 10.

Therefore, EF = 3.7 cm. Therefore, JK = 12.6 units.

ANSWER: ANSWER: 3.7 cm 12.6

Circle J has a radius of 10 units, has a 21. AD radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

18. CK SOLUTION: Since the radius of K is 8 units, BK = 8. Therefore, AD = 30.6 units. ANSWER: 30.6 Therefore, CK = 2.6 units. 22. PIZZA Find the radius and circumference of the ANSWER: pizza shown. Round to the nearest hundredth, if 2.6 necessary.

19. AB SOLUTION: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units. SOLUTION: The diameter of the pizza is 16 inches. ANSWER: 14.6 20. JK SOLUTION: . First find CK. Since circle K has a radius of 8 units, So, the radius of the pizza is 8 inches. BK = 8. The circumference C of a circle of diameter d is given by

Since circle J has a radius of 10 units, JC = 10. Therefore, the circumference of the pizza is about 50.27 inches.

Therefore, JK = 12.6 units. ANSWER: 8 in.; 50.27 in. ANSWER: 12.6 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a 21. AD tire. Round to the nearest hundredth, if necessary. SOLUTION: SOLUTION: We can find AD using AD = AC + CK + KD. The The diameter of a tire is 26 inches. The radius is half radius of circle K is 8 units and the radius of circle J the diameter. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by Therefore, AD = 30.6 units.

ANSWER: 30.6 Therefore, the circumference of the bicycle tire is 22. PIZZA Find the radius and circumference of the about 81.68 inches. pizza shown. Round to the nearest hundredth, if necessary. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by Here, C = 18 in. Use the formula to find the The diameter of the pizza is 16 inches. diameter. Then find the radius.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the diameter is about 5.73 inches and the Therefore, the circumference of the pizza is about radius is about 2.86 inches. 50.27 inches. ANSWER: ANSWER: 10-1 Circles and Circumference 5.73 in.; 2.86 in. 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 25. C = 124 ft 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The diameter of a tire is 26 inches. The radius is half Here, C = 124 ft. Use the formula to find the the diameter. diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the diameter is about 39.47 feet and the Therefore, the circumference of the bicycle tire is radius is about 19.74 feet. about 81.68 inches. ANSWER: ANSWER: 39.47 ft; 19.74 ft 13 in.; 81.68 in. 26. C = 375.3 cm Find the diameter and radius of a circle by with the given circumference. Round to the nearest SOLUTION: hundredth. The circumference C of a circle with diameter d is 24. C = 18 in. given by SOLUTION: Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the diameter is about 5.73 inches and the 119.46 cm; 59.73 cm radius is about 2.86 inches. 27. C = 2608.25 m ANSWER: 5.73 in.; 2.86 in. SOLUTION: The circumference C of a circle with diameter d is

25. C = 124 ft given by eSolutions Manual - Powered by Cognero Here, C = 2608.25 meters. Use the formula to findPage 5 SOLUTION: the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given ANSWER: inscribed or 39.47 ft; 19.74 ft circumscribed polygon.

26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the 28. diameter. Then find the radius. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the circumference of the circle is 17 119.46 cm; 59.73 cm centimeters.

27. C = 2608.25 m ANSWER: SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and

the radius is about 415.12 meters. The diameter of the circle is 12 feet. The ANSWER: circumference C of a circle is given by 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given

inscribed or Therefore, the circumference of the circle is 12 circumscribed polygon. feet.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of 30. the circle. Use the Pythagorean Theorem to find the diameter. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

The diameter of the circle is inches. The Therefore, the circumference of the circle is 17 circumference C of a circle is given by centimeters.

ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle.

The diameter of the circle is 12 feet. The Use the Pythagorean Theorem to find the diameter of the circle. circumference C of a circle is given by

Therefore, the circumference of the circle is 12 The diameter of the circle is 10 inches. The feet. circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER: 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d

= 25 millimeters. The diameter of the circle is inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 25 millimeters. Therefore, the circumference of the circle is inches. ANSWER:

ANSWER:

31. 33. SOLUTION: SOLUTION: Each diagonal of the inscribed rectangle will pass A diameter perpendicular to a side of the square and through the origin, so it is a diameter of the circle. the length of each side will measure the same. So, d Use the Pythagorean Theorem to find the diameter = 14 yards. of the circle. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 The diameter of the circle is 10 inches. The yards. circumference C of a circle is given by ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, Therefore, the circumference of the circle is 10 except that a flying disc is used instead of a ball and inches. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in ANSWER: centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the 32. diameter. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or Therefore, the circumference of the circle is 25 about 176.8 grams. millimeters. ANSWER: ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by Therefore, the circumference of the circle is 14 yards. Use the formula to find the circumference.

ANSWER:

Therefore, the circumference is about 31.42 feet. 34. DISC GOLF Disc golf is similar to regular golf, b. Here, C = 25 ft. Use the formula to find the except that a flying disc is used instead of a ball and inner clubs. For professional competitions, the maximum radius of the inner circle. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is So, the radius of the inner circle is about 4 feet. given by ANSWER: Here, C = 66.92 cm. Use the formula to find the a. diameter. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

The diameter of the disc is about 21.3 centimeters 36. and the maximum weight allowed is 8.3 times the diameter in centimeters. SOLUTION: Therefore, the maximum weight is 8.3 × 21.3 or The radius is half the diameter and the about 176.8 grams. circumference C of a circle with diameter d is given by ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown.

a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

37. SOLUTION: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by The diameter is twice the radius and the circumference C of a circle with diameter d is given

Use the formula to find the circumference. by

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft So, the radius of the inner circle is about 4 feet. 38. ANSWER: a. 31.42 ft SOLUTION: b. 4 ft The circumference C of a circle with diameter d is given by The radius, diameter, or circumference of a Here, C = 35x cm. Use the formula to find the circle is given. Find each missing measure to diameter.Then find the radius. the nearest hundredth.

36.

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 11.14x cm; 5.57x cm

Therefore, the radius is 4.25 inches and the 39. circumference is about 26.70 inches.

ANSWER: SOLUTION: 4.25 in.; 26.70 in. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. ANSWER: 0.25x; 0.79x

So, the diameter is 22.80 feet and the circumference Determine whether the circles in the figures is about 71.63 feet. below appear to be congruent, concentric, or neither. ANSWER: 40. Refer to the photo on page 703. 22.80 ft; 71.63 ft SOLUTION: 38. The circles have the same center and they are in the same plane. So, they are concentric circles. SOLUTION: The circumference C of a circle with diameter d is ANSWER: given by concentric Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. 41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION:

The circles appear to have the same radius. So, they Therefore, the diameter is about 11.14x centimeters appear to be congruent. and the radius is about 5.57x centimeters. ANSWER: ANSWER: congruent 11.14x cm; 5.57x cm 43. HISTORY The Indian Shell Ring on Hilton Head

39. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the SOLUTION: ring? Round to the nearest tenth. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. SOLUTION: ANSWER: The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a 0.25x; 0.79x circle with diameter d is given by Determine whether the circles in the figures So, the circumference is below appear to be congruent, concentric, or Therefore, someone have to neither. walk about 471.2 feet to go completely around the 40. Refer to the photo on page 703. ring.

SOLUTION: ANSWER: The circles have the same center and they are in the 471.2 ft same plane. So, they are concentric circles. 44. CCSS MODELING A brick path is being installed ANSWER: around a circular pond. The pond has a concentric circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way 41. Refer to the photo on page 703. around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: SOLUTION: The circles neither have the same radius nor are they concentric. The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the ANSWER: formula to find the radius of the pond. neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they appear to be congruent. So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 ANSWER: = 14.82 ft. Use the radius to find the circumference congruent of the brick path. 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the Therefore, the circumference of the path is about ring? Round to the nearest tenth. 93.13 feet. ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. SOLUTION: c. VERBAL Explain why these three circles are The radius of the circle is 3 units, that is 75 feet. So, geometrically similar. the diameter is 150 ft. The circumference C of a d. VERBAL Make a conjecture about the ratio circle with diameter d is given by between the circumferences of two circles when the So, the circumference is ratio between their radii is 2. Therefore, someone have to e. ANALYTICAL The scale factor from to walk about 471.2 feet to go completely around the is . Write an equation relating the ring. circumference (C ) of to the circumference ANSWER: A 471.2 ft (CB) of . f. NUMERICAL If the scale factor from to 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a is , and the circumference of is 12 circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way inches, what is the circumference of ? around. What is the approximate circumference of SOLUTION: the path? Round to the nearest hundredth. a. Sample answer: SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

b. Sample answer.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference

of the brick path. c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. Therefore, the circumference of the path is about So, if the radii are in the ratio b : a, so will the 93.13 feet. circumference. Therefore, . ANSWER: f. Use the formula from part e. Here, the 93.13 ft circumference of is 12 inches and . 45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to Therefore, the circumference of is 4 inches. the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest ANSWER: tenth) and circumference (to the nearest hundredth) a. Sample answer: of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to b. Sample answer. is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to is , and the circumference of is 12 c. They all have the same shape—circular. inches, what is the circumference of ? d. The ratio of their circumferences is also 2.

SOLUTION: e. a. Sample answer: f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

b. Sample answer.

c. They all have the same shape–circular. a. Drop the needle onto the paper. When the needle d. The ratio of their circumferences is also 2. lands, record whether it touches one of the lines as a e. The circumference is directly proportional to radii. hit. Record the number of hits after 25, 50, and 100 So, if the radii are in the ratio b : a, so will the drops.

circumference. Therefore, . f. Use the formula from part e. Here, the b. Calculate the ratio of two times the total number circumference of is 12 inches and . of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? Therefore, the circumference of is 4 inches. SOLUTION: ANSWER: a. See students’ work. b. See students’ work. a. Sample answer: c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. b. Sample answer. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper. a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle. a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number The difference of the circumferences of the of drops to the number of hits after 25, 50, and 100 π - 10π = drops. outermost circle and the center circle is 60 50π or about 157.1 miles. c. How are the values you found in part b related to b. Let the radius of one circle be x miles and the π? radius of the next larger circle be x + 5 miles. Find the circumference of each circle. SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: The difference of the two circumferences is (2xπ + a. See students’ work. 10π) - (2xπ) = 10π or about 31.4 miles. b. See students’ work. Therefore, as the radius of each circle increases by 5 c. Sample answer: The values are approaching 3.14, miles, the circumference increases by about 31.4 which is approximately equal to π. miles.

ANSWER: M APS 47. The concentric circles on the map below a. 157.1 mi show the areas that are 5, 10, 15, 20, 25, and 30 b. by about 31.4 mi miles from downtown Phoenix. 48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. a. How much greater is the circumference of the A line segment with endpoints on a circle can be outermost circle than the circumference of the center described as a chord. If the chord passes through circle? the center of the circle, it can be described as a b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? diameter. A line segment with endpoints at the center and on the circle can be described as a SOLUTION: radius. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the ANSWER: circumference of each circle. Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be The difference of the circumferences of the described as a radius. outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. 49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed

about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or The difference of the two circumferences is (2xπ + less than the perimeter of the circumscribed polygon? 10π) - (2xπ) = 10π or about 31.4 miles. the inscribed polygon? Write a compound inequality Therefore, as the radius of each circle increases by 5 comparing C to these perimeters. miles, the circumference increases by about 31.4 c. Rewrite the inequality from part b in terms of the miles. diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed ANSWER: and inscribed polygons increase, what will happen to a. 157.1 mi the upper and lower limits of the inequality from part c b. by about 31.4 mi , and what does this imply? SOLUTION: WRITING IN MATH How can we describe the 48. a. The circumscribed polygon is a square with sides relationships that exist between of length 2r. circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A The inscribed polygon is a hexagon with sides of line that intersects a circle in one point can be length r. (Each triangle formed using a central angle described as a tangent. A line that intersects a circle will be equilateral.) in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER:

Sample answer: A line that intersects a circle in one So, the perimeters are 8r and 6r. point can be described as a tangent. A line that b. intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, The circumference is less than the perimeter of the it can be described as a diameter. A line segment circumscribed polygon and greater than the with endpoints at the center and on the circle can be perimeter of the inscribed polygon. described as a radius. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. 49. d. These limits will approach a value of πd, implying REASONING In the figure, a circle with radius r is that C = πd. inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and ANSWER: inscribed polygons in terms of r? Explain. a. 8r and 6r ; Twice the radius of the circle, 2r is b. Is the circumference C of the circle greater or the side length of the square, so the perimeter of the less than the perimeter of the circumscribed polygon? square is 4(2r) or 8r. The the inscribed polygon? Write a compound inequality comparing C to these perimeters. regular hexagon is made up of six equilateral c. Rewrite the inequality from part b in terms of the triangles with side length r, so the perimeter of the diameter d of the circle and interpret its meaning. hexagon is 6(r) or 6r. d. As the number of sides of both the circumscribed b. less; greater; 6r < C < 8r and inscribed polygons increase, what will happen to c. 3d < C < 4d; The circumference of the circle is the upper and lower limits of the inequality from part between 3 and 4 times its diameter. c, and what does this imply? d. These limits will approach a value of πd, implying SOLUTION: that C = πd. a. The circumscribed polygon is a square with sides 50. CHALLENGE The sum of the circumferences of of length 2r. circles H, J, and K shown at the right is units. Find KJ.

The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) The sum of the circumferences is 56π units. or 4d and 6r = 3(2r) or 3d. Substitute the three circumferences and solve for x. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying So, the radius of that C = πd. is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. ANSWER: The measure of KJ is equal to the sum of the radii a. 8r and 6r ; Twice the radius of the circle, 2r is of and . Therefore, KJ = 16 + 8 or 24 units. the side length of the square, so the perimeter of the square is 4(2r) or 8r. The ANSWER: regular hexagon is made up of six equilateral 24 units triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. 51. REASONING Is the distance from the center of a b. less; greater; 6r < C < 8r circle to a point in the interior of a circle sometimes, c. 3d < C < 4d; The circumference of the circle is always, or never less than the radius of the circle?

between 3 and 4 times its diameter. Explain. d. These limits will approach a value of πd, implying SOLUTION: that C = πd. A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn 50. CHALLENGE The sum of the circumferences of from the center to a point inside the circle will always circles H, J, and K shown at the right is units. have a length less than the radius of the circle. Find KJ. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and

dilations to prove that all circles are similar. SOLUTION: The radius of is 4x units, the radius of is x SOLUTION: units, and the radius of is 2x units. Find the A circle is a locus of points in a plane equidistant circumference of each circle. from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

The sum of the circumferences is 56π units.

Substitute the three circumferences and solve for x.

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are So, the radius of is 4 units, the radius of is similar. Thus, all circles are similar. 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii of and . ANSWER: Therefore, KJ = 16 + 8 or 24 units. A circle is a locus of points in a plane equidistant from a given point. For any two circles and ANSWER: , there exists a translation that maps center A 24 units onto center B, moving so that it is concentric with . There also exists a dilation with scale 51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, factor k such that each point that makes up is always, or never less than the radius of the circle? moved to be the same distance from center A as the Explain. points that make up are from center B. Therefore, is mapped onto . Since there SOLUTION: exists a rigid motion followed by a scaling that maps A radius is a segment drawn between the center of onto , the circles are similar. Thus, all the circle and a point on the circle. A segment drawn circles are similar. from the center to a point inside the circle will always have a length less than the radius of the circle. 53. CHALLENGE In the figure, is inscribed in ANSWER: equilateral triangle LMN. What is the circumference Always; a radius is a segment drawn between the of ? center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: SOLUTION: A circle is a locus of points in a plane equidistant Draw a perpendicular to the side Also join from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B.

For some value k, r = 2 Since is equilateral triangle, MN = 8 inches kr1. and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. Therefore, the radius of is The circumference C of a circle with radius r is

ANSWER: given by A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale

factor k such that each point that makes up is The circumference of is inches. moved to be the same distance from center A as the points that make up are from center B. ANSWER: Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps in. onto , the circles are similar. Thus, all circles are similar. 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of 53. CHALLENGE In the figure, is inscribed in geometry. equilateral triangle LMN. What is the circumference of ? SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. SOLUTION: Draw a perpendicular to the side Also join

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

Since is equilateral triangle, MN = 8 inches The circumference C of a circle with diameter d is and RN = 4 inches. Also, bisects given by Use the definition of Use the formula to find the circumference. tangent of an angle to find PR, which is the radius of

ANSWER: Therefore, the radius of is 40.8 The circumference C of a circle with radius r is 56. What is the radius of a table with a circumference of given by 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft SOLUTION:

The circumference of is inches. The circumference C of a circle with radius r is given by ANSWER: Here, C = 10 ft. Use the formula to find the radius.

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. The radius of the circle is about 1.6 feet, so the correct choice is A. SOLUTION: See students’ work. ANSWER: A ANSWER: See students’ work. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use 55. GRIDDED RESPONSE What is the up to 50 feet of fence, what radius can he use for the circumference of ? Round to the nearest tenth. garden? F 10 G 9 H 8 J 7 SOLUTION: SOLUTION: The circumference C of a circle with radius r is given by Use the Pythagorean Theorem to find the diameter Because 50 feet of fence can be used to border the of the circle. garden, Cmax = 50. Use the circumference formula to find the maximum radius. The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: ANSWER: J 40.8 58. SAT/ACT What is the radius of a circle with an 56. What is the radius of a table with a circumference of 10 feet? area of square units? A 1.6 ft A 0.4 units B 2.5 ft B 0.5 units C 3.2 ft C 2 units D 5 ft D 4 units SOLUTION: E 16 units The circumference C of a circle with radius r is SOLUTION:

given by The area of a circle of radius r is given by the Here, C = 10 ft. Use the formula to find the radius. formula Here A = square units. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the correct choice is A. Therefore, the correct choice is B. ANSWER: ANSWER: A B 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use Copy each figure and point B. Then use a ruler up to 50 feet of fence, what radius can he use for the to draw the image of the figure under a dilation garden? with center B and the scale factor r indicated.

F 10 59. G 9 H 8 J 7 SOLUTION: The circumference C of a circle with radius r is given by SOLUTION: Because 50 feet of fence can be used to border the garden, C = 50. Use the circumference formula to Draw a ray from point B through a vertex of the max hexagon. Use the ruler to measure the distance from find the maximum radius. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units?

A 0.4 units B 0.5 units ANSWER: C 2 units D 4 units

E 16 units SOLUTION: The area of a circle of radius r is given by the 60. formula Here A = square units. Use the formula to find the radius.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from Therefore, the correct choice is B. B to the vertex. Mark a point on the ray that is of ANSWER: that distance from B. Repeat the process for rays B through the other two vertices. Connect the three points on the rays to construct the dilated image of Copy each figure and point B. Then use a ruler the triangle. to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

SOLUTION: 61. Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other five vertices. Connect the six points Draw a ray from point B through a vertex of the on the rays to construct the dilated image of the trapezoid. Use the ruler to measure the distance from hexagon. B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER: ANSWER:

60.

62.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is of Draw a ray from point B through a vertex of the that distance from B. Repeat the process for rays triangle. Use the ruler to measure the distance from through the other two vertices. Connect the three B to the vertex. Mark a point on the ray that is 3 points on the rays to construct the dilated image of times that distance from B. Repeat the process for the triangle. rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

61.

ANSWER: SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

63. Refer to the image on page 705. ANSWER: SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

62. 64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is 3 Yes; 4, 90 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

65. SOLUTION: The given figure cannot be mapped onto itself by any

rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: No 66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. State whether each figure has rotational If you are over 18 years old, then you vote in all symmetry. Write yes or no. If so, copy the elections. figure, locate the center of symmetry, and state 67. You are 19 years old and you vote. the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true The figure cannot be mapped onto itself by any for the conditions. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: True; sample answer: Since the hypothesis is true No and the conclusion is true, then the statement is true for the conditions. 64. Refer to the image on page 705. SOLUTION: 68. You are 21 years old and do not vote. The figure can be mapped onto itself by a rotation of SOLUTION: 90º. So, it has a rotational symmetry. So, the False; sample answer: Since the hypothesis is true magnitude of symmetry is 90º and order of symmetry and the conclusion is false, then the statement is false is 4. for the conditions.

ANSWER: ANSWER: Yes; 4, 90 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. 69. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any The sum of the measures of angles around a point is rotation. So, it does not have a rotational symmetry. 360º. ANSWER: No

66. Refer to the image on page 705. ANSWER: SOLUTION: 90 The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain 70. your reasoning. If you are over 18 years old, then you vote in all SOLUTION: elections. The angles with the measures xº and 2xº form a 67. You are 19 years old and you vote. straight line. So, the sum is 180.

SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. ANSWER: ANSWER: 60 True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: 71. False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false SOLUTION: for the conditions. The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180. ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. ANSWER: Find x. 20

69. SOLUTION: 72. The sum of the measures of angles around a point is SOLUTION: 360º. The sum of the three angles is 180.

ANSWER: ANSWER: 90 45

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. For Exercises 1–4, refer to . A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

1. Name the circle. ANSWER: a. SOLUTION: b. The center of the circle is N. So, the circle is c.

ANSWER: 3. If CN = 8 centimeters, find DN.

SOLUTION: 2. Identify each. Here, are radii of the same circle. So,

a. a chord they are equal in length. Therefore, DN = 8 cm. b. a diameter ANSWER: c. a radius 8 cm SOLUTION: 4. If EN = 13 feet, what is the diameter of the circle? a. A chord is a segment with endpoints on the circle. So, here are chords. SOLUTION: b. A diameter of a circle is a chord that passes Here, is a radius and the diameter is twice the through the center. Here, is a diameter. radius. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius. Therefore, the diameter is 26 ft.

ANSWER: ANSWER: a. 26 ft b. The diameters of , , and are 8 c. inches, 18 inches, and 11 inches, respectively. Find each measure. 3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 5. FG 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: SOLUTION: Since the diameter of B is 18 inches and A is 8 Here, is a radius and the diameter is twice the inches, AG = 18 and AF = (8) or 4. radius.

Therefore, the diameter is 26 ft. Therefore, FG = 14 in. ANSWER: 26 ft ANSWER: 14 in. The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

5. FG Therefore, FB = 5 inches. SOLUTION: Since the diameter of B is 18 inches and A is 8 ANSWER: 5 in. inches, AG = 18 and AF = (8) or 4. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. Therefore, FG = 14 in. SOLUTION: ANSWER: The radius is half the diameter. So, the radius of the 14 in. circular ride is 22 feet.

6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 about inches, AB = (18) or 9 and AF = (8) or 4. Therefore, the circumference of the ride is 138.23 feet.

ANSWER: 22 ft; 138.23 ft

Therefore, FB = 5 inches. 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. ANSWER: What are the diameter and radius of the pool? Round 5 in. to the nearest hundredth. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. SOLUTION:

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round The diameter of the pool is about 17.98 feet and the to the nearest hundredth. radius of the pool is about 8.99 feet.

ANSWER: 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of .

SOLUTION:

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. The diameter of the circle is 4 centimeters.

ANSWER: 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is The circumference of the circle is 4π inscribed in . Find the exact circumference of centimeters. . ANSWER:

For Exercises 10–13, refer to .

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

10. Name the center of the circle. SOLUTION: R The diameter of the circle is 4 centimeters. ANSWER: R

11. Identify a chord that is also a diameter. The circumference of the circle is 4π SOLUTION: centimeters. Two chords are shown: and . goes ANSWER: through the center, R, so is a diameter.

For Exercises 10–13, refer to . ANSWER:

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on 10. Name the center of the circle. the circle, so it is a chord. SOLUTION: ANSWER: R No; it is a chord. ANSWER: R 13. If SU = 16.2 centimeters, what is RT? SOLUTION: 11. Identify a chord that is also a diameter. Here, is a diameter and is a radius. SOLUTION: Two chords are shown: and . goes through the center, R, so is a diameter.

Therefore, RT = 8.1 cm. ANSWER: ANSWER: 8.1 cm

12. Is a radius? Explain. For Exercises 14–17, refer to . SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord.

ANSWER:

No; it is a chord. 14. Identify a chord that is not a diameter. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: SOLUTION: The chords do not pass through the center. So, they are not diameters. Here, is a diameter and is a radius. ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? Therefore, RT = 8.1 cm. SOLUTION: ANSWER: Here, is a radius and the diameter is twice the 8.1 cm radius. .

For Exercises 14–17, refer to .

Therefore, the diameter of the circle is 28 inches.

ANSWER: 28 in.

14. Identify a chord that is not a diameter. 16. Is ? Explain. SOLUTION: SOLUTION: All radii of a circle are congruent. Since The chords do not pass through the

center. So, they are not diameters. are both radii of , .

ANSWER: ANSWER: Yes; they are both radii of . 17. If DA = 7.4 centimeters, what is EF? 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Here, is a diameter and is a radius. Here, is a radius and the diameter is twice the radius. .

Therefore, the diameter of the circle is 28 inches. Therefore, EF = 3.7 cm.

ANSWER: ANSWER: 28 in. 3.7 cm

16. Is ? Explain. Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. All radii of a circle are congruent. Since are both radii of , .

ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? 18. CK SOLUTION: SOLUTION: Here, is a diameter and is a radius. Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units.

Therefore, EF = 3.7 cm. ANSWER: ANSWER: 2.6 3.7 cm 19. AB Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units.

ANSWER:

18. CK 14.6 SOLUTION: 20. JK Since the radius of K is 8 units, BK = 8. SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8. Therefore, CK = 2.6 units.

ANSWER: Since circle J has a radius of 10 units, JC = 10. 2.6

19. AB SOLUTION: Therefore, JK = 12.6 units. Since is a diameter of circle J and the radius is ANSWER: 10, AC = 2(10) or 20 units. 12.6 21. AD

Therefore, AB = 14.6 units. SOLUTION: We can find AD using AD = AC + CK + KD. The ANSWER: radius of circle K is 8 units and the radius of circle J 14.6 is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Therefore, AD = 30.6 units.

Since circle J has a radius of 10 units, JC = 10. ANSWER: 30.6

22. PIZZA Find the radius and circumference of the Therefore, JK = 12.6 units. pizza shown. Round to the nearest hundredth, if necessary. ANSWER: 12.6 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. SOLUTION: The diameter of the pizza is 16 inches.

Therefore, AD = 30.6 units. . So, the radius of the pizza is 8 inches. ANSWER: The circumference C of a circle of diameter d is 30.6 given by 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary. Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: SOLUTION: The diameter of the pizza is 16 inches. The diameter of a tire is 26 inches. The radius is half the diameter.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is So, the radius of the tires is 13 inches. given by The circumference C of a circle with diameter d is given by

Therefore, the circumference of the pizza is about 50.27 inches. Therefore, the circumference of the bicycle tire is about 81.68 inches. ANSWER: 8 in.; 50.27 in. ANSWER: 13 in.; 81.68 in. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a Find the diameter and radius of a circle by with tire. Round to the nearest hundredth, if necessary. the given circumference. Round to the nearest hundredth. SOLUTION: 24. C = 18 in. The diameter of a tire is 26 inches. The radius is half the diameter. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: Therefore, the diameter is about 5.73 inches and the 13 in.; 81.68 in. radius is about 2.86 inches. Find the diameter and radius of a circle by with ANSWER: the given circumference. Round to the nearest hundredth. 5.73 in.; 2.86 in. 24. C = 18 in. 25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 18 in. Use the formula to find the given by diameter. Then find the radius. Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. ANSWER: 5.73 in.; 2.86 in. ANSWER: 39.47 ft; 19.74 ft 25. C = 124 ft 26. C = 375.3 cm SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is given by given by Here, C = 124 ft. Use the formula to find the Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the Therefore, the diameter is about 119.46 centimeters

radius is about 19.74 feet. and the radius is about 59.73 centimeters.

ANSWER: ANSWER: 39.47 ft; 19.74 ft 119.46 cm; 59.73 cm

26. C = 375.3 cm 27. C = 2608.25 m SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is

given by given by Here, C = 375.3 cm. Use the formula to find the Here, C = 2608.25 meters. Use the formula to find

diameter. Then find the radius. the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters Therefore, the diameter is about 830.23 meters and and the radius is about 59.73 centimeters. the radius is about 415.12 meters.

10-1ANSWER: Circles and Circumference ANSWER: 119.46 cm; 59.73 cm 830.23 m; 415.12 m

27. C = 2608.25 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given SOLUTION: inscribed or The circumference C of a circle with diameter d is circumscribed polygon. given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: The diameter of the circle is 17 cm. The 830.23 m; 415.12 m circumference C of a circle is given by CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon. Therefore, the circumference of the circle is 17 centimeters.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the The diameter of the circle is 17 cm. The diameter. circumference C of a circle is given by

The diameter of the circle is 12 feet. The Therefore, the circumference of the circle is 17 circumference C of a circle is given by centimeters. eSolutionsANSWER: Manual - Powered by Cognero Page 6

Therefore, the circumference of the circle is 12 feet.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of 30. the circle. Use the Pythagorean Theorem to find the diameter. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. The diameter of the circle is 12 feet. The circumference C of a circle is given by

The diameter of the circle is inches. The Therefore, the circumference of the circle is 12 circumference C of a circle is given by feet.

ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER: 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter The diameter of the circle is inches. The of the circle. circumference C of a circle is given by

The diameter of the circle is 10 inches. The Therefore, the circumference of the circle is circumference C of a circle is given by inches.

ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter 32. of the circle. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by The diameter of the circle is 10 inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 25 millimeters. Therefore, the circumference of the circle is 10 inches. ANSWER:

ANSWER:

32. 33. SOLUTION: SOLUTION: A diameter perpendicular to a side of the square and A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d the length of each side will measure the same. So, d = 25 millimeters. = 14 yards. The circumference C of a circle is given by The circumference C of a circle is given by

Therefore, the circumference of the circle is 25 Therefore, the circumference of the circle is 14 millimeters. yards.

ANSWER: ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. 33. SOLUTION: SOLUTION: The circumference C of a circle with diameter d is A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d given by = 14 yards. Here, C = 66.92 cm. Use the formula to find the diameter. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 yards. The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the ANSWER: diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and ANSWER: clubs. For professional competitions, the maximum 176.8 g weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight 35. PATIOS Mr. Martinez is going to build the patio for a disc with circumference 66.92 centimeters? shown. Round to the nearest tenth. a. What is the patio’s approximate circumference? SOLUTION: b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 The circumference C of a circle with diameter d is feet, what should the radius of the circle be to the given by nearest foot? Here, C = 66.92 cm. Use the formula to find the diameter.

The diameter of the disc is about 21.3 centimeters SOLUTION: and the maximum weight allowed is 8.3 times the diameter in centimeters. a. The radius of the patio is 3 + 2 = 5 ft. The Therefore, the maximum weight is 8.3 × 21.3 or circumference C of a circle with radius r is given by about 176.8 grams. Use the formula to find the circumference. ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio Therefore, the circumference is about 31.42 feet. shown. b. a. What is the patio’s approximate circumference? Here, Cinner = 25 ft. Use the formula to find the b. If Mr. Martinez changes the plans so that the radius of the inner circle. inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft b. 4 ft SOLUTION: The radius, diameter, or circumference of a a. The radius of the patio is 3 + 2 = 5 ft. The circle is given. Find each missing measure to circumference C of a circle with radius r is given by the nearest hundredth.

Use the formula to find the circumference. 36.

SOLUTION: The radius is half the diameter and the Therefore, the circumference is about 31.42 feet. circumference C of a circle with diameter d is given by b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the radius of the inner circle is about 4 feet.

ANSWER: Therefore, the radius is 4.25 inches and the a. 31.42 ft circumference is about 26.70 inches. b. 4 ft ANSWER: The radius, diameter, or circumference of a 4.25 in.; 26.70 in. circle is given. Find each missing measure to the nearest hundredth. 37. 36. SOLUTION: SOLUTION: The diameter is twice the radius and the The radius is half the diameter and the circumference C of a circle with diameter d is given circumference C of a circle with diameter d is given by by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet. Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. ANSWER: 22.80 ft; 71.63 ft ANSWER: 4.25 in.; 26.70 in. 38.

SOLUTION: 37. The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 35x cm. Use the formula to find the The diameter is twice the radius and the diameter.Then find the radius. circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet. Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 22.80 ft; 71.63 ft ANSWER: 11.14x cm; 5.57x cm 38.

SOLUTION: 39. The circumference C of a circle with diameter d is SOLUTION: given by Here, C = 35x cm. Use the formula to find the The diameter is twice the radius and the diameter.Then find the radius. circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 0.25x; 0.79x ANSWER: 11.14x cm; 5.57x cm Determine whether the circles in the figures below appear to be congruent, concentric, or

39. neither. 40. Refer to the photo on page 703. SOLUTION: SOLUTION: The diameter is twice the radius and the The circles have the same center and they are in the circumference C of a circle with diameter d is given same plane. So, they are concentric circles. by ANSWER: concentric

41. Refer to the photo on page 703.

SOLUTION: The circles neither have the same radius nor are they concentric. So, the diameter is 0.25x units and the circumference ANSWER: is about 0.79x units. neither ANSWER: 42. Refer to the photo on page 703. 0.25x; 0.79x SOLUTION: Determine whether the circles in the figures The circles appear to have the same radius. So, they below appear to be congruent, concentric, or appear to be congruent. neither. 40. Refer to the photo on page 703. ANSWER: congruent SOLUTION: The circles have the same center and they are in the 43. HISTORY The Indian Shell Ring on Hilton Head same plane. So, they are concentric circles. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would ANSWER: someone have to walk to go completely around the concentric ring? Round to the nearest tenth.

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they SOLUTION:

appear to be congruent. The radius of the circle is 3 units, that is 75 feet. So, ANSWER: the diameter is 150 ft. The circumference C of a

congruent circle with diameter d is given by So, the circumference is 43. HISTORY The Indian Shell Ring on Hilton Head Therefore, someone have to Island approximates a circle. If each unit on the walk about 471.2 feet to go completely around the coordinate grid represents 25 feet, how far would ring. someone have to walk to go completely around the ring? Round to the nearest tenth. ANSWER: 471.2 ft

44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a circle with diameter d is given by So, the circumference is Therefore, someone have to So, the radius of the pond is about 10.82 feet and the

walk about 471.2 feet to go completely around the radius to the outer edge of the path will be 10.82 + 4

ring. = 14.82 ft. Use the radius to find the circumference ANSWER: of the brick path. 471.2 ft

44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a Therefore, the circumference of the path is about circumference of 68 feet. The outer edge of the path 93.13 feet. is going to be 4 feet from the pond all the way around. What is the approximate circumference of ANSWER: the path? Round to the nearest hundredth. 93.13 ft

SOLUTION: 45. MULTIPLE REPRESENTATIONS In this The circumference C of a circle with radius r is problem, you will explore changing dimensions in given by Here, Cpond = 68 ft. Use the circles. formula to find the radius of the pond. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are So, the radius of the pond is about 10.82 feet and the geometrically similar. radius to the outer edge of the path will be 10.82 + 4 d. VERBAL Make a conjecture about the ratio = 14.82 ft. Use the radius to find the circumference between the circumferences of two circles when the

of the brick path. ratio between their radii is 2. e. ANALYTICAL The scale factor from to

is . Write an equation relating the Therefore, the circumference of the path is about circumference (C ) of to the circumference 93.13 feet. A (CB) of . ANSWER: f. NUMERICAL If the scale factor from to 93.13 ft is , and the circumference of is 12 45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in inches, what is the circumference of ? circles. SOLUTION: a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to a. Sample answer: the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio b. Sample answer. between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to is . Write an equation relating the circumference (C ) of to the circumference A (CB) of . c. They all have the same shape–circular. f. NUMERICAL If the scale factor from to d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. is , and the circumference of is 12 So, if the radii are in the ratio b : a, so will the

inches, what is the circumference of ? circumference. Therefore, . SOLUTION: f. Use the formula from part e. Here, the a. Sample answer: circumference of is 12 inches and .

Therefore, the circumference of is 4 inches.

b. Sample answer. ANSWER: a. Sample answer:

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. b. Sample answer. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . f. Use the formula from part e. Here, the circumference of is 12 inches and . c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e.

Therefore, the circumference of is 4 inches. f. 4 in.

ANSWER: 46. BUFFON’S NEEDLE Measure the length of a a. Sample answer: needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

b. Sample answer.

a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops. c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. b. Calculate the ratio of two times the total number f. 4 in. of drops to the number of hits after 25, 50, and 100 drops. 46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set c. How are the values you found in part b related to of horizontal lines that are centimeters apart on a π? sheet of plain white paper. SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. a. Drop the needle onto the paper. When the needle b. See students’ work. lands, record whether it touches one of the lines as a c. Sample answer: The values are approaching 3.14, hit. Record the number of hits after 25, 50, and 100 which is approximately equal to π. drops.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 b. Calculate the ratio of two times the total number miles from downtown Phoenix. of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. a. How much greater is the circumference of the

outermost circle than the circumference of the center ANSWER: circle? b. As the radii of the circles increases by 5 miles, by a. See students’ work. how much does the circumference increase? b. See students’ work. c. Sample answer: The values are approaching 3.14, SOLUTION: which is approximately equal to π. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

The difference of the circumferences of the outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the The difference of the two circumferences is (2xπ + circumference of each circle. 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles. ANSWER: a. 157.1 mi b. by about 31.4 mi The difference of the circumferences of the outermost circle and the center circle is 60π - 10π = 48. WRITING IN MATH How can we describe the 50π or about 157.1 miles. relationships that exist between b. Let the radius of one circle be x miles and the circles and lines? radius of the next larger circle be x + 5 miles. Find SOLUTION: the circumference of each circle. Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through The difference of the two circumferences is (2xπ + the center of the circle, it can be described as a 10π) - (2xπ) = 10π or about 31.4 miles. diameter. A line segment with endpoints at the Therefore, as the radius of each circle increases by 5 center and on the circle can be described as a miles, the circumference increases by about 31.4 radius. miles. ANSWER: ANSWER: Sample answer: A line that intersects a circle in one a. 157.1 mi point can be described as a tangent. A line that b. by about 31.4 mi intersects a circle in exactly two points can be described as a secant. A line segment with 48. WRITING IN MATH How can we describe the endpoints on a circle can be described as a chord. relationships that exist between If the chord passes through the center of the circle, circles and lines? it can be described as a diameter. A line segment SOLUTION: with endpoints at the center and on the circle can be Sample answer: A line and a circle may intersect in described as a radius. one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be 49. described as a chord. If the chord passes through REASONING In the figure, a circle with radius r is the center of the circle, it can be described as a inscribed in a regular polygon and circumscribed diameter. A line segment with endpoints at the about another. center and on the circle can be described as a a. What are the perimeters of the circumscribed and radius. inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or ANSWER: less than the perimeter of the circumscribed polygon? Sample answer: A line that intersects a circle in one the inscribed polygon? Write a compound inequality point can be described as a tangent. A line that comparing C to these perimeters. intersects a circle in exactly two points can be c. Rewrite the inequality from part b in terms of the described as a secant. A line segment with diameter d of the circle and interpret its meaning. d. endpoints on a circle can be described as a chord. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to If the chord passes through the center of the circle, the upper and lower limits of the inequality from part it can be described as a diameter. A line segment c, and what does this imply? with endpoints at the center and on the circle can be described as a radius. SOLUTION: a. The circumscribed polygon is a square with sides of length 2r.

49. The inscribed polygon is a hexagon with sides of REASONING In the figure, a circle with radius r is length r. (Each triangle formed using a central angle inscribed in a regular polygon and circumscribed about another. will be equilateral.) a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part So, the perimeters are 8r and 6r. c, and what does this imply? b. SOLUTION:

a. The circumscribed polygon is a square with sides The circumference is less than the perimeter of the of length 2r. circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) The inscribed polygon is a hexagon with sides of or 4d and 6r = 3(2r) or 3d. length r. (Each triangle formed using a central angle The inequality is then 4d < C < 3d. Therefore, the will be equilateral.) circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is

the side length of the square, so the perimeter of the square is 4(2r) or 8r. The So, the perimeters are 8r and 6r. regular hexagon is made up of six equilateral b. triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r The circumference is less than the perimeter of the c. 3d < C < 4d; The circumference of the circle is circumscribed polygon and greater than the between 3 and 4 times its diameter. perimeter of the inscribed polygon. d. These limits will approach a value of πd, implying 8r < C < 6r that C = πd. c. Since the diameter is twice the radius, 8r = 4(2r) 50. CHALLENGE The sum of the circumferences of or 4d and 6r = 3(2r) or 3d. circles H, J, and K shown at the right is units. The inequality is then 4d < C < 3d. Therefore, the Find KJ. circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the SOLUTION: square is 4(2r) or 8r. The The radius of is 4x units, the radius of is x regular hexagon is made up of six equilateral units, and the radius of is 2x units. Find the triangles with side length r, so the perimeter of the circumference of each circle. hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ.

The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

So, the radius of is 4 units, the radius of is SOLUTION: 2(4) or 8 units, and the radius of is 4(4) or 16 The radius of is 4x units, the radius of is x units. units, and the radius of is 2x units. Find the The measure of KJ is equal to the sum of the radii circumference of each circle. of and . Therefore, KJ = 16 + 8 or 24 units.

ANSWER: 24 units

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Explain. SOLUTION: A radius is a segment drawn between the center of The sum of the circumferences is 56π units. the circle and a point on the circle. A segment drawn Substitute the three circumferences and solve for x. from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A So, the radius of is 4 units, the radius of is segment drawn from the center to a point inside the 2(4) or 8 units, and the radius of is 4(4) or 16 circle will always have a length less than the radius units. of the circle. The measure of KJ is equal to the sum of the radii of and . 52. PROOF Use the locus definition of a circle and Therefore, KJ = 16 + 8 or 24 units. dilations to prove that all circles are similar. ANSWER: SOLUTION: 24 units A circle is a locus of points in a plane equidistant from a given point. For any two circles and 51. REASONING Is the distance from the center of a , there exists a translation that circle to a point in the interior of a circle sometimes, maps center A onto center B, moving so that it always, or never less than the radius of the circle? is concentric with . There also exists a dilation Explain. with scale factor k such that each point that makes up is moved to be the SOLUTION: same distance from center A as the points that make A radius is a segment drawn between the center of up are from center B. the circle and a point on the circle. A segment drawn For some value k, r = from the center to a point inside the circle will always 2 have a length less than the radius of the circle. kr1.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps 52. PROOF Use the locus definition of a circle and onto , the circles are dilations to prove that all circles are similar. similar. Thus, all circles are similar.

SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and ANSWER: , there exists a translation that A circle is a locus of points in a plane equidistant maps center A onto center B, moving so that it from a given point. For any two circles and is concentric with . There also exists a dilation , there exists a translation that maps center A with scale factor k such onto center B, moving so that it is concentric that each point that makes up is moved to be the same distance from center A as the points that make with . There also exists a dilation with scale factor k such that each point that makes up is up are from center B. moved to be the same distance from center A as the For some value k, r2 = kr . points that make up are from center B. 1 Therefore, is mapped onto . Since there

exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

53. CHALLENGE In the figure, is inscribed in Therefore, is mapped onto . Since there equilateral triangle LMN. What is the circumference exists a rigid motion followed by a scaling that maps of ? onto , the circles are similar. Thus, all circles are similar.

ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and SOLUTION: , there exists a translation that maps center A onto center B, moving so that it is concentric Draw a perpendicular to the side Also join with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects 53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference Use the definition of tangent of an angle to find PR, which is the radius of of ?

Therefore, the radius of is SOLUTION: The circumference C of a circle with radius r is Draw a perpendicular to the side Also join given by

The circumference of is inches. Since is equilateral triangle, MN = 8 inches ANSWER: and RN = 4 inches. Also, bisects

Use the definition of in. tangent of an angle to find PR, which is the radius of

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

Therefore, the radius of is ANSWER: The circumference C of a circle with radius r is See students’ work. given by 55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

The circumference of is inches. ANSWER: SOLUTION: Use the Pythagorean Theorem to find the diameter in. of the circle.

54. WRITING IN MATH Research and write about The circumference C of a circle with diameter d is the history of pi and its importance to the study of given by geometry. Use the formula to find the circumference.

SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. ANSWER: 40.8 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft B 2.5 ft C SOLUTION: 3.2 ft D 5 ft Use the Pythagorean Theorem to find the diameter of the circle. SOLUTION: The circumference C of a circle with radius r is The circumference C of a circle with diameter d is given by given by Here, C = 10 ft. Use the formula to find the radius. Use the formula to find the circumference.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: ANSWER: A 40.8 57. ALGEBRA Bill is planning a circular vegetable 56. What is the radius of a table with a circumference of garden with a fence around the border. If he can use 10 feet? up to 50 feet of fence, what radius can he use for the A 1.6 ft garden? B 2.5 ft F 10 C 3.2 ft G 9 D 5 ft H 8 SOLUTION: J 7 The circumference C of a circle with radius r is SOLUTION: given by The circumference C of a circle with radius r is Here, C = 10 ft. Use the formula to find the radius. given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: A Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only 57. ALGEBRA Bill is planning a circular vegetable correct choice is J. garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the ANSWER: garden? J F 10 G 9 58. SAT/ACT What is the radius of a circle with an

H 8 area of square units? J 7 A 0.4 units SOLUTION: B 0.5 units The circumference C of a circle with radius r is C 2 units

given by D 4 units Because 50 feet of fence can be used to border the E 16 units garden, Cmax = 50. Use the circumference formula to find the maximum radius. SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J. Therefore, the correct choice is B. ANSWER: J ANSWER: B 58. SAT/ACT What is the radius of a circle with an Copy each figure and point B. Then use a ruler area of square units? to draw the image of the figure under a dilation with center B and the scale factor r indicated. A 0.4 units

B 0.5 units 59. C 2 units D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the SOLUTION: formula to find the radius. Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the Therefore, the correct choice is B. hexagon. ANSWER: B Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from

B to the vertex. Mark a point on the ray that is of 60. that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

ANSWER:

60. 61.

SOLUTION: Draw a ray from point B through a vertex of the SOLUTION: triangle. Use the ruler to measure the distance from Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of B to the vertex. Mark a point on the ray that is 2 that distance from B. Repeat the process for rays times that distance from B. Repeat the process for through the other two vertices. Connect the three rays through the other three vertices. Connect the points on the rays to construct the dilated image of four points on the rays to construct the dilated image the triangle. of the trapezoid.

ANSWER:

ANSWER: 61.

SOLUTION: Draw a ray from point B through a vertex of the 62. trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

62.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: ANSWER: No

64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90 State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. 65. ANSWER: No SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. 64. Refer to the image on page 705. ANSWER: SOLUTION: No The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the 66. Refer to the image on page 705. magnitude of symmetry is 90º and order of symmetry is 4. SOLUTION: The given figure cannot be mapped onto itself by any ANSWER: rotation. So, it does not have a rotational symmetry. Yes; 4, 90 ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote.

65. SOLUTION: SOLUTION: True; sample answer: Since the hypothesis is true The given figure cannot be mapped onto itself by any and the conclusion is true, then the statement is true rotation. So, it does not have a rotational symmetry. for the conditions.

ANSWER: ANSWER: No True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true 66. Refer to the image on page 705. for the conditions. SOLUTION: 68. You are 21 years old and do not vote. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. SOLUTION: False; sample answer: Since the hypothesis is true ANSWER: and the conclusion is false, then the statement is false No for the conditions.

Determine the truth value of the following ANSWER: statement for each set of conditions. Explain False; sample answer: Since the hypothesis is true your reasoning. and the conclusion is false, then the statement is false If you are over 18 years old, then you vote in all for the conditions. elections. 67. You are 19 years old and you vote. Find x. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

ANSWER: 69. True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true SOLUTION: for the conditions. The sum of the measures of angles around a point is 360º. 68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false ANSWER: for the conditions. 90 ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x. 70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

69. SOLUTION: The sum of the measures of angles around a point is 360º. ANSWER: 60

ANSWER: 90

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180. ANSWER: 20

ANSWER: 60

72. SOLUTION: The sum of the three angles is 180.

71. SOLUTION: The angles with the measures 6xº and 3xº form a ANSWER: straight line. So, the sum is 180. 45

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. For Exercises 1–4, refer to . c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c. 1. Name the circle. 3. If CN = 8 centimeters, find DN. SOLUTION: SOLUTION: The center of the circle is N. So, the circle is Here, are radii of the same circle. So, ANSWER: they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 2. Identify each. a. a chord 4. If EN = 13 feet, what is the diameter of the circle? b. a diameter SOLUTION: c. a radius Here, is a radius and the diameter is twice the SOLUTION: radius. a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes Therefore, the diameter is 26 ft.

through the center. Here, is a diameter. ANSWER: c. A radius is a segment with endpoints at the center 26 ft and on the circle. Here, is radius. The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. ANSWER: Find each measure. a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION:

Here, are radii of the same circle. So, 5. FG they are equal in length. Therefore, DN = 8 cm. SOLUTION: ANSWER: Since the diameter of B is 18 inches and A is 8 8 cm inches, AG = 18 and AF = (8) or 4. 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius. Therefore, FG = 14 in.

ANSWER:

Therefore, the diameter is 26 ft. 14 in.

ANSWER: 6. FB 26 ft SOLUTION: The diameters of , , and are 8 Since the diameter of B is 18 inches and A is 4 inches, 18 inches, and 11 inches, respectively. inches, AB = (18) or 9 and AF = (8) or 4. Find each measure.

Therefore, FB = 5 inches.

ANSWER: 5 in. 5. FG SOLUTION: 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the Since the diameter of B is 18 inches and A is 8 radius and circumference of the ride? Round to the inches, AG = 18 and AF = (8) or 4. nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet.

Therefore, FG = 14 in.

ANSWER: 14 in.

Therefore, the circumference of the ride is about 6. FB 138.23 feet. SOLUTION: ANSWER: Since the diameter of B is 18 inches and A is 4 22 ft; 138.23 ft inches, AB = (18) or 9 and AF = (8) or 4. 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. 8. CCSS MODELING The circumference of the ANSWER: circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round 17.98 ft; 8.99 ft to the nearest hundredth. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of .

SOLUTION: SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

The diameter of the circle is 4 centimeters.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. The circumference of the circle is 4π ANSWER: centimeters. 17.98 ft; 8.99 ft ANSWER: 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of . For Exercises 10–13, refer to .

SOLUTION: 10. Name the center of the circle. The diameter of the circle is the hypotenuse of the right triangle. SOLUTION: R ANSWER: R

The diameter of the circle is 4 centimeters. 11. Identify a chord that is also a diameter. SOLUTION: Two chords are shown: and . goes

through the center, R, so is a diameter. The circumference of the circle is 4π centimeters. ANSWER: ANSWER:

12. Is a radius? Explain. For Exercises 10–13, refer to . SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord.

ANSWER: No; it is a chord. 10. Name the center of the circle. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: R SOLUTION: Here, is a diameter and is a radius. ANSWER: R

11. Identify a chord that is also a diameter.

SOLUTION: Therefore, RT = 8.1 cm. Two chords are shown: and . goes ANSWER: through the center, R, so is a diameter. 8.1 cm

ANSWER: For Exercises 14–17, refer to .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on 14. Identify a chord that is not a diameter. the circle, so it is a chord. SOLUTION: ANSWER: The chords do not pass through the No; it is a chord. center. So, they are not diameters. 13. If SU = 16.2 centimeters, what is RT? ANSWER:

SOLUTION: Here, is a diameter and is a radius. 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius. .

Therefore, RT = 8.1 cm.

ANSWER: 8.1 cm Therefore, the diameter of the circle is 28 inches.

For Exercises 14–17, refer to . ANSWER: 28 in.

16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since are both radii of , . 14. Identify a chord that is not a diameter. SOLUTION: ANSWER: Yes; they are both radii of . The chords do not pass through the center. So, they are not diameters. 17. If DA = 7.4 centimeters, what is EF? ANSWER: SOLUTION: Here, is a diameter and is a radius.

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius. . Therefore, EF = 3.7 cm.

ANSWER: 3.7 cm

Therefore, the diameter of the circle is 28 inches. Circle J has a radius of 10 units, has a ANSWER: radius of 8 units, and BC = 5.4 units. Find each measure. 28 in.

16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since are both radii of , .

ANSWER: 18. CK Yes; they are both radii of . SOLUTION: Since the radius of K is 8 units, BK = 8. 17. If DA = 7.4 centimeters, what is EF? SOLUTION:

Here, is a diameter and is a radius. Therefore, CK = 2.6 units.

ANSWER: 2.6

19. AB Therefore, EF = 3.7 cm. SOLUTION: ANSWER: Since is a diameter of circle J and the radius is 3.7 cm 10, AC = 2(10) or 20 units. Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each

measure. Therefore, AB = 14.6 units.

ANSWER: 14.6 20. JK

SOLUTION: 18. CK First find CK. Since circle K has a radius of 8 units, BK = 8. SOLUTION: Since the radius of K is 8 units, BK = 8.

Since circle J has a radius of 10 units, JC = 10.

Therefore, CK = 2.6 units.

ANSWER: Therefore, JK = 12.6 units. 2.6 ANSWER: 19. AB 12.6 SOLUTION: 21. AD Since is a diameter of circle J and the radius is SOLUTION: 10, AC = 2(10) or 20 units. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. Therefore, AB = 14.6 units.

ANSWER: 14.6 20. JK

SOLUTION: Therefore, AD = 30.6 units. First find CK. Since circle K has a radius of 8 units, BK = 8. ANSWER: 30.6

22. PIZZA Find the radius and circumference of the Since circle J has a radius of 10 units, JC = 10. pizza shown. Round to the nearest hundredth, if necessary.

Therefore, JK = 12.6 units.

ANSWER: 12.6 21. AD

SOLUTION: SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J The diameter of the pizza is 16 inches. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by Therefore, AD = 30.6 units.

ANSWER: 30.6 Therefore, the circumference of the pizza is about 22. PIZZA Find the radius and circumference of the 50.27 inches. pizza shown. Round to the nearest hundredth, if ANSWER: necessary. 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter. SOLUTION: The diameter of the pizza is 16 inches.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is . given by So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is

given by Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: Therefore, the circumference of the pizza is about 50.27 inches. 13 in.; 81.68 in.

ANSWER: Find the diameter and radius of a circle by with 8 in.; 50.27 in. the given circumference. Round to the nearest hundredth. 23. BICYCLES A bicycle has tires with a diameter of 24. C = 18 in. 26 inches. Find the radius and circumference of a SOLUTION: tire. Round to the nearest hundredth, if necessary. The circumference C of a circle with diameter d is SOLUTION: given by The diameter of a tire is 26 inches. The radius is half Here, C = 18 in. Use the formula to find the the diameter. diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the diameter is about 5.73 inches and the Therefore, the circumference of the bicycle tire is radius is about 2.86 inches. about 81.68 inches. ANSWER: ANSWER: 5.73 in.; 2.86 in. 13 in.; 81.68 in. Find the diameter and radius of a circle by with 25. C = 124 ft the given circumference. Round to the nearest SOLUTION: hundredth. The circumference C of a circle with diameter d is 24. C = 18 in. given by SOLUTION: Here, C = 124 ft. Use the formula to find the The circumference C of a circle with diameter d is diameter. Then find the radius. given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet.

ANSWER: Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. 39.47 ft; 19.74 ft

ANSWER: 26. C = 375.3 cm 5.73 in.; 2.86 in. SOLUTION: The circumference C of a circle with diameter d is 25. C = 124 ft given by Here, C = 375.3 cm. Use the formula to find the SOLUTION: diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the diameter is about 39.47 feet and the 119.46 cm; 59.73 cm radius is about 19.74 feet. 27. C = 2608.25 m ANSWER: SOLUTION: 39.47 ft; 19.74 ft The circumference C of a circle with diameter d is 26. C = 375.3 cm given by Here, C = 2608.25 meters. Use the formula to find SOLUTION: the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER:

830.23 m; 415.12 m Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. CCSS SENSE-MAKING Find the exact ANSWER: circumference of each circle by using the given inscribed or 119.46 cm; 59.73 cm circumscribed polygon. 27. C = 2608.25 m SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find 28. the diameter. Then find the radius. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m Therefore, the circumference of the circle is 17 centimeters. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given ANSWER: inscribed or circumscribed polygon.

28. 29. SOLUTION: The hypotenuse of the right triangle is a diameter of SOLUTION: the circle. Use the Pythagorean Theorem to find the The hypotenuse of the right triangle is a diameter of diameter. the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The The diameter of the circle is 12 feet. The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 17 Therefore, the circumference of the circle is 12 centimeters. feet. ANSWER: ANSWER: 10-1 Circles and Circumference

30. 29. SOLUTION: SOLUTION: Each diagonal of the inscribed rectangle will pass The hypotenuse of the right triangle is a diameter of through the origin, so it is a diameter of the circle. the circle. Use the Pythagorean Theorem to find the Use the Pythagorean Theorem to find the diameter diameter. of the circle.

The diameter of the circle is 12 feet. The circumference C of a circle is given by The diameter of the circle is inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 12 feet. Therefore, the circumference of the circle is ANSWER: inches.

ANSWER:

30. SOLUTION: Each diagonal of the inscribed rectangle will pass 31. through the origin, so it is a diameter of the circle. SOLUTION: Use the Pythagorean Theorem to find the diameter Each diagonal of the inscribed rectangle will pass of the circle. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is inches. The circumference C of a circle is given by The diameter of the circle is 10 inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is Therefore, the circumference of the circle is 10 eSolutionsinches.Manual - Powered by Cognero Page 7 inches. ANSWER: ANSWER:

31. 32. SOLUTION: SOLUTION: Each diagonal of the inscribed rectangle will pass A diameter perpendicular to a side of the square and through the origin, so it is a diameter of the circle. the length of each side will measure the same. So, d Use the Pythagorean Theorem to find the diameter = 25 millimeters. of the circle. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The Therefore, the circumference of the circle is 25 circumference C of a circle is given by millimeters. ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER:

33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d 32. = 14 yards. SOLUTION: The circumference C of a circle is given by A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by Therefore, the circumference of the circle is 14 yards.

ANSWER: Therefore, the circumference of the circle is 25 millimeters. 34. DISC GOLF Disc golf is similar to regular golf, ANSWER: except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by 33. Here, C = 66.92 cm. Use the formula to find the diameter. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the circumference of the circle is 14 Therefore, the maximum weight is 8.3 × 21.3 or yards. about 176.8 grams.

ANSWER: ANSWER: 176.8 g PATIOS 34. DISC GOLF Disc golf is similar to regular golf, 35. Mr. Martinez is going to build the patio

except that a flying disc is used instead of a ball and shown. clubs. For professional competitions, the maximum a. What is the patio’s approximate circumference? weight of a disc in grams is 8.3 times its diameter in b. If Mr. Martinez changes the plans so that the centimeters. What is the maximum allowable weight inner circle has a circumference of approximately 25 for a disc with circumference 66.92 centimeters? feet, what should the radius of the circle be to the Round to the nearest tenth. nearest foot? SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter.

SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by

The diameter of the disc is about 21.3 centimeters Use the formula to find the circumference. and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. Therefore, the circumference is about 31.42 feet. ANSWER: b. Here, C = 25 ft. Use the formula to find the 176.8 g inner radius of the inner circle. 35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the So, the radius of the inner circle is about 4 feet. nearest foot? ANSWER: a. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

SOLUTION: 36. a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by SOLUTION: The radius is half the diameter and the Use the formula to find the circumference. circumference C of a circle with diameter d is given by

Therefore, the circumference is about 31.42 feet. b. Here, C = 25 ft. Use the formula to find the inner radius of the inner circle.

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

So, the radius of the inner circle is about 4 feet. ANSWER: ANSWER: 4.25 in.; 26.70 in. a. 31.42 ft b. 4 ft 37. The radius, diameter, or circumference of a circle is given. Find each missing measure to SOLUTION: the nearest hundredth. The diameter is twice the radius and the circumference C of a circle with diameter d is given 36. by

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given

by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet. ANSWER: 22.80 ft; 71.63 ft

Therefore, the radius is 4.25 inches and the 38. circumference is about 26.70 inches. SOLUTION: ANSWER: The circumference C of a circle with diameter d is 4.25 in.; 26.70 in. given by Here, C = 35x cm. Use the formula to find the diameter.Then find the radius.

37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters.

ANSWER: So, the diameter is 22.80 feet and the circumference 11.14x cm; 5.57x cm is about 71.63 feet.

ANSWER: 39. 22.80 ft; 71.63 ft SOLUTION: 38. The diameter is twice the radius and the SOLUTION: circumference C of a circle with diameter d is given by The circumference C of a circle with diameter d is given by Here, C = 35x cm. Use the formula to find the diameter.Then find the radius.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x

Therefore, the diameter is about 11.14x centimeters Determine whether the circles in the figures and the radius is about 5.57x centimeters. below appear to be congruent, concentric, or neither. ANSWER: 40. Refer to the photo on page 703. 11.14x cm; 5.57x cm SOLUTION: The circles have the same center and they are in the 39. same plane. So, they are concentric circles.

SOLUTION: ANSWER: The diameter is twice the radius and the concentric circumference C of a circle with diameter d is given by 41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. So, the diameter is 0.25x units and the circumference is about 0.79x units. SOLUTION: The circles appear to have the same radius. So, they ANSWER: appear to be congruent. 0.25x; 0.79x ANSWER: Determine whether the circles in the figures congruent below appear to be congruent, concentric, or neither. 43. HISTORY The Indian Shell Ring on Hilton Head 40. Refer to the photo on page 703. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would SOLUTION: someone have to walk to go completely around the The circles have the same center and they are in the ring? Round to the nearest tenth. same plane. So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither SOLUTION: 42. Refer to the photo on page 703. The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a SOLUTION: circle with diameter d is given by The circles appear to have the same radius. So, they So, the circumference is appear to be congruent. Therefore, someone have to ANSWER: walk about 471.2 feet to go completely around the congruent ring. ANSWER: 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the 471.2 ft coordinate grid represents 25 feet, how far would someone have to walk to go completely around the 44. CCSS MODELING A brick path is being installed ring? Round to the nearest tenth. around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 circle with diameter d is given by = 14.82 ft. Use the radius to find the circumference So, the circumference is of the brick path. Therefore, someone have to walk about 471.2 feet to go completely around the

ring. ANSWER: Therefore, the circumference of the path is about 93.13 feet. 471.2 ft ANSWER: 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a 93.13 ft circumference of 68 feet. The outer edge of the path 45. MULTIPLE REPRESENTATIONS In this is going to be 4 feet from the pond all the way problem, you will explore changing dimensions in around. What is the approximate circumference of circles. the path? Round to the nearest hundredth. a. GEOMETRIC Use a compass to draw three SOLUTION: circles in which the scale factor from each circle to The circumference C of a circle with radius r is the next larger circle is 1 : 2. b. TABULAR given by Here, Cpond = 68 ft. Use the Calculate the radius (to the nearest formula to find the radius of the pond. tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. So, the radius of the pond is about 10.82 feet and the e. ANALYTICAL The scale factor from to radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference is . Write an equation relating the of the brick path. circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to Therefore, the circumference of the path is about 93.13 feet. is , and the circumference of is 12

ANSWER: inches, what is the circumference of ? 93.13 ft SOLUTION: a. Sample answer: 45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) b. Sample answer. of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to c. They all have the same shape–circular. is . Write an equation relating the d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. circumference (CA) of to the circumference So, if the radii are in the ratio b : a, so will the (C ) of . B circumference. Therefore, . f. NUMERICAL If the scale factor from to f. Use the formula from part e. Here, the is , and the circumference of is 12 circumference of is 12 inches and . inches, what is the circumference of ? SOLUTION: a. Sample answer:

Therefore, the circumference of is 4 inches.

ANSWER: a. Sample answer:

b. Sample answer.

b. Sample answer.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . c. They all have the same shape—circular. d. The ratio of their circumferences is also 2. f. Use the formula from part e. Here, the e. circumference of is 12 inches and . f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set

Therefore, the circumference of is 4 inches. of horizontal lines that are centimeters apart on a sheet of plain white paper. ANSWER: a. Sample answer:

b. Sample answer. a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number c. They all have the same shape—circular. of drops to the number of hits after 25, 50, and 100 d. The ratio of their circumferences is also 2. drops.

e. c. How are the values you found in part b related to π? f. 4 in. SOLUTION: BUFFON’S NEEDLE 46. Measure the length of a a. See students’ work. needle (or toothpick) in centimeters. Next, draw a set b. See students’ work. of horizontal lines that are centimeters apart on a c. Sample answer: The values are approaching 3.14, sheet of plain white paper. which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14,

which is approximately equal to π.

a. Drop the needle onto the paper. When the needle M APS lands, record whether it touches one of the lines as a 47. The concentric circles on the map below hit. Record the number of hits after 25, 50, and 100 show the areas that are 5, 10, 15, 20, 25, and 30

drops. miles from downtown Phoenix.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π?

SOLUTION: a. How much greater is the circumference of the a. See students’ work. outermost circle than the circumference of the center b. See students’ work. circle? c. Sample answer: The values are approaching 3.14, b. As the radii of the circles increases by 5 miles, by which is approximately equal to π. how much does the circumference increase?

SOLUTION: ANSWER: a. The radius of the outermost circle is 30 miles and a. See students’ work. that of the center circle is 5 miles. Find the b. See students’ work. circumference of each circle. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. The difference of the circumferences of the outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

a. How much greater is the circumference of the outermost circle than the circumference of the center circle?

b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? The difference of the two circumferences is (2xπ + 10π) - (2xπ) = 10π or about 31.4 miles. SOLUTION: Therefore, as the radius of each circle increases by 5 a. The radius of the outermost circle is 30 miles and miles, the circumference increases by about 31.4 that of the center circle is 5 miles. Find the miles. circumference of each circle. ANSWER: a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? The difference of the circumferences of the SOLUTION: outermost circle and the center circle is 60π - 10π = Sample answer: A line and a circle may intersect in 50π or about 157.1 miles. one point, two points, or may not intersect at all. A b. Let the radius of one circle be x miles and the line that intersects a circle in one point can be radius of the next larger circle be x + 5 miles. Find described as a tangent. A line that intersects a circle the circumference of each circle. in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius. π + The difference of the two circumferences is (2x ANSWER: 10π) - (2xπ) = 10π or about 31.4 miles. Sample answer: A line that intersects a circle in one Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 point can be described as a tangent. A line that miles. intersects a circle in exactly two points can be described as a secant. A line segment with ANSWER: endpoints on a circle can be described as a chord. a. 157.1 mi If the chord passes through the center of the circle, b. by about 31.4 mi it can be described as a diameter. A line segment with endpoints at the center and on the circle can be 48. WRITING IN MATH How can we describe the described as a radius. relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A 49. line that intersects a circle in one point can be REASONING In the figure, a circle with radius r is described as a tangent. A line that intersects a circle inscribed in a regular polygon and circumscribed in exactly two points can be described as a secant. about another. A line segment with endpoints on a circle can be a. What are the perimeters of the circumscribed and described as a chord. If the chord passes through inscribed polygons in terms of r? Explain. the center of the circle, it can be described as a b. Is the circumference C of the circle greater or diameter. A line segment with endpoints at the less than the perimeter of the circumscribed polygon? center and on the circle can be described as a the inscribed polygon? Write a compound inequality radius. comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the ANSWER: diameter d of the circle and interpret its meaning. Sample answer: A line that intersects a circle in one d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to point can be described as a tangent. A line that the upper and lower limits of the inequality from part intersects a circle in exactly two points can be c, and what does this imply? described as a secant. A line segment with endpoints on a circle can be described as a chord. SOLUTION: If the chord passes through the center of the circle, a. The circumscribed polygon is a square with sides it can be described as a diameter. A line segment of length 2r. with endpoints at the center and on the circle can be described as a radius.

The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.) 49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality comparing C to these perimeters. So, the perimeters are 8r and 6r. c. Rewrite the inequality from part b in terms of the b. diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part The circumference is less than the perimeter of the c, and what does this imply? circumscribed polygon and greater than the SOLUTION: perimeter of the inscribed polygon. a. The circumscribed polygon is a square with sides 8r < C < 6r of length 2r. c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. The inscribed polygon is a hexagon with sides of d. These limits will approach a value of πd, implying length r. (Each triangle formed using a central angle that C = πd. will be equilateral.)

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r So, the perimeters are 8r and 6r. c. 3d < C < 4d; The circumference of the circle is b. between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. The circumference is less than the perimeter of the CHALLENGE circumscribed polygon and greater than the 50. The sum of the circumferences of perimeter of the inscribed polygon. circles H, J, and K shown at the right is units. Find KJ. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. SOLUTION: ANSWER: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the a. 8r and 6r ; Twice the radius of the circle, 2r is circumference of each circle. the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

50. CHALLENGE The sum of the circumferences of The sum of the circumferences is 56π units. circles H, J, and K shown at the right is units. Substitute the three circumferences and solve for x. Find KJ.

So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii of and . SOLUTION: Therefore, KJ = 16 + 8 or 24 units. The radius of is 4x units, the radius of is x ANSWER: units, and the radius of is 2x units. Find the circumference of each circle. 24 units 51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Explain. SOLUTION: A radius is a segment drawn between the center of

the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: The sum of the circumferences is 56π units. Always; a radius is a segment drawn between the Substitute the three circumferences and solve for x. center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

So, the radius of is 4 units, the radius of is 52. PROOF Use the locus definition of a circle and 2(4) or 8 units, and the radius of is 4(4) or 16 dilations to prove that all circles are similar. units. The measure of KJ is equal to the sum of the radii SOLUTION: A circle is a locus of points in a plane equidistant of and . from a given point. For any two circles and Therefore, KJ = 16 + 8 or 24 units. , there exists a translation that ANSWER: maps center A onto center B, moving so that it 24 units is concentric with . There also exists a dilation with scale factor k such 51. REASONING Is the distance from the center of a that each point that makes up is moved to be the circle to a point in the interior of a circle sometimes, same distance from center A as the points that make always, or never less than the radius of the circle? up are from center B. Explain. For some value k, r2 = kr . SOLUTION: 1

A radius is a segment drawn between the center of

the circle and a point on the circle. A segment drawn

from the center to a point inside the circle will always

have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the Therefore, is mapped onto . Since there center of the circle and a point on the circle. A exists a rigid motion followed by a scaling that maps segment drawn from the center to a point inside the onto , the circles are circle will always have a length less than the radius similar. Thus, all circles are similar. of the circle.

PROOF Use the locus definition of a circle and 52. ANSWER: dilations to prove that all circles are similar. A circle is a locus of points in a plane equidistant SOLUTION: from a given point. For any two circles and A circle is a locus of points in a plane equidistant , there exists a translation that maps center A from a given point. For any two circles and onto center B, moving so that it is concentric , there exists a translation that with . There also exists a dilation with scale maps center A onto center B, moving so that it factor k such that each point that makes up is is concentric with . There also exists a dilation moved to be the same distance from center A as the with scale factor k such that each point that makes up is moved to be the points that make up are from center B. same distance from center A as the points that make Therefore, is mapped onto . Since there up are from center B. exists a rigid motion followed by a scaling that maps For some value k, r2 = onto , the circles are similar. Thus, all circles are similar. kr1. 53. CHALLENGE In the figure, is inscribed in

equilateral triangle LMN. What is the circumference

of ?

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

SOLUTION: ANSWER: Draw a perpendicular to the side Also join A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Since is equilateral triangle, MN = 8 inches Therefore, is mapped onto . Since there and RN = 4 inches. Also, bisects exists a rigid motion followed by a scaling that maps Use the definition of onto , the circles are similar. Thus, all tangent of an angle to find PR, which is the radius of circles are similar.

53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference of ?

Therefore, the radius of is The circumference C of a circle with radius r is given by

SOLUTION: Draw a perpendicular to the side Also join

The circumference of is inches.

ANSWER:

in.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects 54. WRITING IN MATH Research and write about Use the definition of the history of pi and its importance to the study of

tangent of an angle to find PR, which is the radius of geometry. SOLUTION: See students’ work.

ANSWER: See students’ work.

GRIDDED RESPONSE 55. What is the Therefore, the radius of is circumference of ? Round to the nearest tenth. The circumference C of a circle with radius r is given by

SOLUTION: Use the Pythagorean Theorem to find the diameter

The circumference of is inches. of the circle.

ANSWER: The circumference C of a circle with diameter d is given by in. Use the formula to find the circumference.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

ANSWER: ANSWER: See students’ work. 40.8

55. GRIDDED RESPONSE What is the 56. What is the radius of a table with a circumference of

circumference of ? Round to the nearest tenth. 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft SOLUTION:

The circumference C of a circle with radius r is SOLUTION: given by Use the Pythagorean Theorem to find the diameter Here, C = 10 ft. Use the formula to find the radius. of the circle.

The circumference C of a circle with diameter d is given by Use the formula to find the circumference. The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use ANSWER: up to 50 feet of fence, what radius can he use for the 40.8 garden? F 10 56. What is the radius of a table with a circumference of G 9 10 feet? H 8 A 1.6 ft J 7 B 2.5 ft C 3.2 ft SOLUTION: D 5 ft The circumference C of a circle with radius r is given by SOLUTION: Because 50 feet of fence can be used to border the The circumference C of a circle with radius r is garden, Cmax = 50. Use the circumference formula to

given by find the maximum radius. Here, C = 10 ft. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the Therefore, the radius of the garden must be less than correct choice is A. or equal to about 7.96 feet. Therefore, the only correct choice is J. ANSWER: ANSWER: A J 57. ALGEBRA Bill is planning a circular vegetable SAT/ACT garden with a fence around the border. If he can use 58. What is the radius of a circle with an

up to 50 feet of fence, what radius can he use for the area of square units? garden? F 10 A 0.4 units G 9 B 0.5 units H 8 C 2 units J 7 D 4 units E 16 units SOLUTION: The circumference C of a circle with radius r is SOLUTION: given by The area of a circle of radius r is given by the Because 50 feet of fence can be used to border the formula Here A = square units. Use the garden, Cmax = 50. Use the circumference formula to formula to find the radius. find the maximum radius.

Therefore, the correct choice is B.

Therefore, the radius of the garden must be less than ANSWER: or equal to about 7.96 feet. Therefore, the only B correct choice is J. Copy each figure and point B. Then use a ruler ANSWER: to draw the image of the figure under a dilation J with center B and the scale factor r indicated.

58. SAT/ACT What is the radius of a circle with an 59.

area of square units? A 0.4 units B 0.5 units C 2 units

D 4 units E 16 units SOLUTION: Draw a ray from point B through a vertex of the SOLUTION: hexagon. Use the ruler to measure the distance from The area of a circle of radius r is given by the B to the vertex. Mark a point on the ray that is of formula Here A = square units. Use the that distance from B. Repeat the process for rays formula to find the radius. through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the correct choice is B.

ANSWER: B Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59. ANSWER:

60. SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other five vertices. Connect the six points Draw a ray from point B through a vertex of the on the rays to construct the dilated image of the triangle. Use the ruler to measure the distance from hexagon. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER: ANSWER:

61.

60.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is 2 Draw a ray from point B through a vertex of the times that distance from B. Repeat the process for triangle. Use the ruler to measure the distance from rays through the other three vertices. Connect the four points on the rays to construct the dilated image B to the vertex. Mark a point on the ray that is of of the trapezoid. that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER: ANSWER:

61. 62.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 SOLUTION: times that distance from B. Repeat the process for Draw a ray from point B through a vertex of the rays through the other three vertices. Connect the triangle. Use the ruler to measure the distance from four points on the rays to construct the dilated image B to the vertex. Mark a point on the ray that is 3 of the trapezoid. times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

ANSWER: 62.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the State whether each figure has rotational three points on the rays to construct the dilated image symmetry. Write yes or no. If so, copy the of the triangle. figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

64. Refer to the image on page 705. ANSWER: SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. 65. SOLUTION: SOLUTION: The figure cannot be mapped onto itself by any The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. rotation. So, it does not have a rotational symmetry.

ANSWER: ANSWER: No No

64. Refer to the image on page 705. 66. Refer to the image on page 705. SOLUTION: SOLUTION: The figure can be mapped onto itself by a rotation of The given figure cannot be mapped onto itself by any 90º. So, it has a rotational symmetry. So, the rotation. So, it does not have a rotational symmetry. magnitude of symmetry is 90º and order of symmetry ANSWER: is 4. No ANSWER: Yes; 4, 90 Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. 65. ANSWER: SOLUTION: True; sample answer: Since the hypothesis is true The given figure cannot be mapped onto itself by any and the conclusion is true, then the statement is true rotation. So, it does not have a rotational symmetry. for the conditions. ANSWER: No 68. You are 21 years old and do not vote. SOLUTION: 66. Refer to the image on page 705. False; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is false, then the statement is false The given figure cannot be mapped onto itself by any for the conditions. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: False; sample answer: Since the hypothesis is true No and the conclusion is false, then the statement is false for the conditions. Determine the truth value of the following statement for each set of conditions. Explain Find x. your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true 69. and the conclusion is true, then the statement is true SOLUTION: for the conditions. The sum of the measures of angles around a point is ANSWER: 360º. True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

ANSWER: 68. You are 21 years old and do not vote. 90 SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true 70. and the conclusion is false, then the statement is false for the conditions. SOLUTION: The angles with the measures xº and 2xº form a Find x. straight line. So, the sum is 180.

ANSWER: 69. 60 SOLUTION: The sum of the measures of angles around a point is 360º.

71. ANSWER: SOLUTION: 90 The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 70. 20 SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

72. ANSWER: SOLUTION: 60 The sum of the three angles is 180.

ANSWER: 45 71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius. For Exercises 1–4, refer to .

Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 1. Name the circle. inches, 18 inches, and 11 inches, respectively. SOLUTION: Find each measure. The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord 5. FG b. a diameter SOLUTION: c. a radius Since the diameter of B is 18 inches and A is 8 SOLUTION: inches, AG = 18 and AF = (8) or 4. a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center Therefore, FG = 14 in. and on the circle. Here, is ANSWER: radius. 14 in. ANSWER: a. 6. FB b. SOLUTION: c. Since the diameter of B is 18 inches and A is 4 3. If CN = 8 centimeters, find DN. inches, AB = (18) or 9 and AF = (8) or 4. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm. Therefore, FB = 5 inches. ANSWER: 8 cm ANSWER: 5 in. 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the Here, is a radius and the diameter is twice the radius and circumference of the ride? Round to the radius. nearest hundredth, if necessary.

SOLUTION: Therefore, the diameter is 26 ft. The radius is half the diameter. So, the radius of the circular ride is 22 feet. ANSWER: 26 ft

The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet.

5. FG What are the diameter and radius of the pool? Round to the nearest hundredth. SOLUTION: Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4.

Therefore, FG = 14 in. SOLUTION: ANSWER: 14 in.

6. FB SOLUTION:

Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

The diameter of the pool is about 17.98 feet and the Therefore, FB = 5 inches. radius of the pool is about 8.99 feet.

ANSWER: ANSWER: 5 in. 17.98 ft; 8.99 ft

7. RIDES The circular ride described at the beginning 9. SHORT RESPONSE The right triangle shown is of the lesson has a diameter of 44 feet. What are the inscribed in . Find the exact circumference of radius and circumference of the ride? Round to the . nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet.

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle. Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft The diameter of the circle is 4 centimeters. 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to . SOLUTION:

10. Name the center of the circle. SOLUTION: R ANSWER:

R The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. 11. Identify a chord that is also a diameter. ANSWER: SOLUTION: 17.98 ft; 8.99 ft Two chords are shown: and . goes 9. SHORT RESPONSE The right triangle shown is through the center, R, so is a diameter. inscribed in . Find the exact circumference of . ANSWER:

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center SOLUTION: The diameter of the circle is the hypotenuse of the and on the circle. But has both the end points on right triangle. the circle, so it is a chord. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT?

The diameter of the circle is 4 centimeters. SOLUTION: Here, is a diameter and is a radius.

The circumference of the circle is 4π centimeters. Therefore, RT = 8.1 cm. ANSWER: ANSWER:

8.1 cm

For Exercises 10–13, refer to . For Exercises 14–17, refer to .

10. Name the center of the circle. 14. Identify a chord that is not a diameter. SOLUTION: SOLUTION: R The chords do not pass through the ANSWER: center. So, they are not diameters. R ANSWER:

11. Identify a chord that is also a diameter. SOLUTION: 15. If CF = 14 inches, what is the diameter of the circle? Two chords are shown: and . goes SOLUTION:

through the center, R, so is a diameter. Here, is a radius and the diameter is twice the radius. . ANSWER:

Therefore, the diameter of the circle is 28 inches. 12. Is a radius? Explain. SOLUTION: ANSWER: A radius is a segment with endpoints at the center 28 in. and on the circle. But has both the end points on 16. Is ? Explain. the circle, so it is a chord. SOLUTION: ANSWER: All radii of a circle are congruent. Since No; it is a chord. are both radii of , . 13. If SU = 16.2 centimeters, what is RT? ANSWER: SOLUTION: Yes; they are both radii of . Here, is a diameter and is a radius. 17. If DA = 7.4 centimeters, what is EF? SOLUTION: Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm.

ANSWER: 8.1 cm Therefore, EF = 3.7 cm. For Exercises 14–17, refer to . ANSWER: 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

14. Identify a chord that is not a diameter. SOLUTION: The chords do not pass through the center. So, they are not diameters.

ANSWER: 18. CK

SOLUTION: 15. If CF = 14 inches, what is the diameter of the circle? Since the radius of K is 8 units, BK = 8. SOLUTION: Here, is a radius and the diameter is twice the

radius. . Therefore, CK = 2.6 units. ANSWER: 2.6

Therefore, the diameter of the circle is 28 inches. 19. AB ANSWER: SOLUTION: 28 in. Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. 16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since Therefore, AB = 14.6 units. are both radii of , . ANSWER: ANSWER: 14.6 Yes; they are both radii of . 20. JK 17. If DA = 7.4 centimeters, what is EF? SOLUTION: SOLUTION: First find CK. Since circle K has a radius of 8 units, Here, is a diameter and is a radius. BK = 8.

Since circle J has a radius of 10 units, JC = 10.

Therefore, EF = 3.7 cm.

ANSWER: Therefore, JK = 12.6 units. 3.7 cm ANSWER: Circle J has a radius of 10 units, has a 12.6 radius of 8 units, and BC = 5.4 units. Find each measure. 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

18. CK

SOLUTION: Since the radius of K is 8 units, BK = 8.

Therefore, AD = 30.6 units.

ANSWER: Therefore, CK = 2.6 units. 30.6 ANSWER: PIZZA 2.6 22. Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary. 19. AB SOLUTION: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units. SOLUTION: ANSWER: The diameter of the pizza is 16 inches. 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, . BK = 8. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Since circle J has a radius of 10 units, JC = 10.

Therefore, the circumference of the pizza is about Therefore, JK = 12.6 units. 50.27 inches.

ANSWER: ANSWER: 12.6 8 in.; 50.27 in.

21. AD 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a SOLUTION: tire. Round to the nearest hundredth, if necessary. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J SOLUTION: is 10 units, so KD = 8 and AC = 2(10) or 20. Before The diameter of a tire is 26 inches. The radius is half finding AD, we need to find CK. the diameter.

So, the radius of the tires is 13 inches. Therefore, AD = 30.6 units. The circumference C of a circle with diameter d is given by ANSWER: 30.6

22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if Therefore, the circumference of the bicycle tire is necessary. about 81.68 inches. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in.

SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The diameter of the pizza is 16 inches. given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the circumference of the pizza is about 50.27 inches. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. ANSWER: 8 in.; 50.27 in. ANSWER: 5.73 in.; 2.86 in. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. 25. C = 124 ft SOLUTION: SOLUTION: The diameter of a tire is 26 inches. The radius is half The circumference C of a circle with diameter d is the diameter. given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches. Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. ANSWER: 13 in.; 81.68 in. ANSWER: 39.47 ft; 19.74 ft Find the diameter and radius of a circle by with the given circumference. Round to the nearest 26. C = 375.3 cm hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 375.3 cm. Use the formula to find the given by diameter. Then find the radius. Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. ANSWER: 119.46 cm; 59.73 cm ANSWER: 5.73 in.; 2.86 in. 27. C = 2608.25 m SOLUTION: 25. C = 124 ft The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

Therefore, the diameter is about 39.47 feet and the ANSWER: radius is about 19.74 feet. 830.23 m; 415.12 m ANSWER: CCSS SENSE-MAKING Find the exact 39.47 ft; 19.74 ft circumference of each circle by using the given inscribed or 26. C = 375.3 cm circumscribed polygon. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. 28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The Therefore, the diameter is about 119.46 centimeters circumference C of a circle is given by and the radius is about 59.73 centimeters.

ANSWER: 119.46 cm; 59.73 cm Therefore, the circumference of the circle is 17 27. C = 2608.25 m centimeters. SOLUTION: ANSWER: The circumference C of a circle with diameter d is

given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER:

830.23 m; 415.12 m The diameter of the circle is 12 feet. The circumference C of a circle is given by CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon. Therefore, the circumference of the circle is 12 feet.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter The diameter of the circle is 17 cm. The of the circle. circumference C of a circle is given by

Therefore, the circumference of the circle is 17 centimeters. The diameter of the circle is inches. The circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

31. SOLUTION: Each diagonal of the inscribed rectangle will pass The diameter of the circle is 12 feet. The through the origin, so it is a diameter of the circle. circumference C of a circle is given by Use the Pythagorean Theorem to find the diameter of the circle.

Therefore, the circumference of the circle is 12 feet. The diameter of the circle is 10 inches. The ANSWER: circumference C of a circle is given by

Therefore, the circumference of the circle is 10 inches.

30. ANSWER: SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

32. SOLUTION: A diameter perpendicular to a side of the square and The diameter of the circle is inches. The the length of each side will measure the same. So, d circumference C of a circle is given by = 25 millimeters. The circumference C of a circle is given by

Therefore, the circumference of the circle is Therefore, the circumference of the circle is 25 inches. millimeters. ANSWER: 10-1 Circles and Circumference ANSWER:

31. SOLUTION: 33. Each diagonal of the inscribed rectangle will pass SOLUTION: through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter A diameter perpendicular to a side of the square and of the circle. the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The circumference C of a circle is given by Therefore, the circumference of the circle is 14 yards.

ANSWER:

Therefore, the circumference of the circle is 10 inches. 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and ANSWER: clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is 32. given by SOLUTION: Here, C = 66.92 cm. Use the formula to find the diameter. A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the Therefore, the circumference of the circle is 25 diameter in centimeters. millimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. ANSWER: ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. eSolutions Manual - Powered by Cognero Page 8 a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the 33. nearest foot? SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

SOLUTION:

Therefore, the circumference of the circle is 14 a. The radius of the patio is 3 + 2 = 5 ft. The yards. circumference C of a circle with radius r is given by

ANSWER: Use the formula to find the circumference.

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and Therefore, the circumference is about 31.42 feet. clubs. For professional competitions, the maximum b. Here, C = 25 ft. Use the formula to find the weight of a disc in grams is 8.3 times its diameter in inner centimeters. What is the maximum allowable weight radius of the inner circle. for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is

given by So, the radius of the inner circle is about 4 feet. Here, C = 66.92 cm. Use the formula to find the diameter. ANSWER: a. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to The diameter of the disc is about 21.3 centimeters the nearest hundredth. and the maximum weight allowed is 8.3 times the 36. diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. SOLUTION: The radius is half the diameter and the ANSWER: circumference C of a circle with diameter d is given 176.8 g by

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

SOLUTION: 37. a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by SOLUTION: The diameter is twice the radius and the Use the formula to find the circumference. circumference C of a circle with diameter d is given by

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

So, the radius of the inner circle is about 4 feet. ANSWER: 22.80 ft; 71.63 ft ANSWER: a. 31.42 ft 38. b. 4 ft SOLUTION: The radius, diameter, or circumference of a The circumference C of a circle with diameter d is circle is given. Find each missing measure to given by the nearest hundredth. Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. 36.

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters.

ANSWER: Therefore, the radius is 4.25 inches and the 11.14x cm; 5.57x cm circumference is about 26.70 inches.

39. ANSWER: 4.25 in.; 26.70 in. SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given 37. by SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: So, the diameter is 22.80 feet and the circumference 0.25x; 0.79x is about 71.63 feet. Determine whether the circles in the figures ANSWER: below appear to be congruent, concentric, or 22.80 ft; 71.63 ft neither. 40. Refer to the photo on page 703. 38. SOLUTION: SOLUTION: The circles have the same center and they are in the The circumference C of a circle with diameter d is same plane. So, they are concentric circles. given by ANSWER: Here, C = 35x cm. Use the formula to find the concentric diameter.Then find the radius.

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. Therefore, the diameter is about 11.14x centimeters SOLUTION: and the radius is about 5.57x centimeters. The circles appear to have the same radius. So, they appear to be congruent. ANSWER: 11.14x cm; 5.57x cm ANSWER: congruent

39. 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the SOLUTION: coordinate grid represents 25 feet, how far would The diameter is twice the radius and the someone have to walk to go completely around the circumference C of a circle with diameter d is given ring? Round to the nearest tenth. by

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: SOLUTION: 0.25x; 0.79x The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a Determine whether the circles in the figures circle with diameter d is given by below appear to be congruent, concentric, or So, the circumference is neither. Therefore, someone have to 40. Refer to the photo on page 703. walk about 471.2 feet to go completely around the SOLUTION: ring. The circles have the same center and they are in the same plane. So, they are concentric circles. ANSWER: 471.2 ft ANSWER: concentric 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path 41. Refer to the photo on page 703. is going to be 4 feet from the pond all the way SOLUTION: around. What is the approximate circumference of The circles neither have the same radius nor are they the path? Round to the nearest hundredth. concentric. SOLUTION: ANSWER: The circumference C of a circle with radius r is neither given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. 42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they appear to be congruent.

ANSWER: So, the radius of the pond is about 10.82 feet and the congruent radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference 43. HISTORY The Indian Shell Ring on Hilton Head of the brick path. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth. Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest SOLUTION: tenth) and circumference (to the nearest hundredth) The radius of the circle is 3 units, that is 75 feet. So, of each circle. Record your results in a table. the diameter is 150 ft. The circumference C of a c. VERBAL Explain why these three circles are circle with diameter d is given by geometrically similar. So, the circumference is d. VERBAL Make a conjecture about the ratio Therefore, someone have to between the circumferences of two circles when the ratio between their radii is 2. walk about 471.2 feet to go completely around the e. ANALYTICAL The scale factor from to ring. is . Write an equation relating the ANSWER:

471.2 ft circumference (CA) of to the circumference

44. CCSS MODELING A brick path is being installed (CB) of . around a circular pond. The pond has a f. NUMERICAL If the scale factor from to circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way is , and the circumference of is 12 around. What is the approximate circumference of inches, what is the circumference of ? the path? Round to the nearest hundredth. SOLUTION: SOLUTION: a. The circumference C of a circle with radius r is Sample answer: given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

b. Sample answer.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

c. They all have the same shape–circular. Therefore, the circumference of the path is about d. The ratio of their circumferences is also 2. 93.13 feet. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the ANSWER: 93.13 ft circumference. Therefore, . f. Use the formula from part e. Here, the MULTIPLE REPRESENTATIONS 45. In this problem, you will explore changing dimensions in circumference of is 12 inches and . circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest Therefore, the circumference of is 4 inches. tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. ANSWER: c. VERBAL Explain why these three circles are a. Sample answer: geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to is . Write an equation relating the b. Sample answer. circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to is , and the circumference of is 12 inches, what is the circumference of ? c. They all have the same shape—circular. SOLUTION: d. The ratio of their circumferences is also 2. a. Sample answer: e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a b. Sample answer. sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. a. Drop the needle onto the paper. When the needle So, if the radii are in the ratio b : a, so will the lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 circumference. Therefore, . drops.

f. Use the formula from part e. Here, the circumference of is 12 inches and . b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to Therefore, the circumference of is 4 inches. π? ANSWER: SOLUTION: a. Sample answer: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER:

b. Sample answer. a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 c. They all have the same shape—circular. miles from downtown Phoenix. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase?

SOLUTION: a. The radius of the outermost circle is 30 miles and a. Drop the needle onto the paper. When the needle that of the center circle is 5 miles. Find the lands, record whether it touches one of the lines as a circumference of each circle. hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops. The difference of the circumferences of the outermost circle and the center circle is 60π - 10π = c. How are the values you found in part b related to 50π or about 157.1 miles. π? b. Let the radius of one circle be x miles and the SOLUTION: radius of the next larger circle be x + 5 miles. Find a. See students’ work. the circumference of each circle. b. See students’ work. c. Sample answer: The values are approaching 3.14,

which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. The difference of the two circumferences is (2xπ + c. Sample answer: The values are approaching 3.14, 10π) - (2xπ) = 10π or about 31.4 miles. which is approximately equal to π. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 ANSWER: miles from downtown Phoenix. a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be a. How much greater is the circumference of the described as a tangent. A line that intersects a circle outermost circle than the circumference of the center in exactly two points can be described as a secant. circle? A line segment with endpoints on a circle can be b. As the radii of the circles increases by 5 miles, by described as a chord. If the chord passes through how much does the circumference increase? the center of the circle, it can be described as a SOLUTION: diameter. A line segment with endpoints at the a. The radius of the outermost circle is 30 miles and center and on the circle can be described as a that of the center circle is 5 miles. Find the radius. circumference of each circle. ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, The difference of the circumferences of the it can be described as a diameter. A line segment outermost circle and the center circle is 60π - 10π = with endpoints at the center and on the circle can be 50π or about 157.1 miles. described as a radius. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. The difference of the two circumferences is (2xπ + b. Is the circumference C of the circle greater or 10π) - (2xπ) = 10π or about 31.4 miles. less than the perimeter of the circumscribed polygon? Therefore, as the radius of each circle increases by 5 the inscribed polygon? Write a compound inequality miles, the circumference increases by about 31.4 comparing C to these perimeters. miles. c. Rewrite the inequality from part b in terms of the ANSWER: diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed a. 157.1 mi and inscribed polygons increase, what will happen to b. by about 31.4 mi the upper and lower limits of the inequality from part c, and what does this imply? 48. WRITING IN MATH How can we describe the relationships that exist between SOLUTION: circles and lines? a. The circumscribed polygon is a square with sides SOLUTION: of length 2r. Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle The inscribed polygon is a hexagon with sides of in exactly two points can be described as a secant. length r. (Each triangle formed using a central angle A line segment with endpoints on a circle can be will be equilateral.) described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER:

Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be So, the perimeters are 8r and 6r. described as a secant. A line segment with b. endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment The circumference is less than the perimeter of the with endpoints at the center and on the circle can be circumscribed polygon and greater than the described as a radius. perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the 49. circumference of the circle is between 3 and 4 times REASONING In the figure, a circle with radius r is its diameter. inscribed in a regular polygon and circumscribed d. These limits will approach a value of πd, implying about another. that C = πd. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or ANSWER: less than the perimeter of the circumscribed polygon? a. 8r and 6r ; Twice the radius of the circle, 2r is the inscribed polygon? Write a compound inequality the side length of the square, so the perimeter of the comparing C to these perimeters. square is 4(2r) or 8r. The c. Rewrite the inequality from part b in terms of the regular hexagon is made up of six equilateral diameter d of the circle and interpret its meaning. triangles with side length r, so the perimeter of the d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to hexagon is 6(r) or 6r. the upper and lower limits of the inequality from part b. less; greater; 6r < C < 8r c, and what does this imply? c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. SOLUTION: d. These limits will approach a value of πd, implying a. The circumscribed polygon is a square with sides that C = πd. of length 2r. 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units.

Find KJ. The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The sum of the circumferences is 56π units. . Therefore, the The inequality is then 4d < C < 3d Substitute the three circumferences and solve for x. circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 ANSWER: units. a. 8r and 6r ; Twice the radius of the circle, 2r is The measure of KJ is equal to the sum of the radii the side length of the square, so the perimeter of the of and . square is 4(2r) or 8r. The Therefore, KJ = 16 + 8 or 24 units. regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the ANSWER: hexagon is 6(r) or 6r. 24 units b. less; greater; 6r < C < 8r 51. REASONING Is the distance from the center of a c. 3d < C < 4d; The circumference of the circle is circle to a point in the interior of a circle sometimes, between 3 and 4 times its diameter. always, or never less than the radius of the circle? d. These limits will approach a value of πd, implying Explain. that C = πd. SOLUTION: 50. CHALLENGE The sum of the circumferences of A radius is a segment drawn between the center of circles H, J, and K shown at the right is units. the circle and a point on the circle. A segment drawn Find KJ. from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

SOLUTION: 52. PROOF Use the locus definition of a circle and The radius of is 4x units, the radius of is x dilations to prove that all circles are similar. units, and the radius of is 2x units. Find the SOLUTION: circumference of each circle. A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr . 1

The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

Therefore, is mapped onto . Since there So, the radius of is 4 units, the radius of is exists a rigid motion followed by a scaling that maps 2(4) or 8 units, and the radius of is 4(4) or 16 onto , the circles are units. similar. Thus, all circles are similar. The measure of KJ is equal to the sum of the radii of and . Therefore, KJ = 16 + 8 or 24 units. ANSWER: ANSWER: A circle is a locus of points in a plane equidistant 24 units from a given point. For any two circles and , there exists a translation that maps center A 51. REASONING Is the distance from the center of a onto center B, moving so that it is concentric circle to a point in the interior of a circle sometimes, with . There also exists a dilation with scale always, or never less than the radius of the circle? factor k such that each point that makes up is Explain. moved to be the same distance from center A as the SOLUTION: points that make up are from center B. A radius is a segment drawn between the center of Therefore, is mapped onto . Since there the circle and a point on the circle. A segment drawn exists a rigid motion followed by a scaling that maps from the center to a point inside the circle will always onto , the circles are similar. Thus, all have a length less than the radius of the circle. circles are similar. ANSWER: 53. CHALLENGE In the figure, is inscribed in Always; a radius is a segment drawn between the equilateral triangle LMN. What is the circumference center of the circle and a point on the circle. A

segment drawn from the center to a point inside the of ? circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar.

SOLUTION: A circle is a locus of points in a plane equidistant SOLUTION: from a given point. For any two circles and , there exists a translation that Draw a perpendicular to the side Also join maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1. Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

Therefore, the radius of is ANSWER: The circumference C of a circle with radius r is A circle is a locus of points in a plane equidistant given by from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the

points that make up are from center B. The circumference of is inches. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps ANSWER: onto , the circles are similar. Thus, all circles are similar. in.

53. CHALLENGE In the figure, is inscribed in 54. WRITING IN MATH Research and write about equilateral triangle LMN. What is the circumference the history of pi and its importance to the study of of ? geometry. SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the SOLUTION: circumference of ? Round to the nearest tenth. Draw a perpendicular to the side Also join

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle. Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects The circumference C of a circle with diameter d is Use the definition of given by tangent of an angle to find PR, which is the radius of Use the formula to find the circumference.

Therefore, the radius of is ANSWER: The circumference C of a circle with radius r is 40.8 given by 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft

D 5 ft The circumference of is inches. SOLUTION: ANSWER: The circumference C of a circle with radius r is given by in. Here, C = 10 ft. Use the formula to find the radius.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry.

SOLUTION: The radius of the circle is about 1.6 feet, so the See students’ work. correct choice is A.

ANSWER: ANSWER: See students’ work. A

55. GRIDDED RESPONSE What is the 57. ALGEBRA Bill is planning a circular vegetable circumference of ? Round to the nearest tenth. garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? F 10 G 9 H 8 J 7 SOLUTION: SOLUTION: Use the Pythagorean Theorem to find the diameter The circumference C of a circle with radius r is of the circle. given by Because 50 feet of fence can be used to border the The circumference C of a circle with diameter d is garden, Cmax = 50. Use the circumference formula to given by find the maximum radius. Use the formula to find the circumference.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J. ANSWER: 40.8 ANSWER: J 56. What is the radius of a table with a circumference of 10 feet? 58. SAT/ACT What is the radius of a circle with an A 1.6 ft area of square units? B 2.5 ft C 3.2 ft A 0.4 units D 5 ft B 0.5 units C 2 units SOLUTION: D 4 units The circumference C of a circle with radius r is E 16 units given by Here, C = 10 ft. Use the formula to find the radius. SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER:

A Therefore, the correct choice is B. ANSWER: 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use B up to 50 feet of fence, what radius can he use for the Copy each figure and point B. Then use a ruler garden? to draw the image of the figure under a dilation F 10 with center B and the scale factor r indicated. G 9 H 8 59. J 7 SOLUTION: The circumference C of a circle with radius r is given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to SOLUTION: find the maximum radius. Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the Therefore, the radius of the garden must be less than hexagon. or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units ANSWER: D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the

formula Here A = square units. Use the 60. formula to find the radius.

SOLUTION:

Therefore, the correct choice is B. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is of B that distance from B. Repeat the process for rays Copy each figure and point B. Then use a ruler through the other two vertices. Connect the three to draw the image of the figure under a dilation points on the rays to construct the dilated image of with center B and the scale factor r indicated. the triangle.

59.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the 61. hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the SOLUTION: hexagon. Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

ANSWER:

60.

62. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from

B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other two vertices. Connect the three Draw a ray from point B through a vertex of the points on the rays to construct the dilated image of triangle. Use the ruler to measure the distance from the triangle. B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

61.

SOLUTION: ANSWER: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state ANSWER: the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No 62.

64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of SOLUTION: 90º. So, it has a rotational symmetry. So, the Draw a ray from point B through a vertex of the magnitude of symmetry is 90º and order of symmetry triangle. Use the ruler to measure the distance from is 4. B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for ANSWER: rays through the other two vertices. Connect the Yes; 4, 90 three points on the rays to construct the dilated image of the triangle.

65. SOLUTION: The given figure cannot be mapped onto itself by any

rotation. So, it does not have a rotational symmetry.

ANSWER: ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

Determine the truth value of the following State whether each figure has rotational statement for each set of conditions. Explain symmetry. Write yes or no. If so, copy the your reasoning. figure, locate the center of symmetry, and state If you are over 18 years old, then you vote in all the order and magnitude of symmetry. elections. 63. Refer to the image on page 705. 67. You are 19 years old and you vote. SOLUTION: SOLUTION: The figure cannot be mapped onto itself by any True; sample answer: Since the hypothesis is true rotation. So, it does not have a rotational symmetry. and the conclusion is true, then the statement is true for the conditions. ANSWER: No ANSWER: True; sample answer: Since the hypothesis is true 64. Refer to the image on page 705. and the conclusion is true, then the statement is true for the conditions. SOLUTION: The figure can be mapped onto itself by a rotation of 68. You are 21 years old and do not vote. 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false ANSWER: for the conditions. Yes; 4, 90 ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. SOLUTION:

The given figure cannot be mapped onto itself by any 69. rotation. So, it does not have a rotational symmetry. SOLUTION: ANSWER: The sum of the measures of angles around a point is 360º. No 66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any ANSWER: rotation. So, it does not have a rotational symmetry. 90 ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all 70. elections. SOLUTION: 67. You are 19 years old and you vote. The angles with the measures xº and 2xº form a SOLUTION: straight line. So, the sum is 180. True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

ANSWER: ANSWER: True; sample answer: Since the hypothesis is true 60 and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false 71. for the conditions. SOLUTION: ANSWER: The angles with the measures 6xº and 3xº form a False; sample answer: Since the hypothesis is true straight line. So, the sum is 180. and the conclusion is false, then the statement is false for the conditions.

Find x. ANSWER: 20

69. SOLUTION: The sum of the measures of angles around a point is 72. 360º. SOLUTION: The sum of the three angles is 180.

ANSWER: 90 ANSWER: 45

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: For Exercises 1–4, refer to . Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle?

1. Name the circle. SOLUTION: SOLUTION: Here, is a radius and the diameter is twice the radius. The center of the circle is N. So, the circle is

ANSWER: Therefore, the diameter is 26 ft.

ANSWER: 2. Identify each. 26 ft a. a chord The diameters of , , and are 8 b. a diameter inches, 18 inches, and 11 inches, respectively. c. a radius Find each measure. SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center

and on the circle. Here, is 5. FG radius. SOLUTION: ANSWER: Since the diameter of B is 18 inches and A is 8 a. inches, AG = 18 and AF = (8) or 4. b. c.

3. If CN = 8 centimeters, find DN.

SOLUTION: Therefore, FG = 14 in. Here, are radii of the same circle. So, ANSWER:

they are equal in length. Therefore, DN = 8 cm. 14 in. ANSWER: 8 cm 6. FB 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Since the diameter of B is 18 inches and A is 4 SOLUTION: inches, AB = (18) or 9 and AF = (8) or 4. Here, is a radius and the diameter is twice the radius.

Therefore, the diameter is 26 ft. Therefore, FB = 5 inches.

ANSWER: ANSWER: 26 ft 5 in.

The diameters of , , and are 8 7. RIDES The circular ride described at the beginning inches, 18 inches, and 11 inches, respectively. of the lesson has a diameter of 44 feet. What are the Find each measure. radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet.

5. FG SOLUTION: Since the diameter of B is 18 inches and A is 8 Therefore, the circumference of the ride is about inches, AG = 18 and AF = (8) or 4. 138.23 feet. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the Therefore, FG = 14 in. circular swimming pool shown is about 56.5 feet. ANSWER: What are the diameter and radius of the pool? Round to the nearest hundredth. 14 in.

6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

SOLUTION:

Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The diameter of the pool is about 17.98 feet and the The radius is half the diameter. So, the radius of the radius of the pool is about 8.99 feet. circular ride is 22 feet. ANSWER: 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of Therefore, the circumference of the ride is about . 138.23 feet.

ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round SOLUTION: to the nearest hundredth. The diameter of the circle is the hypotenuse of the right triangle.

The diameter of the circle is 4 centimeters.

SOLUTION:

The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to .

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

ANSWER: 17.98 ft; 8.99 ft 10. Name the center of the circle. 9. SHORT RESPONSE The right triangle shown is SOLUTION: inscribed in . Find the exact circumference of R . ANSWER: R

11. Identify a chord that is also a diameter. SOLUTION: SOLUTION: Two chords are shown: and . goes The diameter of the circle is the hypotenuse of the through the center, R, so is a diameter. right triangle.

ANSWER:

The diameter of the circle is 4 centimeters. 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord. The circumference of the circle is 4π ANSWER: centimeters. No; it is a chord. ANSWER: 13. If SU = 16.2 centimeters, what is RT?

SOLUTION: For Exercises 10–13, refer to . Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm. 10. Name the center of the circle. ANSWER: SOLUTION: 8.1 cm R For Exercises 14–17, refer to . ANSWER: R

11. Identify a chord that is also a diameter. SOLUTION:

Two chords are shown: and . goes 14. Identify a chord that is not a diameter. through the center, R, so is a diameter. SOLUTION:

The chords do not pass through the ANSWER: center. So, they are not diameters.

ANSWER:

12. Is a radius? Explain.

SOLUTION: 15. If CF = 14 inches, what is the diameter of the circle? A radius is a segment with endpoints at the center SOLUTION: and on the circle. But has both the end points on the circle, so it is a chord. Here, is a radius and the diameter is twice the radius. . ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? Therefore, the diameter of the circle is 28 inches. SOLUTION: ANSWER: Here, is a diameter and is a radius. 28 in.

16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since Therefore, RT = 8.1 cm. are both radii of , . ANSWER: ANSWER: 8.1 cm Yes; they are both radii of . For Exercises 14–17, refer to . 17. If DA = 7.4 centimeters, what is EF? SOLUTION: Here, is a diameter and is a radius.

14. Identify a chord that is not a diameter. SOLUTION: Therefore, EF = 3.7 cm. The chords do not pass through the center. So, they are not diameters. ANSWER: 3.7 cm ANSWER: Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each 15. If CF = 14 inches, what is the diameter of the circle? measure. SOLUTION: Here, is a radius and the diameter is twice the radius. .

18. CK Therefore, the diameter of the circle is 28 inches. SOLUTION: ANSWER: Since the radius of K is 8 units, BK = 8. 28 in.

16. Is ? Explain. Therefore, CK = 2.6 units. SOLUTION: All radii of a circle are congruent. Since ANSWER: are both radii of , . 2.6 ANSWER: 19. AB Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? Since is a diameter of circle J and the radius is SOLUTION: 10, AC = 2(10) or 20 units. Here, is a diameter and is a radius.

Therefore, AB = 14.6 units.

ANSWER: 14.6 Therefore, EF = 3.7 cm. 20. JK ANSWER: 3.7 cm SOLUTION: First find CK. Since circle K has a radius of 8 units, Circle J has a radius of 10 units, has a BK = 8. radius of 8 units, and BC = 5.4 units. Find each measure.

Since circle J has a radius of 10 units, JC = 10.

Therefore, JK = 12.6 units.

ANSWER: 18. CK 12.6 SOLUTION: Since the radius of K is 8 units, BK = 8. 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J

Therefore, CK = 2.6 units. is 10 units, so KD = 8 and AC = 2(10) or 20. Before ANSWER: finding AD, we need to find CK. 2.6

19. AB SOLUTION:

Since is a diameter of circle J and the radius is Therefore, AD = 30.6 units. 10, AC = 2(10) or 20 units. ANSWER: 30.6

Therefore, AB = 14.6 units. 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if ANSWER: necessary. 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Since circle J has a radius of 10 units, JC = 10. SOLUTION: The diameter of the pizza is 16 inches.

Therefore, JK = 12.6 units.

ANSWER: . 12.6 So, the radius of the pizza is 8 inches. 21. AD The circumference C of a circle of diameter d is given by SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before Therefore, the circumference of the pizza is about finding AD, we need to find CK. 50.27 inches.

ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a Therefore, AD = 30.6 units. tire. Round to the nearest hundredth, if necessary. ANSWER: SOLUTION: 30.6 The diameter of a tire is 26 inches. The radius is half the diameter. 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

SOLUTION: Therefore, the circumference of the bicycle tire is The diameter of the pizza is 16 inches. about 81.68 inches. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with . the given circumference. Round to the nearest So, the radius of the pizza is 8 inches. hundredth. The circumference C of a circle of diameter d is 24. C = 18 in.

given by SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the Therefore, the circumference of the pizza is about diameter. Then find the radius. 50.27 inches.

ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: 5.73 in.; 2.86 in.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is 25. C = 124 ft given by SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the Therefore, the circumference of the bicycle tire is diameter. Then find the radius. about 81.68 inches.

ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is Therefore, the diameter is about 39.47 feet and the given by radius is about 19.74 feet. Here, C = 18 in. Use the formula to find the diameter. Then find the radius. ANSWER: 39.47 ft; 19.74 ft

26. C = 375.3 cm SOLUTION:

The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: 5.73 in.; 2.86 in.

25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is Therefore, the diameter is about 119.46 centimeters given by and the radius is about 59.73 centimeters. Here, C = 124 ft. Use the formula to find the diameter. Then find the radius. ANSWER: 119.46 cm; 59.73 cm

27. C = 2608.25 m SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet.

ANSWER: 39.47 ft; 19.74 ft

26. C = 375.3 cm SOLUTION: Therefore, the diameter is about 830.23 meters and The circumference C of a circle with diameter d is the radius is about 415.12 meters. given by Here, C = 375.3 cm. Use the formula to find the ANSWER: diameter. Then find the radius. 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon.

Therefore, the diameter is about 119.46 centimeters 28. and the radius is about 59.73 centimeters. SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of 119.46 cm; 59.73 cm the circle. Use the Pythagorean Theorem to find the diameter. 27. C = 2608.25 m SOLUTION: The circumference C of a circle with diameter d is given by The diameter of the circle is 17 cm. The Here, C = 2608.25 meters. Use the formula to find

the diameter. Then find the radius. circumference C of a circle is given by

Therefore, the circumference of the circle is 17 centimeters.

ANSWER:

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m 29. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given SOLUTION: inscribed or The hypotenuse of the right triangle is a diameter of circumscribed polygon. the circle. Use the Pythagorean Theorem to find the diameter.

28. The diameter of the circle is 12 feet. The circumference C of a circle is given by SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. Therefore, the circumference of the circle is 12 feet.

ANSWER:

The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the circumference of the circle is 17 30. centimeters. SOLUTION: ANSWER: Each diagonal of the inscribed rectangle will pass

through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is inches. The 29. circumference C of a circle is given by SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the circumference of the circle is inches.

ANSWER: The diameter of the circle is 12 feet. The

circumference C of a circle is given by

Therefore, the circumference of the circle is 12 feet. 31. ANSWER: SOLUTION:

Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

30. SOLUTION: Each diagonal of the inscribed rectangle will pass The diameter of the circle is 10 inches. The through the origin, so it is a diameter of the circle. circumference C of a circle is given by Use the Pythagorean Theorem to find the diameter of the circle.

Therefore, the circumference of the circle is 10 inches.

ANSWER: The diameter of the circle is inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 32. inches. SOLUTION: ANSWER: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by

31. Therefore, the circumference of the circle is 25 millimeters. SOLUTION: Each diagonal of the inscribed rectangle will pass ANSWER: through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is 10 inches. The circumference C of a circle is given by 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d Therefore, the circumference of the circle is 10 = 14 yards. inches. The circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 14 yards.

ANSWER:

32. SOLUTION: 34. DISC GOLF Disc golf is similar to regular golf, A diameter perpendicular to a side of the square and except that a flying disc is used instead of a ball and the length of each side will measure the same. So, d clubs. For professional competitions, the maximum = 25 millimeters. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight The circumference C of a circle is given by for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION:

Therefore, the circumference of the circle is 25 The circumference C of a circle with diameter d is

millimeters. given by Here, C = 66.92 cm. Use the formula to find the ANSWER: diameter.

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or 33. about 176.8 grams. SOLUTION: ANSWER: A diameter perpendicular to a side of the square and 176.8 g the length of each side will measure the same. So, d = 14 yards. 35. PATIOS Mr. Martinez is going to build the patio The circumference C of a circle is given by shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 Therefore, the circumference of the circle is 14 feet, what should the radius of the circle be to the yards. nearest foot?

ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in SOLUTION: centimeters. What is the maximum allowable weight a. The radius of the patio is 3 + 2 = 5 ft. The for a disc with circumference 66.92 centimeters? circumference C of a circle with radius r is given by Round to the nearest tenth.

SOLUTION: Use the formula to find the circumference. The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter. Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or So, the radius of the inner circle is about 4 feet. about 176.8 grams. ANSWER: 10-1ANSWER: Circles and Circumference a. 31.42 ft 176.8 g b. 4 ft

35. PATIOS Mr. Martinez is going to build the patio The radius, diameter, or circumference of a shown. circle is given. Find each missing measure to a. What is the patio’s approximate circumference? the nearest hundredth. b. If Mr. Martinez changes the plans so that the 36. inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by

Use the formula to find the circumference. Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in. Therefore, the circumference is about 31.42 feet. b. Here, C = 25 ft. Use the formula to find the inner radius of the inner circle. 37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth. So, the diameter is 22.80 feet and the circumference is about 71.63 feet. 36. ANSWER: SOLUTION: 22.80 ft; 71.63 ft The radius is half the diameter and the circumference C of a circle with diameter d is given 38. by SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 35x cm. Use the formula to find the eSolutions Manual - Powered by Cognero diameter.Then find the radius. Page 9

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

37. Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters.

SOLUTION: ANSWER: The diameter is twice the radius and the 11.14x cm; 5.57x cm circumference C of a circle with diameter d is given by 39.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft

38. So, the diameter is 0.25x units and the circumference SOLUTION: is about 0.79x units. The circumference C of a circle with diameter d is given by ANSWER: Here, C = 35x cm. Use the formula to find the 0.25x; 0.79x diameter.Then find the radius. Determine whether the circles in the figures below appear to be congruent, concentric, or neither. 40. Refer to the photo on page 703. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: concentric Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. 41. Refer to the photo on page 703.

ANSWER: SOLUTION: 11.14x cm; 5.57x cm The circles neither have the same radius nor are they concentric.

39. ANSWER: neither SOLUTION: Refer to the photo on page 703. The diameter is twice the radius and the 42. circumference C of a circle with diameter d is given SOLUTION: by The circles appear to have the same radius. So, they appear to be congruent.

ANSWER: congruent

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the So, the diameter is 0.25x units and the circumference ring? Round to the nearest tenth. is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or neither. 40. Refer to the photo on page 703. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: SOLUTION: concentric The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a circle with diameter d is given by 41. Refer to the photo on page 703. So, the circumference is SOLUTION: Therefore, someone have to The circles neither have the same radius nor are they walk about 471.2 feet to go completely around the concentric. ring.

ANSWER: ANSWER: neither 471.2 ft

42. Refer to the photo on page 703. 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a SOLUTION: circumference of 68 feet. The outer edge of the path The circles appear to have the same radius. So, they is going to be 4 feet from the pond all the way appear to be congruent. around. What is the approximate circumference of the path? Round to the nearest hundredth. ANSWER: congruent SOLUTION: The circumference C of a circle with radius r is 43. HISTORY The Indian Shell Ring on Hilton Head given by Here, Cpond = 68 ft. Use the Island approximates a circle. If each unit on the formula to find the radius of the pond. coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

Therefore, the circumference of the path is about 93.13 feet. SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, ANSWER: the diameter is 150 ft. The circumference C of a 93.13 ft circle with diameter d is given by MULTIPLE REPRESENTATIONS So, the circumference is 45. In this problem, you will explore changing dimensions in Therefore, someone have to circles. walk about 471.2 feet to go completely around the a. GEOMETRIC Use a compass to draw three ring. circles in which the scale factor from each circle to the next larger circle is 1 : 2. ANSWER: b. TABULAR Calculate the radius (to the nearest 471.2 ft tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. 44. CCSS MODELING A brick path is being installed c. VERBAL Explain why these three circles are around a circular pond. The pond has a geometrically similar. circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way d. VERBAL Make a conjecture about the ratio around. What is the approximate circumference of between the circumferences of two circles when the the path? Round to the nearest hundredth. ratio between their radii is 2. e. ANALYTICAL The scale factor from to SOLUTION: The circumference C of a circle with radius r is is . Write an equation relating the given by Here, Cpond = 68 ft. Use the circumference (CA) of to the circumference formula to find the radius of the pond. (CB) of . f. NUMERICAL If the scale factor from to is , and the circumference of is 12 inches, what is the circumference of ?

So, the radius of the pond is about 10.82 feet and the SOLUTION: radius to the outer edge of the path will be 10.82 + 4 a. Sample answer: = 14.82 ft. Use the radius to find the circumference of the brick path.

Therefore, the circumference of the path is about 93.13 feet. b. Sample answer. ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to c. They all have the same shape–circular. the next larger circle is 1 : 2. d. The ratio of their circumferences is also 2. b. TABULAR Calculate the radius (to the nearest e. The circumference is directly proportional to radii. tenth) and circumference (to the nearest hundredth) So, if the radii are in the ratio b : a, so will the of each circle. Record your results in a table. circumference. Therefore, . c. VERBAL Explain why these three circles are geometrically similar. f. Use the formula from part e. Here, the d. VERBAL Make a conjecture about the ratio circumference of is 12 inches and . between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to

is . Write an equation relating the Therefore, the circumference of is 4 inches. circumference (CA) of to the circumference ANSWER: (CB) of . f. NUMERICAL If the scale factor from to a. Sample answer: is , and the circumference of is 12 inches, what is the circumference of ? SOLUTION: a. Sample answer: b. Sample answer.

b. Sample answer. c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

c. They all have the same shape–circular. 46. BUFFON’S NEEDLE Measure the length of a d. The ratio of their circumferences is also 2. needle (or toothpick) in centimeters. Next, draw a set e. The circumference is directly proportional to radii. of horizontal lines that are centimeters apart on a So, if the radii are in the ratio b : a, so will the sheet of plain white paper.

circumference. Therefore, . f. Use the formula from part e. Here, the circumference of is 12 inches and .

a. Drop the needle onto the paper. When the needle Therefore, the circumference of is 4 inches. lands, record whether it touches one of the lines as a ANSWER: hit. Record the number of hits after 25, 50, and 100 drops. a. Sample answer:

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

b. Sample answer. c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. c. They all have the same shape—circular. d. T he ratio of their circumferences is also 2. ANSWER:

e. a. See students’ work. b. See students’ work. f. 4 in. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. BUFFON’S NEEDLE 46. Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a 47. M APS The concentric circles on the map below sheet of plain white paper. show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops. a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by b. Calculate the ratio of two times the total number how much does the circumference increase? of drops to the number of hits after 25, 50, and 100 SOLUTION: drops. a. The radius of the outermost circle is 30 miles and c. How are the values you found in part b related to that of the center circle is 5 miles. Find the π? circumference of each circle. SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

The difference of the circumferences of the ANSWER: outermost circle and the center circle is 60π - 10π = a. See students’ work. 50π or about 157.1 miles. b. See students’ work. b. Let the radius of one circle be x miles and the c. Sample answer: The values are approaching 3.14, radius of the next larger circle be x + 5 miles. Find which is approximately equal to π. the circumference of each circle.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

The difference of the two circumferences is (2xπ + 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER:

a. How much greater is the circumference of the a. 157.1 mi outermost circle than the circumference of the center b. by about 31.4 mi circle? WRITING IN MATH How can we describe the b. As the radii of the circles increases by 5 miles, by 48. how much does the circumference increase? relationships that exist between circles and lines? SOLUTION: SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the Sample answer: A line and a circle may intersect in circumference of each circle. one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant.

A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the The difference of the circumferences of the center and on the circle can be described as a outermost circle and the center circle is 60π - 10π = radius. 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the ANSWER: radius of the next larger circle be x + 5 miles. Find Sample answer: A line that intersects a circle in one the circumference of each circle. point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius. The difference of the two circumferences is (2xπ + 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER: 49. REASONING In the figure, a circle with radius r is a. 157.1 mi inscribed in a regular polygon and circumscribed b. by about 31.4 mi about another. a. What are the perimeters of the circumscribed and 48. WRITING IN MATH How can we describe the inscribed polygons in terms of r? Explain. relationships that exist between b. Is the circumference C of the circle greater or circles and lines? less than the perimeter of the circumscribed polygon? SOLUTION: the inscribed polygon? Write a compound inequality comparing C to these perimeters. Sample answer: A line and a circle may intersect in c. Rewrite the inequality from part b in terms of the one point, two points, or may not intersect at all. A diameter d of the circle and interpret its meaning. line that intersects a circle in one point can be d. As the number of sides of both the circumscribed described as a tangent. A line that intersects a circle and inscribed polygons increase, what will happen to in exactly two points can be described as a secant. the upper and lower limits of the inequality from part A line segment with endpoints on a circle can be c, and what does this imply? described as a chord. If the chord passes through SOLUTION: the center of the circle, it can be described as a a. The circumscribed polygon is a square with sides diameter. A line segment with endpoints at the of length 2r. center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one The inscribed polygon is a hexagon with sides of point can be described as a tangent. A line that length r. (Each triangle formed using a central angle intersects a circle in exactly two points can be will be equilateral.) described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

So, the perimeters are 8r and 6r. b. 49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. The circumference is less than the perimeter of the a. What are the perimeters of the circumscribed and circumscribed polygon and greater than the inscribed polygons in terms of r? Explain. perimeter of the inscribed polygon. b. Is the circumference C of the circle greater or 8r < C < 6r less than the perimeter of the circumscribed polygon? c. Since the diameter is twice the radius, 8r = 4(2r) the inscribed polygon? Write a compound inequality or 4d and 6r = 3(2r) or 3d. comparing C to these perimeters. The inequality is then 4d < C < 3d. Therefore, the c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. circumference of the circle is between 3 and 4 times d. As the number of sides of both the circumscribed its diameter. and inscribed polygons increase, what will happen to d. These limits will approach a value of πd, implying the upper and lower limits of the inequality from part that C = πd. c, and what does this imply?

SOLUTION: ANSWER: a. The circumscribed polygon is a square with sides a. 8r and 6r ; Twice the radius of the circle, 2r is of length 2r. the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the The inscribed polygon is a hexagon with sides of hexagon is 6(r) or 6r. length r. (Each triangle formed using a central angle b. less; greater; 6r < C < 8r will be equilateral.) c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the SOLUTION: perimeter of the inscribed polygon. The radius of is 4x units, the radius of is x 8r < C < 6r units, and the radius of is 2x units. Find the c. Since the diameter is twice the radius, 8r = 4(2r) circumference of each circle. or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The The sum of the circumferences is 56π units. regular hexagon is made up of six equilateral Substitute the three circumferences and solve for x. triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is So, the radius of is 4 units, the radius of is between 3 and 4 times its diameter. 2(4) or 8 units, and the radius of is 4(4) or 16 d. These limits will approach a value of πd, implying units. that C = πd. The measure of KJ is equal to the sum of the radii of and . 50. CHALLENGE The sum of the circumferences of Therefore, KJ = 16 + 8 or 24 units. circles H, J, and K shown at the right is units. Find KJ. ANSWER: 24 units

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Explain. SOLUTION:

A radius is a segment drawn between the center of SOLUTION: the circle and a point on the circle. A segment drawn The radius of is 4x units, the radius of is x from the center to a point inside the circle will always units, and the radius of is 2x units. Find the have a length less than the radius of the circle. circumference of each circle. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant The sum of the circumferences is 56π units. from a given point. For any two circles and Substitute the three circumferences and solve for x. , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make So, the radius of is 4 units, the radius of is up are from center B. 2(4) or 8 units, and the radius of is 4(4) or 16 For some value k, r = units. 2 The measure of KJ is equal to the sum of the radii kr1. of and . Therefore, KJ = 16 + 8 or 24 units.

ANSWER: 24 units

51. REASONING Is the distance from the center of a Therefore, is mapped onto . Since there circle to a point in the interior of a circle sometimes, exists a rigid motion followed by a scaling that maps always, or never less than the radius of the circle? onto , the circles are Explain. similar. Thus, all circles are similar.

SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn ANSWER: from the center to a point inside the circle will always A circle is a locus of points in a plane equidistant have a length less than the radius of the circle. from a given point. For any two circles and ANSWER: , there exists a translation that maps center A Always; a radius is a segment drawn between the onto center B, moving so that it is concentric center of the circle and a point on the circle. A with . There also exists a dilation with scale segment drawn from the center to a point inside the factor k such that each point that makes up is circle will always have a length less than the radius moved to be the same distance from center A as the of the circle. points that make up are from center B. Therefore, is mapped onto . Since there 52. PROOF Use the locus definition of a circle and exists a rigid motion followed by a scaling that maps dilations to prove that all circles are similar. onto , the circles are similar. Thus, all SOLUTION: circles are similar. A circle is a locus of points in a plane equidistant from a given point. For any two circles and 53. CHALLENGE In the figure, is inscribed in , there exists a translation that equilateral triangle LMN. What is the circumference maps center A onto center B, moving so that it of ? is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 =

kr1. SOLUTION: Draw a perpendicular to the side Also join

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects ANSWER: Use the definition of A circle is a locus of points in a plane equidistant tangent of an angle to find PR, which is the radius of from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there Therefore, the radius of is exists a rigid motion followed by a scaling that maps The circumference C of a circle with radius r is onto , the circles are similar. Thus, all given by circles are similar.

53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference of ?

The circumference of is inches.

ANSWER:

in.

SOLUTION: 54. WRITING IN MATH Research and write about Draw a perpendicular to the side Also join the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the Since is equilateral triangle, MN = 8 inches circumference of ? Round to the nearest tenth. and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

Therefore, the radius of is The circumference C of a circle with diameter d is The circumference C of a circle with radius r is given by given by Use the formula to find the circumference.

The circumference of is inches.

ANSWER: ANSWER: 40.8 in. 56. What is the radius of a table with a circumference of 10 feet? 54. WRITING IN MATH Research and write about A 1.6 ft the history of pi and its importance to the study of B 2.5 ft geometry. C 3.2 ft SOLUTION: D 5 ft See students’ work. SOLUTION: ANSWER: The circumference C of a circle with radius r is See students’ work. given by Here, C = 10 ft. Use the formula to find the radius. 55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: SOLUTION: A Use the Pythagorean Theorem to find the diameter of the circle. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use The circumference C of a circle with diameter d is up to 50 feet of fence, what radius can he use for the garden? given by Use the formula to find the circumference. F 10 G 9 H 8 J 7

SOLUTION: The circumference C of a circle with radius r is given by Because 50 feet of fence can be used to border the ANSWER: garden, Cmax = 50. Use the circumference formula to 40.8 find the maximum radius. 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only SOLUTION: correct choice is J. The circumference C of a circle with radius r is given by ANSWER: Here, C = 10 ft. Use the formula to find the radius. J 58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units

The radius of the circle is about 1.6 feet, so the B 0.5 units correct choice is A. C 2 units D 4 units ANSWER: E 16 units A SOLUTION: 57. ALGEBRA Bill is planning a circular vegetable The area of a circle of radius r is given by the garden with a fence around the border. If he can use formula Here A = square units. Use the up to 50 feet of fence, what radius can he use for the formula to find the radius. garden? F 10 G 9 H 8 J 7 SOLUTION: Therefore, the correct choice is B. The circumference C of a circle with radius r is ANSWER: given by B Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to Copy each figure and point B. Then use a ruler find the maximum radius. to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J. SOLUTION: ANSWER: Draw a ray from point B through a vertex of the J hexagon. Use the ruler to measure the distance from 58. SAT/ACT What is the radius of a circle with an B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays area of square units? through the other five vertices. Connect the six points A 0.4 units on the rays to construct the dilated image of the

B 0.5 units hexagon.

C 2 units D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

ANSWER:

Therefore, the correct choice is B.

ANSWER: B 60. Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other two vertices. Connect the three Draw a ray from point B through a vertex of the points on the rays to construct the dilated image of hexagon. Use the ruler to measure the distance from the triangle. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon. ANSWER:

61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

60.

SOLUTION:

Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. 62.

ANSWER: SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from 61. B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

ANSWER:

62. State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. Refer to the image on page 705. 63. SOLUTION: SOLUTION: Draw a ray from point B through a vertex of the The figure cannot be mapped onto itself by any triangle. Use the ruler to measure the distance from rotation. So, it does not have a rotational symmetry. B to the vertex. Mark a point on the ray that is 3 ANSWER: times that distance from B. Repeat the process for rays through the other two vertices. Connect the No three points on the rays to construct the dilated image of the triangle. 64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

ANSWER:

65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: State whether each figure has rotational No symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state 66. Refer to the image on page 705. the order and magnitude of symmetry. SOLUTION: Refer to the image on page 705. 63. The given figure cannot be mapped onto itself by any SOLUTION: rotation. So, it does not have a rotational symmetry. The figure cannot be mapped onto itself by any ANSWER: rotation. So, it does not have a rotational symmetry. No ANSWER: Determine the truth value of the following No statement for each set of conditions. Explain your reasoning. 64. Refer to the image on page 705. If you are over 18 years old, then you vote in all elections. SOLUTION: 67. You are 19 years old and you vote. The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the SOLUTION: magnitude of symmetry is 90º and order of symmetry True; sample answer: Since the hypothesis is true is 4. and the conclusion is true, then the statement is true for the conditions. ANSWER: Yes; 4, 90 ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false 65. for the conditions. SOLUTION: ANSWER: The given figure cannot be mapped onto itself by any False; sample answer: Since the hypothesis is true rotation. So, it does not have a rotational symmetry. and the conclusion is false, then the statement is false for the conditions. ANSWER: No Find x.

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. 69. ANSWER: No SOLUTION: The sum of the measures of angles around a point is Determine the truth value of the following 360º. statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. ANSWER: SOLUTION: 90 True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true 70. for the conditions. SOLUTION: 68. You are 21 years old and do not vote. The angles with the measures xº and 2xº form a straight line. So, the sum is 180. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. ANSWER: ANSWER: 60 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

71. SOLUTION: The angles with the measures 6xº and 3xº form a 69. straight line. So, the sum is 180. SOLUTION: The sum of the measures of angles around a point is 360º.

ANSWER: 20

ANSWER: 90

72. SOLUTION: The sum of the three angles is 180. 70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180. ANSWER: 45

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? For Exercises 1–4, refer to . SOLUTION: Here, is a radius and the diameter is twice the radius.

Therefore, the diameter is 26 ft.

1. Name the circle. ANSWER: SOLUTION: 26 ft The center of the circle is N. So, the circle is The diameters of , , and are 8 ANSWER: inches, 18 inches, and 11 inches, respectively. Find each measure.

2. Identify each. a. a chord b. a diameter c. a radius

SOLUTION: 5. FG a. A chord is a segment with endpoints on the circle. SOLUTION: So, here are chords. b. A diameter of a circle is a chord that passes Since the diameter of B is 18 inches and A is 8

through the center. Here, is a diameter. inches, AG = 18 and AF = (8) or 4. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: Therefore, FG = 14 in. a. ANSWER: b. 14 in. c.

3. If CN = 8 centimeters, find DN. 6. FB SOLUTION: SOLUTION: Here, are radii of the same circle. So, Since the diameter of B is 18 inches and A is 4 they are equal in length. Therefore, DN = 8 cm. inches, AB = (18) or 9 and AF = (8) or 4.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? Therefore, FB = 5 inches. SOLUTION: ANSWER: Here, is a radius and the diameter is twice the 5 in. radius. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the Therefore, the diameter is 26 ft. radius and circumference of the ride? Round to the nearest hundredth, if necessary. ANSWER: 26 ft SOLUTION: The radius is half the diameter. So, the radius of the The diameters of , , and are 8 circular ride is 22 feet. inches, 18 inches, and 11 inches, respectively. Find each measure.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER:

5. FG 22 ft; 138.23 ft SOLUTION: 8. CCSS MODELING The circumference of the Since the diameter of B is 18 inches and A is 8 circular swimming pool shown is about 56.5 feet. inches, AG = 18 and AF = (8) or 4. What are the diameter and radius of the pool? Round to the nearest hundredth.

Therefore, FG = 14 in.

ANSWER: 14 in. SOLUTION: 6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches.

ANSWER: 5 in. The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the ANSWER: radius and circumference of the ride? Round to the 17.98 ft; 8.99 ft nearest hundredth, if necessary. 9. SHORT RESPONSE The right triangle shown is SOLUTION: inscribed in . Find the exact circumference of The radius is half the diameter. So, the radius of the circular ride is 22 feet. .

Therefore, the circumference of the ride is about 138.23 feet. SOLUTION: The diameter of the circle is the hypotenuse of the ANSWER: right triangle. 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π centimeters.

SOLUTION: ANSWER:

For Exercises 10–13, refer to .

10. Name the center of the circle. SOLUTION: The diameter of the pool is about 17.98 feet and the R radius of the pool is about 8.99 feet. ANSWER: ANSWER: R 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is 11. Identify a chord that is also a diameter. inscribed in . Find the exact circumference of SOLUTION: . Two chords are shown: and . goes through the center, R, so is a diameter.

ANSWER:

SOLUTION:

The diameter of the circle is the hypotenuse of the 12. Is a radius? Explain. right triangle. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord.

ANSWER: The diameter of the circle is 4 centimeters. No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION:

The circumference of the circle is 4π Here, is a diameter and is a radius. centimeters.

ANSWER:

Therefore, RT = 8.1 cm. For Exercises 10–13, refer to . ANSWER: 8.1 cm

For Exercises 14–17, refer to .

10. Name the center of the circle. SOLUTION: R

ANSWER: 14. Identify a chord that is not a diameter. R SOLUTION: The chords do not pass through the 11. Identify a chord that is also a diameter. center. So, they are not diameters. SOLUTION: ANSWER: Two chords are shown: and . goes through the center, R, so is a diameter. 15. If CF = 14 inches, what is the diameter of the circle? ANSWER: SOLUTION: Here, is a radius and the diameter is twice the radius. . 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center Therefore, the diameter of the circle is 28 inches. and on the circle. But has both the end points on the circle, so it is a chord. ANSWER: ANSWER: 28 in. No; it is a chord. 16. Is ? Explain. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: SOLUTION: All radii of a circle are congruent. Since Here, is a diameter and is a radius. are both radii of , . ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF?

Therefore, RT = 8.1 cm. SOLUTION: ANSWER: Here, is a diameter and is a radius. 8.1 cm

For Exercises 14–17, refer to .

Therefore, EF = 3.7 cm.

ANSWER: 3.7 cm

14. Identify a chord that is not a diameter. Circle J has a radius of 10 units, has a SOLUTION: radius of 8 units, and BC = 5.4 units. Find each measure. The chords do not pass through the center. So, they are not diameters.

ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? 18. CK SOLUTION: SOLUTION: Here, is a radius and the diameter is twice the radius. . Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units. Therefore, the diameter of the circle is 28 inches. ANSWER: ANSWER: 2.6 28 in. 19. AB 16. Is ? Explain. SOLUTION: SOLUTION: All radii of a circle are congruent. Since Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. are both radii of , .

ANSWER:

Yes; they are both radii of . Therefore, AB = 14.6 units.

17. If DA = 7.4 centimeters, what is EF? ANSWER: SOLUTION: 14.6 Here, is a diameter and is a radius. 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Therefore, EF = 3.7 cm.

ANSWER: Since circle J has a radius of 10 units, JC = 10. 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each Therefore, JK = 12.6 units. measure. ANSWER: 12.6 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J 18. CK is 10 units, so KD = 8 and AC = 2(10) or 20. Before SOLUTION: finding AD, we need to find CK. Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units.

ANSWER: Therefore, AD = 30.6 units. 2.6 ANSWER: 30.6 19. AB SOLUTION: 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if Since is a diameter of circle J and the radius is necessary. 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units.

ANSWER: 14.6

20. JK SOLUTION: SOLUTION: The diameter of the pizza is 16 inches. First find CK. Since circle K has a radius of 8 units, BK = 8.

Since circle J has a radius of 10 units, JC = 10. . So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, JK = 12.6 units.

ANSWER: 12.6 Therefore, the circumference of the pizza is about 50.27 inches. 21. AD ANSWER: SOLUTION: 8 in.; 50.27 in. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J 23. BICYCLES A bicycle has tires with a diameter of is 10 units, so KD = 8 and AC = 2(10) or 20. Before 26 inches. Find the radius and circumference of a

finding AD, we need to find CK. tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter.

Therefore, AD = 30.6 units.

ANSWER:

30.6 So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is PIZZA 22. Find the radius and circumference of the given by pizza shown. Round to the nearest hundredth, if necessary.

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with SOLUTION: the given circumference. Round to the nearest hundredth. The diameter of the pizza is 16 inches. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is given by . Here, C = 18 in. Use the formula to find the So, the radius of the pizza is 8 inches. diameter. Then find the radius. The circumference C of a circle of diameter d is given by

Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in. Therefore, the diameter is about 5.73 inches and the 23. BICYCLES A bicycle has tires with a diameter of radius is about 2.86 inches. 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. ANSWER: SOLUTION: 5.73 in.; 2.86 in. The diameter of a tire is 26 inches. The radius is half the diameter. 25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the So, the radius of the tires is 13 inches. diameter. Then find the radius. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. Therefore, the diameter is about 39.47 feet and the Find the diameter and radius of a circle by with radius is about 19.74 feet. the given circumference. Round to the nearest hundredth. ANSWER: 24. C = 18 in. 39.47 ft; 19.74 ft SOLUTION: 26. C = 375.3 cm The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 18 in. Use the formula to find the The circumference C of a circle with diameter d is diameter. Then find the radius. given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. Therefore, the diameter is about 119.46 centimeters ANSWER: and the radius is about 59.73 centimeters. 5.73 in.; 2.86 in. ANSWER: 25. C = 124 ft 119.46 cm; 59.73 cm SOLUTION: 27. C = 2608.25 m The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 124 ft. Use the formula to find the diameter. Then find the radius. given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. ANSWER: 39.47 ft; 19.74 ft ANSWER: 830.23 m; 415.12 m 26. C = 375.3 cm CCSS SENSE-MAKING Find the exact SOLUTION: circumference of each circle by using the given The circumference C of a circle with diameter d is inscribed or given by circumscribed polygon. Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

28.

SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: The diameter of the circle is 17 cm. The 119.46 cm; 59.73 cm circumference C of a circle is given by 27. C = 2608.25 m SOLUTION:

The circumference C of a circle with diameter d is Therefore, the circumference of the circle is 17

given by centimeters. Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. ANSWER:

29.

Therefore, the diameter is about 830.23 meters and SOLUTION: the radius is about 415.12 meters. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the ANSWER: diameter. 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or The diameter of the circle is 12 feet. The circumscribed polygon. circumference C of a circle is given by

Therefore, the circumference of the circle is 12 28. feet. SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

30. The diameter of the circle is 17 cm. The SOLUTION: circumference C of a circle is given by Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. Therefore, the circumference of the circle is 17 centimeters.

ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 29. inches. SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 12 feet. The circumference C of a circle is given by 31. SOLUTION: Each diagonal of the inscribed rectangle will pass

through the origin, so it is a diameter of the circle. Therefore, the circumference of the circle is 12 Use the Pythagorean Theorem to find the diameter

feet. of the circle.

ANSWER:

The diameter of the circle is 10 inches. The circumference C of a circle is given by

30.

SOLUTION: Therefore, the circumference of the circle is 10 Each diagonal of the inscribed rectangle will pass inches. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter ANSWER: of the circle.

The diameter of the circle is inches. The 32. circumference C of a circle is given by SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by Therefore, the circumference of the circle is inches.

ANSWER: Therefore, the circumference of the circle is 25 millimeters.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The diameter of the circle is 10 inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 14 Therefore, the circumference of the circle is 10 yards. inches. ANSWER: ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters?

32. Round to the nearest tenth. SOLUTION: SOLUTION: A diameter perpendicular to a side of the square and The circumference C of a circle with diameter d is the length of each side will measure the same. So, d given by = 25 millimeters. Here, C = 66.92 cm. Use the formula to find the

The circumference C of a circle is given by diameter.

Therefore, the circumference of the circle is 25 millimeters. The diameter of the disc is about 21.3 centimeters ANSWER: and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. 33. a. What is the patio’s approximate circumference? SOLUTION: b. If Mr. Martinez changes the plans so that the A diameter perpendicular to a side of the square and inner circle has a circumference of approximately 25 the length of each side will measure the same. So, d feet, what should the radius of the circle be to the = 14 yards. nearest foot? The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 yards.

ANSWER: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and Use the formula to find the circumference. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Therefore, the circumference is about 31.42 feet. Round to the nearest tenth. b. Here, Cinner = 25 ft. Use the formula to find the SOLUTION: radius of the inner circle. The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter.

So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft The diameter of the disc is about 21.3 centimeters b. 4 ft and the maximum weight allowed is 8.3 times the diameter in centimeters. The radius, diameter, or circumference of a Therefore, the maximum weight is 8.3 × 21.3 or circle is given. Find each missing measure to about 176.8 grams. the nearest hundredth.

ANSWER: 36. 176.8 g SOLUTION: 35. PATIOS Mr. Martinez is going to build the patio The radius is half the diameter and the shown. circumference C of a circle with diameter d is given a. What is the patio’s approximate circumference? by b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. SOLUTION: ANSWER: a. The radius of the patio is 3 + 2 = 5 ft. The 4.25 in.; 26.70 in. circumference C of a circle with radius r is given by

Use the formula to find the circumference. 37.

SOLUTION: The diameter is twice the radius and the Therefore, the circumference is about 31.42 feet. circumference C of a circle with diameter d is given by b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the radius of the inner circle is about 4 feet. So, the diameter is 22.80 feet and the circumference ANSWER: is about 71.63 feet. a. 31.42 ft b. 4 ft ANSWER: 22.80 ft; 71.63 ft The radius, diameter, or circumference of a circle is given. Find each missing measure to 38. the nearest hundredth. SOLUTION: 36. The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 35x cm. Use the formula to find the The radius is half the diameter and the diameter.Then find the radius. circumference C of a circle with diameter d is given by

Therefore, the radius is 4.25 inches and the Therefore, the diameter is about 11.14x centimeters circumference is about 26.70 inches. and the radius is about 5.57x centimeters.

ANSWER: ANSWER: 4.25 in.; 26.70 in. 11.14x cm; 5.57x cm

39. 37. SOLUTION: SOLUTION: The diameter is twice the radius and the The diameter is twice the radius and the circumference C of a circle with diameter d is given circumference C of a circle with diameter d is given by by

So, the diameter is 22.80 feet and the circumference So, the diameter is 0.25x units and the circumference is about 71.63 feet. is about 0.79x units.

10-1ANSWER: Circles and Circumference ANSWER: 22.80 ft; 71.63 ft 0.25x; 0.79x

38. Determine whether the circles in the figures below appear to be congruent, concentric, or SOLUTION: neither. The circumference C of a circle with diameter d is 40. Refer to the photo on page 703. given by SOLUTION: Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. neither

ANSWER: 42. Refer to the photo on page 703. 11.14x cm; 5.57x cm SOLUTION: The circles appear to have the same radius. So, they

39. appear to be congruent. ANSWER: SOLUTION: congruent The diameter is twice the radius and the circumference C of a circle with diameter d is given 43. HISTORY The Indian Shell Ring on Hilton Head by Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or SOLUTION: neither. The radius of the circle is 3 units, that is 75 feet. So, 40. Refer to the photo on page 703. the diameter is 150 ft. The circumference C of a SOLUTION: circle with diameter d is given by The circles have the same center and they are in the So, the circumference is eSolutionssameManual plane.- Powered So, theyby areCognero concentric circles. Therefore, someone have to Page 10 walk about 471.2 feet to go completely around the ANSWER: ring. concentric ANSWER: 41. Refer to the photo on page 703. 471.2 ft SOLUTION: 44. CCSS MODELING A brick path is being installed The circles neither have the same radius nor are they around a circular pond. The pond has a concentric. circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way ANSWER: around. What is the approximate circumference of neither the path? Round to the nearest hundredth. SOLUTION: 42. Refer to the photo on page 703. The circumference C of a circle with radius r is SOLUTION: given by Here, Cpond = 68 ft. Use the The circles appear to have the same radius. So, they formula to find the radius of the pond. appear to be congruent.

ANSWER: congruent

HISTORY 43. The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the So, the radius of the pond is about 10.82 feet and the coordinate grid represents 25 feet, how far would radius to the outer edge of the path will be 10.82 + 4 someone have to walk to go completely around the = 14.82 ft. Use the radius to find the circumference ring? Round to the nearest tenth. of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. SOLUTION: a. GEOMETRIC Use a compass to draw three The radius of the circle is 3 units, that is 75 feet. So, circles in which the scale factor from each circle to the diameter is 150 ft. The circumference C of a the next larger circle is 1 : 2. circle with diameter d is given by b. TABULAR Calculate the radius (to the nearest So, the circumference is tenth) and circumference (to the nearest hundredth) Therefore, someone have to of each circle. Record your results in a table. c. VERBAL Explain why these three circles are

walk about 471.2 feet to go completely around the geometrically similar. ring. d. VERBAL Make a conjecture about the ratio ANSWER: between the circumferences of two circles when the

471.2 ft ratio between their radii is 2. e. ANALYTICAL The scale factor from to 44. CCSS MODELING A brick path is being installed is . Write an equation relating the around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path circumference (C ) of to the circumference is going to be 4 feet from the pond all the way A around. What is the approximate circumference of (CB) of . the path? Round to the nearest hundredth. f. NUMERICAL If the scale factor from to SOLUTION: is , and the circumference of is 12 The circumference C of a circle with radius r is inches, what is the circumference of ? given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. SOLUTION: a. Sample answer:

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference b. Sample answer. of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: c. They all have the same shape–circular. 93.13 ft d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. 45. MULTIPLE REPRESENTATIONS In this So, if the radii are in the ratio b : a, so will the problem, you will explore changing dimensions in circles. circumference. Therefore, . a. GEOMETRIC Use a compass to draw three f. Use the formula from part e. Here, the circles in which the scale factor from each circle to the next larger circle is 1 : 2. circumference of is 12 inches and . b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. Therefore, the circumference of is 4 inches. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ANSWER: ratio between their radii is 2. a. Sample answer: e. ANALYTICAL The scale factor from to is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to b. Sample answer. is , and the circumference of is 12 inches, what is the circumference of ? SOLUTION: a. Sample answer:

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e.

f. 4 in. b. Sample answer. 46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . a. Drop the needle onto the paper. When the needle f. e. Use the formula from part Here, the lands, record whether it touches one of the lines as a circumference of is 12 inches and . hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number Therefore, the circumference of is 4 inches. of drops to the number of hits after 25, 50, and 100 ANSWER: drops.

a. Sample answer: c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, b. Sample answer. which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. c. They all have the same shape—circular. d. The ratio of their circumferences is also 2. 47. M APS The concentric circles on the map below e. show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by a. Drop the needle onto the paper. When the needle how much does the circumference increase? lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 SOLUTION: drops. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100

drops.

c. How are the values you found in part b related to π? SOLUTION: The difference of the circumferences of the a. See students’ work. outermost circle and the center circle is 60π - 10π = b. See students’ work. 50π or about 157.1 miles. c. Sample answer: The values are approaching 3.14, b. Let the radius of one circle be x miles and the which is approximately equal to π. radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

The difference of the two circumferences is (2xπ + 47. M APS The concentric circles on the map below 10π) - (2xπ) = 10π or about 31.4 miles. show the areas that are 5, 10, 15, 20, 25, and 30 Therefore, as the radius of each circle increases by 5 miles from downtown Phoenix. miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines?

SOLUTION: a. How much greater is the circumference of the outermost circle than the circumference of the center Sample answer: A line and a circle may intersect in circle? one point, two points, or may not intersect at all. A b. As the radii of the circles increases by 5 miles, by line that intersects a circle in one point can be how much does the circumference increase? described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. SOLUTION: A line segment with endpoints on a circle can be a. The radius of the outermost circle is 30 miles and described as a chord. If the chord passes through that of the center circle is 5 miles. Find the circumference of each circle. the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that The difference of the circumferences of the intersects a circle in exactly two points can be outermost circle and the center circle is 60π - 10π = described as a secant. A line segment with 50π or about 157.1 miles. endpoints on a circle can be described as a chord. b. Let the radius of one circle be x miles and the If the chord passes through the center of the circle, radius of the next larger circle be x + 5 miles. Find it can be described as a diameter. A line segment the circumference of each circle. with endpoints at the center and on the circle can be described as a radius.

49. REASONING In the figure, a circle with radius r is The difference of the two circumferences is (2xπ + inscribed in a regular polygon and circumscribed 10π) - (2xπ) = 10π or about 31.4 miles. about another. Therefore, as the radius of each circle increases by 5 a. What are the perimeters of the circumscribed and miles, the circumference increases by about 31.4 inscribed polygons in terms of r? Explain. miles. b. Is the circumference C of the circle greater or ANSWER: less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality a. 157.1 mi comparing C to these perimeters. b. by about 31.4 mi c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. 48. WRITING IN MATH How can we describe the d. As the number of sides of both the circumscribed relationships that exist between and inscribed polygons increase, what will happen to circles and lines? the upper and lower limits of the inequality from part SOLUTION: c, and what does this imply? Sample answer: A line and a circle may intersect in SOLUTION: one point, two points, or may not intersect at all. A a. The circumscribed polygon is a square with sides line that intersects a circle in one point can be of length 2r. described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be

described as a chord. If the chord passes through The inscribed polygon is a hexagon with sides of the center of the circle, it can be described as a length r. (Each triangle formed using a central angle diameter. A line segment with endpoints at the will be equilateral.) center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment So, the perimeters are 8r and 6r. with endpoints at the center and on the circle can be b. described as a radius.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r 49. c. Since the diameter is twice the radius, 8r = 4(2r) REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed or 4d and 6r = 3(2r) or 3d. about another. The inequality is then 4d < C < 3d. Therefore, the a. What are the perimeters of the circumscribed and circumference of the circle is between 3 and 4 times inscribed polygons in terms of r? Explain. its diameter. b. Is the circumference C of the circle greater or d. These limits will approach a value of πd, implying less than the perimeter of the circumscribed polygon? that C = πd. the inscribed polygon? Write a compound inequality comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the ANSWER: diameter d of the circle and interpret its meaning. a. 8r and 6r ; Twice the radius of the circle, 2r is d. As the number of sides of both the circumscribed the side length of the square, so the perimeter of the and inscribed polygons increase, what will happen to square is 4(2r) or 8r. The the upper and lower limits of the inequality from part regular hexagon is made up of six equilateral c, and what does this imply? triangles with side length r, so the perimeter of the SOLUTION: hexagon is 6(r) or 6r. a. The circumscribed polygon is a square with sides b. less; greater; 6r < C < 8r of length 2r. c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

The inscribed polygon is a hexagon with sides of CHALLENGE length r. (Each triangle formed using a central angle 50. The sum of the circumferences of will be equilateral.) circles H, J, and K shown at the right is units. Find KJ.

SOLUTION: So, the perimeters are 8r and 6r. The radius of is 4x units, the radius of is x b. units, and the radius of is 2x units. Find the circumference of each circle.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying The sum of the circumferences is 56π units. that C = πd. Substitute the three circumferences and solve for x.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the So, the radius of is 4 units, the radius of is square is 4(2r) or 8r. The 2(4) or 8 units, and the radius of is 4(4) or 16 regular hexagon is made up of six equilateral units. triangles with side length r, so the perimeter of the The measure of KJ is equal to the sum of the radii hexagon is 6(r) or 6r. of and . b. less; greater; 6r < C < 8r Therefore, KJ = 16 + 8 or 24 units. c. 3d < C < 4d; The circumference of the circle is ANSWER: between 3 and 4 times its diameter. 24 units d. These limits will approach a value of πd, implying that C = πd. 51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, 50. CHALLENGE The sum of the circumferences of always, or never less than the radius of the circle? circles H, J, and K shown at the right is units. Explain. Find KJ. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the

center of the circle and a point on the circle. A SOLUTION: segment drawn from the center to a point inside the The radius of is 4x units, the radius of is x circle will always have a length less than the radius units, and the radius of is 2x units. Find the of the circle. circumference of each circle. 52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar.

SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make The sum of the circumferences is 56π units. up are from center B. Substitute the three circumferences and solve for x. For some value k, r2 = kr1.

So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii Therefore, is mapped onto . Since there of and . exists a rigid motion followed by a scaling that maps Therefore, KJ = 16 + 8 or 24 units. onto , the circles are similar. Thus, all circles are similar. ANSWER: 24 units

51. REASONING Is the distance from the center of a ANSWER: circle to a point in the interior of a circle sometimes, A circle is a locus of points in a plane equidistant always, or never less than the radius of the circle? from a given point. For any two circles and Explain. , there exists a translation that maps center A SOLUTION: onto center B, moving so that it is concentric A radius is a segment drawn between the center of with . There also exists a dilation with scale the circle and a point on the circle. A segment drawn factor k such that each point that makes up is from the center to a point inside the circle will always moved to be the same distance from center A as the have a length less than the radius of the circle. points that make up are from center B. Therefore, is mapped onto . Since there ANSWER: exists a rigid motion followed by a scaling that maps Always; a radius is a segment drawn between the onto , the circles are similar. Thus, all center of the circle and a point on the circle. A circles are similar. segment drawn from the center to a point inside the circle will always have a length less than the radius 53. CHALLENGE In the figure, is inscribed in of the circle. equilateral triangle LMN. What is the circumference 52. PROOF Use the locus definition of a circle and of ? dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such SOLUTION: that each point that makes up is moved to be the Draw a perpendicular to the side Also join same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects Therefore, is mapped onto . Since there Use the definition of exists a rigid motion followed by a scaling that maps tangent of an angle to find PR, which is the radius of onto , the circles are similar. Thus, all circles are similar.

ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A Therefore, the radius of is onto center B, moving so that it is concentric The circumference C of a circle with radius r is with . There also exists a dilation with scale given by factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. The circumference of is inches.

53. CHALLENGE In the figure, is inscribed in ANSWER: equilateral triangle LMN. What is the circumference in. of ?

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work. SOLUTION: ANSWER: Draw a perpendicular to the side Also join See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects SOLUTION: Use the definition of Use the Pythagorean Theorem to find the diameter tangent of an angle to find PR, which is the radius of of the circle.

The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of is

The circumference C of a circle with radius r is

given by

ANSWER: 40.8 56. What is the radius of a table with a circumference of 10 feet?

The circumference of is inches. A 1.6 ft B 2.5 ft ANSWER: C 3.2 ft D 5 ft in. SOLUTION: The circumference C of a circle with radius r is 54. WRITING IN MATH Research and write about given by the history of pi and its importance to the study of Here, C = 10 ft. Use the formula to find the radius. geometry. SOLUTION: See students’ work.

ANSWER:

See students’ work. The radius of the circle is about 1.6 feet, so the correct choice is A. 55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? F 10 SOLUTION: G 9 Use the Pythagorean Theorem to find the diameter H 8 of the circle. J 7 The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with radius r is Use the formula to find the circumference. given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

ANSWER: 40.8 Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only 56. What is the radius of a table with a circumference of correct choice is J. 10 feet? A 1.6 ft ANSWER: B 2.5 ft J C 3.2 ft SAT/ACT D 5 ft 58. What is the radius of a circle with an

SOLUTION: area of square units? The circumference C of a circle with radius r is A 0.4 units given by B 0.5 units Here, C = 10 ft. Use the formula to find the radius. C 2 units D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the The radius of the circle is about 1.6 feet, so the formula Here A = square units. Use the correct choice is A. formula to find the radius. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the Therefore, the correct choice is B. garden? ANSWER: F 10 B G 9 H 8 Copy each figure and point B. Then use a ruler J 7 to draw the image of the figure under a dilation with center B and the scale factor r indicated. SOLUTION:

The circumference C of a circle with radius r is 59. given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from

Therefore, the radius of the garden must be less than B to the vertex. Mark a point on the ray that is of or equal to about 7.96 feet. Therefore, the only that distance from B. Repeat the process for rays correct choice is J. through the other five vertices. Connect the six points on the rays to construct the dilated image of the ANSWER: hexagon. J

58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units D 4 units E 16 units

SOLUTION: The area of a circle of radius r is given by the ANSWER: formula Here A = square units. Use the formula to find the radius.

60.

Therefore, the correct choice is B.

ANSWER: SOLUTION: B Draw a ray from point B through a vertex of the Copy each figure and point B. Then use a ruler triangle. Use the ruler to measure the distance from to draw the image of the figure under a dilation with center B and the scale factor r indicated. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays 59. through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the ANSWER: hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of

that distance from B. Repeat the process for rays through the other five vertices. Connect the six points 61. on the rays to construct the dilated image of the hexagon.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

60. ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from 62. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the ANSWER: three points on the rays to construct the dilated image of the triangle.

61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the ANSWER: four points on the rays to construct the dilated image of the trapezoid.

ANSWER: State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: 62. The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

SOLUTION: 64. Refer to the image on page 705. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is 3 The figure can be mapped onto itself by a rotation of times that distance from B. Repeat the process for 90º. So, it has a rotational symmetry. So, the rays through the other two vertices. Connect the magnitude of symmetry is 90º and order of symmetry three points on the rays to construct the dilated image is 4. of the triangle. ANSWER: Yes; 4, 90

65. ANSWER: SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

State whether each figure has rotational ANSWER: symmetry. Write yes or no. If so, copy the No figure, locate the center of symmetry, and state the order and magnitude of symmetry. Determine the truth value of the following 63. Refer to the image on page 705. statement for each set of conditions. Explain SOLUTION: your reasoning. If you are over 18 years old, then you vote in all The figure cannot be mapped onto itself by any elections. rotation. So, it does not have a rotational symmetry. 67. You are 19 years old and you vote. ANSWER: SOLUTION: No True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true 64. Refer to the image on page 705. for the conditions. SOLUTION: ANSWER: The figure can be mapped onto itself by a rotation of True; sample answer: Since the hypothesis is true 90º. So, it has a rotational symmetry. So, the and the conclusion is true, then the statement is true magnitude of symmetry is 90º and order of symmetry for the conditions. is 4.

ANSWER: 68. You are 21 years old and do not vote. Yes; 4, 90 SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

65. Find x. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No 69. SOLUTION: 66. Refer to the image on page 705. The sum of the measures of angles around a point is SOLUTION: 360º. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No ANSWER: Determine the truth value of the following 90 statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote.

SOLUTION: 70. True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true SOLUTION: for the conditions. The angles with the measures xº and 2xº form a straight line. So, the sum is 180. ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. ANSWER: 68. You are 21 years old and do not vote. 60 SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true 71. and the conclusion is false, then the statement is false for the conditions. SOLUTION: The angles with the measures 6xº and 3xº form a Find x. straight line. So, the sum is 180.

ANSWER: 69. 20 SOLUTION: The sum of the measures of angles around a point is 360º.

72.

ANSWER: SOLUTION: 90 The sum of the three angles is 180.

ANSWER: 45 70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius. For Exercises 1–4, refer to . ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN.

1. Name the circle. SOLUTION: Here, are radii of the same circle. So, SOLUTION: they are equal in length. Therefore, DN = 8 cm. The center of the circle is N. So, the circle is ANSWER: ANSWER: 8 cm

4. If EN = 13 feet, what is the diameter of the circle? 2. Identify each. SOLUTION: a. a chord Here, is a radius and the diameter is twice the b. a diameter radius. c. a radius SOLUTION: Therefore, the diameter is 26 ft. a. A chord is a segment with endpoints on the circle. So, here are chords. ANSWER: b. A diameter of a circle is a chord that passes 26 ft through the center. Here, is a diameter. The diameters of , , and are 8 c. A radius is a segment with endpoints at the center inches, 18 inches, and 11 inches, respectively. and on the circle. Here, is radius. Find each measure.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. 5. FG SOLUTION: SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm. Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4. ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle?

SOLUTION: Therefore, FG = 14 in. Here, is a radius and the diameter is twice the ANSWER: radius. 14 in.

Therefore, the diameter is 26 ft. 6. FB SOLUTION: ANSWER: Since the diameter of B is 18 inches and A is 4 26 ft inches, AB = (18) or 9 and AF = (8) or 4. The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure.

Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning

5. FG of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the SOLUTION: nearest hundredth, if necessary. Since the diameter of B is 18 inches and A is 8 SOLUTION:

inches, AG = 18 and AF = (8) or 4. The radius is half the diameter. So, the radius of the circular ride is 22 feet.

Therefore, FG = 14 in.

ANSWER: Therefore, the circumference of the ride is about 14 in. 138.23 feet.

6. FB ANSWER: 22 ft; 138.23 ft SOLUTION: Since the diameter of B is 18 inches and A is 4 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. inches, AB = (18) or 9 and AF = (8) or 4. What are the diameter and radius of the pool? Round to the nearest hundredth.

Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the SOLUTION: radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet.

Therefore, the circumference of the ride is about 138.23 feet. The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. ANSWER: 22 ft; 138.23 ft ANSWER: 17.98 ft; 8.99 ft 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. 9. SHORT RESPONSE The right triangle shown is What are the diameter and radius of the pool? Round inscribed in . Find the exact circumference of to the nearest hundredth. .

SOLUTION: SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π The diameter of the pool is about 17.98 feet and the centimeters. radius of the pool is about 8.99 feet.

ANSWER: ANSWER: 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is For Exercises 10–13, refer to . inscribed in . Find the exact circumference of .

10. Name the center of the circle.

SOLUTION: SOLUTION: R The diameter of the circle is the hypotenuse of the right triangle. ANSWER: R

11. Identify a chord that is also a diameter. SOLUTION: The diameter of the circle is 4 centimeters. Two chords are shown: and . goes through the center, R, so is a diameter.

ANSWER: The circumference of the circle is 4π centimeters. 12. Is a radius? Explain. ANSWER: SOLUTION: A radius is a segment with endpoints at the center For Exercises 10–13, refer to . and on the circle. But has both the end points on the circle, so it is a chord.

ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT?

10. Name the center of the circle. SOLUTION: SOLUTION: Here, is a diameter and is a radius. R ANSWER: R

Therefore, RT = 8.1 cm. 11. Identify a chord that is also a diameter. ANSWER: SOLUTION: 8.1 cm Two chords are shown: and . goes through the center, R, so is a diameter. For Exercises 14–17, refer to .

ANSWER:

12. Is a radius? Explain. 14. Identify a chord that is not a diameter. SOLUTION: A radius is a segment with endpoints at the center SOLUTION: and on the circle. But has both the end points on The chords do not pass through the the circle, so it is a chord. center. So, they are not diameters.

ANSWER: ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Here, is a diameter and is a radius. Here, is a radius and the diameter is twice the radius. .

Therefore, RT = 8.1 cm. Therefore, the diameter of the circle is 28 inches.

ANSWER: ANSWER: 8.1 cm 28 in. For Exercises 14–17, refer to . 16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since are both radii of , .

ANSWER:

14. Identify a chord that is not a diameter. Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? The chords do not pass through the SOLUTION: center. So, they are not diameters. Here, is a diameter and is a radius. ANSWER:

15. If CF = 14 inches, what is the diameter of the circle?

SOLUTION: Therefore, EF = 3.7 cm. Here, is a radius and the diameter is twice the ANSWER: radius. . 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each Therefore, the diameter of the circle is 28 inches. measure.

ANSWER: 28 in.

16. Is ? Explain. SOLUTION:

All radii of a circle are congruent. Since 18. CK are both radii of , . SOLUTION: ANSWER: Since the radius of K is 8 units, BK = 8. Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? Therefore, CK = 2.6 units. SOLUTION: ANSWER: Here, is a diameter and is a radius. 2.6

19. AB SOLUTION:

Therefore, EF = 3.7 cm. Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. ANSWER: 3.7 cm

Circle J has a radius of 10 units, has a Therefore, AB = 14.6 units. radius of 8 units, and BC = 5.4 units. Find each measure. ANSWER: 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

18. CK SOLUTION: Since circle J has a radius of 10 units, JC = 10. Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units. Therefore, JK = 12.6 units.

ANSWER: ANSWER: 2.6 12.6 21. AD 19. AB SOLUTION: SOLUTION: We can find AD using AD = AC + CK + KD. The Since is a diameter of circle J and the radius is radius of circle K is 8 units and the radius of circle J 10, AC = 2(10) or 20 units. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

Therefore, AB = 14.6 units. ANSWER: 14.6 Therefore, AD = 30.6 units. 20. JK ANSWER: SOLUTION: 30.6 First find CK. Since circle K has a radius of 8 units, BK = 8. 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary.

Since circle J has a radius of 10 units, JC = 10.

Therefore, JK = 12.6 units.

ANSWER: 12.6 21. AD SOLUTION: The diameter of the pizza is 16 inches. SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. . So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, AD = 30.6 units. Therefore, the circumference of the pizza is about ANSWER: 50.27 inches. 30.6 ANSWER: 22. PIZZA Find the radius and circumference of the 8 in.; 50.27 in. pizza shown. Round to the nearest hundredth, if necessary. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter.

SOLUTION: The diameter of the pizza is 16 inches. So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is Therefore, the circumference of the bicycle tire is given by about 81.68 inches. ANSWER: 13 in.; 81.68 in.

Therefore, the circumference of the pizza is about Find the diameter and radius of a circle by with 50.27 inches. the given circumference. Round to the nearest hundredth. ANSWER: 24. C = 18 in. 8 in.; 50.27 in. SOLUTION: 23. BICYCLES A bicycle has tires with a diameter of The circumference C of a circle with diameter d is 26 inches. Find the radius and circumference of a given by tire. Round to the nearest hundredth, if necessary. Here, C = 18 in. Use the formula to find the diameter. Then find the radius. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: Therefore, the circumference of the bicycle tire is 5.73 in.; 2.86 in. about 81.68 inches.

ANSWER: 25. C = 124 ft 13 in.; 81.68 in. SOLUTION: Find the diameter and radius of a circle by with The circumference C of a circle with diameter d is the given circumference. Round to the nearest given by hundredth. Here, C = 124 ft. Use the formula to find the 24. C = 18 in. diameter. Then find the radius. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet.

ANSWER: 39.47 ft; 19.74 ft

Therefore, the diameter is about 5.73 inches and the 26. C = 375.3 cm radius is about 2.86 inches. SOLUTION: ANSWER: The circumference C of a circle with diameter d is 5.73 in.; 2.86 in. given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. 25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: 119.46 cm; 59.73 cm

27. C = 2608.25 m Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. SOLUTION: The circumference C of a circle with diameter d is ANSWER: given by 39.47 ft; 19.74 ft Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. 26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m

CCSS SENSE-MAKING Find the exact Therefore, the diameter is about 119.46 centimeters circumference of each circle by using the given and the radius is about 59.73 centimeters. inscribed or circumscribed polygon. ANSWER: 119.46 cm; 59.73 cm

27. C = 2608.25 m SOLUTION: The circumference C of a circle with diameter d is 28.

given by SOLUTION: Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. Therefore, the circumference of the circle is 17 ANSWER: centimeters. 830.23 m; 415.12 m ANSWER: CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon.

29. 28. SOLUTION: The hypotenuse of the right triangle is a diameter of SOLUTION: the circle. Use the Pythagorean Theorem to find the The hypotenuse of the right triangle is a diameter of diameter. the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 12 feet. The circumference C of a circle is given by The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the circumference of the circle is 12 feet. Therefore, the circumference of the circle is 17 centimeters. ANSWER:

ANSWER:

30. SOLUTION: Each diagonal of the inscribed rectangle will pass 29. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter SOLUTION: of the circle. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is inches. The circumference C of a circle is given by The diameter of the circle is 12 feet. The circumference C of a circle is given by

Therefore, the circumference of the circle is Therefore, the circumference of the circle is 12 inches. feet. ANSWER: ANSWER:

30. 31. SOLUTION: SOLUTION: Each diagonal of the inscribed rectangle will pass Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter Use the Pythagorean Theorem to find the diameter of the circle. of the circle.

The diameter of the circle is 10 inches. The The diameter of the circle is inches. The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 10 inches. Therefore, the circumference of the circle is inches. ANSWER:

ANSWER:

32. SOLUTION: 31. A diameter perpendicular to a side of the square and SOLUTION: the length of each side will measure the same. So, d Each diagonal of the inscribed rectangle will pass = 25 millimeters. through the origin, so it is a diameter of the circle. The circumference C of a circle is given by Use the Pythagorean Theorem to find the diameter of the circle.

Therefore, the circumference of the circle is 25 millimeters.

The diameter of the circle is 10 inches. The ANSWER: circumference C of a circle is given by

Therefore, the circumference of the circle is 10 inches.

ANSWER: 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d Therefore, the circumference of the circle is 14 = 25 millimeters. yards. The circumference C of a circle is given by ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, Therefore, the circumference of the circle is 25 except that a flying disc is used instead of a ball and millimeters. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in ANSWER: centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter. 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The diameter of the disc is about 21.3 centimeters The circumference C of a circle is given by and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

Therefore, the circumference of the circle is 14 ANSWER: yards. 176.8 g

ANSWER: 35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? 34. DISC GOLF Disc golf is similar to regular golf, b. If Mr. Martinez changes the plans so that the except that a flying disc is used instead of a ball and inner circle has a circumference of approximately 25 clubs. For professional competitions, the maximum feet, what should the radius of the circle be to the weight of a disc in grams is 8.3 times its diameter in nearest foot? centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter. SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by

Use the formula to find the circumference.

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the circumference is about 31.42 feet. Therefore, the maximum weight is 8.3 × 21.3 or b. Here, C = 25 ft. Use the formula to find the about 176.8 grams. inner radius of the inner circle. ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? So, the radius of the inner circle is about 4 feet. b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 ANSWER: feet, what should the radius of the circle be to the a. 31.42 ft nearest foot? b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

36.

SOLUTION: SOLUTION: The radius is half the diameter and the a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with diameter d is given circumference C of a circle with radius r is given by by

Use the formula to find the circumference.

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle. Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in. So, the radius of the inner circle is about 4 feet.

ANSWER: 37. a. 31.42 ft b. 4 ft SOLUTION: The diameter is twice the radius and the The radius, diameter, or circumference of a circumference C of a circle with diameter d is given circle is given. Find each missing measure to by the nearest hundredth.

36.

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft

38.

SOLUTION: Therefore, the radius is 4.25 inches and the The circumference C of a circle with diameter d is circumference is about 26.70 inches. given by Here, C = 35x cm. Use the formula to find the ANSWER: diameter.Then find the radius. 4.25 in.; 26.70 in.

37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 11.14x cm; 5.57x cm

So, the diameter is 22.80 feet and the circumference 39. is about 71.63 feet.

ANSWER: SOLUTION: 22.80 ft; 71.63 ft The diameter is twice the radius and the circumference C of a circle with diameter d is given 38. by

SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 35x cm. Use the formula to find the diameter.Then find the radius.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or neither. Therefore, the diameter is about 11.14x centimeters 40. Refer to the photo on page 703. and the radius is about 5.57x centimeters. SOLUTION: ANSWER: The circles have the same center and they are in the 11.14x cm; 5.57x cm same plane. So, they are concentric circles. ANSWER: 39. concentric SOLUTION: The diameter is twice the radius and the 41. Refer to the photo on page 703. circumference C of a circle with diameter d is given SOLUTION: by The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they So, the diameter is 0.25x units and the circumference appear to be congruent. is about 0.79x units. ANSWER: ANSWER: congruent 0.25x; 0.79x 43. HISTORY The Indian Shell Ring on Hilton Head Determine whether the circles in the figures Island approximates a circle. If each unit on the below appear to be congruent, concentric, or coordinate grid represents 25 feet, how far would neither. someone have to walk to go completely around the 40. Refer to the photo on page 703. ring? Round to the nearest tenth. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric. SOLUTION: ANSWER: The radius of the circle is 3 units, that is 75 feet. So, neither the diameter is 150 ft. The circumference C of a circle with diameter d is given by 42. Refer to the photo on page 703. So, the circumference is SOLUTION: Therefore, someone have to The circles appear to have the same radius. So, they walk about 471.2 feet to go completely around the appear to be congruent. ring.

10-1ANSWER: Circles and Circumference ANSWER: congruent 471.2 ft

43. HISTORY The Indian Shell Ring on Hilton Head 44. CCSS MODELING A brick path is being installed Island approximates a circle. If each unit on the around a circular pond. The pond has a coordinate grid represents 25 feet, how far would circumference of 68 feet. The outer edge of the path someone have to walk to go completely around the is going to be 4 feet from the pond all the way ring? Round to the nearest tenth. around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

So, the radius of the pond is about 10.82 feet and the SOLUTION: radius to the outer edge of the path will be 10.82 + 4 The radius of the circle is 3 units, that is 75 feet. So, = 14.82 ft. Use the radius to find the circumference the diameter is 150 ft. The circumference C of a of the brick path. circle with diameter d is given by So, the circumference is Therefore, someone have to walk about 471.2 feet to go completely around the Therefore, the circumference of the path is about ring. 93.13 feet.

ANSWER: ANSWER: 471.2 ft 93.13 ft

44. CCSS MODELING A brick path is being installed 45. MULTIPLE REPRESENTATIONS In this around a circular pond. The pond has a problem, you will explore changing dimensions in circumference of 68 feet. The outer edge of the path circles. is going to be 4 feet from the pond all the way a. GEOMETRIC Use a compass to draw three around. What is the approximate circumference of circles in which the scale factor from each circle to the path? Round to the nearest hundredth. the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest SOLUTION: tenth) and circumference (to the nearest hundredth) The circumference C of a circle with radius r is of each circle. Record your results in a table. given by Here, Cpond = 68 ft. Use the c. VERBAL Explain why these three circles are formula to find the radius of the pond. geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to is . Write an equation relating the So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 circumference (CA) of to the circumference = 14.82 ft. Use the radius to find the circumference (C ) of . of the brick path. B f. NUMERICAL If the scale factor from to eSolutions Manual - Powered by Cognero is , and the circumference of is 12 Page 11

Therefore, the circumference of the path is about inches, what is the circumference of ? 93.13 feet. SOLUTION: ANSWER: a. Sample answer: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three

circles in which the scale factor from each circle to b. Sample answer. the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the c. They all have the same shape–circular. ratio between their radii is 2. d. The ratio of their circumferences is also 2. e. ANALYTICAL The scale factor from to e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the is . Write an equation relating the

circumference. Therefore, . circumference (CA) of to the circumference (C ) of . f. Use the formula from part e. Here, the B f. NUMERICAL If the scale factor from to circumference of is 12 inches and . is , and the circumference of is 12 inches, what is the circumference of ?

SOLUTION: Therefore, the circumference of is 4 inches. a. Sample answer: ANSWER: a. Sample answer:

b. Sample answer.

b. Sample answer.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. c. They all have the same shape—circular. So, if the radii are in the ratio b : a, so will the d. The ratio of their circumferences is also 2.

circumference. Therefore, . e. f. Use the formula from part e. Here, the f. 4 in. circumference of is 12 inches and . 46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

Therefore, the circumference of is 4 inches.

ANSWER: a. Sample answer:

a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 b. Sample answer. drops.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. They all have the same shape—circular. c. How are the values you found in part b related to d. The ratio of their circumferences is also 2. π?

e. SOLUTION: f. 4 in. a. See students’ work. b. See students’ work. 46. BUFFON’S NEEDLE Measure the length of a c. Sample answer: The values are approaching 3.14, needle (or toothpick) in centimeters. Next, draw a set which is approximately equal to π. of horizontal lines that are centimeters apart on a sheet of plain white paper. ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 a. Drop the needle onto the paper. When the needle miles from downtown Phoenix. lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. b How are the values you found in part related to a. How much greater is the circumference of the

π? outermost circle than the circumference of the center SOLUTION: circle? a. See students’ work. b. As the radii of the circles increases by 5 miles, by b. See students’ work. how much does the circumference increase? c. Sample answer: The values are approaching 3.14, SOLUTION: which is approximately equal to π. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the

ANSWER: circumference of each circle. a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 The difference of the circumferences of the miles from downtown Phoenix. outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

a. How much greater is the circumference of the outermost circle than the circumference of the center The difference of the two circumferences is (2xπ + circle? 10π) - (2xπ) = 10π or about 31.4 miles. b. As the radii of the circles increases by 5 miles, by Therefore, as the radius of each circle increases by 5 how much does the circumference increase? miles, the circumference increases by about 31.4

SOLUTION: miles. a. The radius of the outermost circle is 30 miles and ANSWER: that of the center circle is 5 miles. Find the a. 157.1 mi circumference of each circle. b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in The difference of the circumferences of the one point, two points, or may not intersect at all. A outermost circle and the center circle is 60π - 10π = line that intersects a circle in one point can be 50π or about 157.1 miles. described as a tangent. A line that intersects a circle b. Let the radius of one circle be x miles and the in exactly two points can be described as a secant. radius of the next larger circle be x + 5 miles. Find A line segment with endpoints on a circle can be the circumference of each circle. described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one The difference of the two circumferences is (2xπ + point can be described as a tangent. A line that 10π) - (2xπ) = 10π or about 31.4 miles. intersects a circle in exactly two points can be Therefore, as the radius of each circle increases by 5 described as a secant. A line segment with miles, the circumference increases by about 31.4 endpoints on a circle can be described as a chord.

miles. If the chord passes through the center of the circle, ANSWER: it can be described as a diameter. A line segment a. 157.1 mi with endpoints at the center and on the circle can be b. by about 31.4 mi described as a radius.

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: 49. Sample answer: A line and a circle may intersect in REASONING In the figure, a circle with radius r is one point, two points, or may not intersect at all. A inscribed in a regular polygon and circumscribed line that intersects a circle in one point can be about another. described as a tangent. A line that intersects a circle a. What are the perimeters of the circumscribed and in exactly two points can be described as a secant. inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or A line segment with endpoints on a circle can be less than the perimeter of the circumscribed polygon? described as a chord. If the chord passes through the inscribed polygon? Write a compound inequality the center of the circle, it can be described as a comparing C to these perimeters. diameter. A line segment with endpoints at the c. Rewrite the inequality from part b in terms of the center and on the circle can be described as a diameter d of the circle and interpret its meaning. radius. d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to ANSWER: the upper and lower limits of the inequality from part Sample answer: A line that intersects a circle in one c, and what does this imply? point can be described as a tangent. A line that SOLUTION: intersects a circle in exactly two points can be a. The circumscribed polygon is a square with sides described as a secant. A line segment with of length 2r. endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius. The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? So, the perimeters are 8r and 6r. the inscribed polygon? Write a compound inequality b. comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. The circumference is less than the perimeter of the d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to circumscribed polygon and greater than the the upper and lower limits of the inequality from part perimeter of the inscribed polygon. c, and what does this imply? 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) SOLUTION: or 4d and 6r = 3(2r) or 3d. a. The circumscribed polygon is a square with sides The inequality is then 4d < C < 3d. Therefore, the of length 2r. circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.) ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. So, the perimeters are 8r and 6r. d. These limits will approach a value of πd, implying b. that C = πd. 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. The circumference is less than the perimeter of the Find KJ. circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying SOLUTION: that C = πd. The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle. ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying The sum of the circumferences is 56π units. π that C = d. Substitute the three circumferences and solve for x. 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ.

So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii of and . Therefore, KJ = 16 + 8 or 24 units.

ANSWER: SOLUTION: 24 units The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the 51. REASONING Is the distance from the center of a circumference of each circle. circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Explain.

SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A The sum of the circumferences is 56π units. segment drawn from the center to a point inside the Substitute the three circumferences and solve for x. circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar.

So, the radius of is 4 units, the radius of is SOLUTION: 2(4) or 8 units, and the radius of is 4(4) or 16 A circle is a locus of points in a plane equidistant units. from a given point. For any two circles and The measure of KJ is equal to the sum of the radii , there exists a translation that of and . maps center A onto center B, moving so that it Therefore, KJ = 16 + 8 or 24 units. is concentric with . There also exists a dilation with scale factor k such ANSWER: that each point that makes up is moved to be the 24 units same distance from center A as the points that make up are from center B. 51. REASONING Is the distance from the center of a For some value k, r2 = circle to a point in the interior of a circle sometimes, kr . always, or never less than the radius of the circle? 1 Explain.

SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps ANSWER: onto , the circles are Always; a radius is a segment drawn between the similar. Thus, all circles are similar. center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle. ANSWER: A circle is a locus of points in a plane equidistant 52. PROOF Use the locus definition of a circle and from a given point. For any two circles and dilations to prove that all circles are similar. , there exists a translation that maps center A SOLUTION: onto center B, moving so that it is concentric A circle is a locus of points in a plane equidistant with . There also exists a dilation with scale from a given point. For any two circles and factor k such that each point that makes up is , there exists a translation that moved to be the same distance from center A as the maps center A onto center B, moving so that it points that make up are from center B. is concentric with . There also exists a dilation Therefore, is mapped onto . Since there with scale factor k such exists a rigid motion followed by a scaling that maps that each point that makes up is moved to be the onto , the circles are similar. Thus, all same distance from center A as the points that make circles are similar. up are from center B. For some value k, r2 = 53. CHALLENGE In the figure, is inscribed in kr1. equilateral triangle LMN. What is the circumference of ?

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. SOLUTION: Draw a perpendicular to the side Also join

ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is Since is equilateral triangle, MN = 8 inches moved to be the same distance from center A as the and RN = 4 inches. Also, bisects points that make up are from center B. Use the definition of Therefore, is mapped onto . Since there tangent of an angle to find PR, which is the radius of exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference of ? Therefore, the radius of is The circumference C of a circle with radius r is given by

SOLUTION:

Draw a perpendicular to the side Also join The circumference of is inches.

ANSWER:

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of Since is equilateral triangle, MN = 8 inches geometry. and RN = 4 inches. Also, bisects Use the definition of SOLUTION: tangent of an angle to find PR, which is the radius of See students’ work. ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

Therefore, the radius of is The circumference C of a circle with radius r is given by

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

The circumference C of a circle with diameter d is The circumference of is inches. given by Use the formula to find the circumference. ANSWER:

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: ANSWER: See students’ work. 40.8 ANSWER: 56. What is the radius of a table with a circumference of See students’ work. 10 feet? A 1.6 ft 55. GRIDDED RESPONSE What is the B 2.5 ft circumference of ? Round to the nearest tenth. C 3.2 ft D 5 ft SOLUTION: The circumference C of a circle with radius r is given by Here, C = 10 ft. Use the formula to find the radius. SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

The circumference C of a circle with diameter d is The radius of the circle is about 1.6 feet, so the given by correct choice is A. Use the formula to find the circumference. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? F 10 ANSWER: G 9 40.8 H 8 56. What is the radius of a table with a circumference of J 7 10 feet? SOLUTION: A 1.6 ft The circumference C of a circle with radius r is B 2.5 ft given by C 3.2 ft Because 50 feet of fence can be used to border the D 5 ft garden, Cmax = 50. Use the circumference formula to SOLUTION: find the maximum radius. The circumference C of a circle with radius r is given by Here, C = 10 ft. Use the formula to find the radius.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J. The radius of the circle is about 1.6 feet, so the correct choice is A. ANSWER: J ANSWER: A 58. SAT/ACT What is the radius of a circle with an

57. ALGEBRA Bill is planning a circular vegetable area of square units? garden with a fence around the border. If he can use A 0.4 units up to 50 feet of fence, what radius can he use for the B 0.5 units garden? C 2 units F 10 D 4 units G 9 E 16 units H 8 J 7 SOLUTION: The area of a circle of radius r is given by the SOLUTION: The circumference C of a circle with radius r is formula Here A = square units. Use the given by formula to find the radius. Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

Therefore, the correct choice is B.

ANSWER: B Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only Copy each figure and point B. Then use a ruler correct choice is J. to draw the image of the figure under a dilation with center B and the scale factor r indicated. ANSWER: 59. J

58. SAT/ACT What is the radius of a circle with an

area of square units?

A 0.4 units B 0.5 units SOLUTION: C 2 units Draw a ray from point B through a vertex of the D 4 units hexagon. Use the ruler to measure the distance from E 16 units B to the vertex. Mark a point on the ray that is of SOLUTION: that distance from B. Repeat the process for rays The area of a circle of radius r is given by the through the other five vertices. Connect the six points formula Here A = square units. Use the on the rays to construct the dilated image of the formula to find the radius. hexagon.

Therefore, the correct choice is B.

ANSWER: B Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation with center B and the scale factor r indicated. ANSWER:

59.

60.

SOLUTION:

Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is of Draw a ray from point B through a vertex of the that distance from B. Repeat the process for rays triangle. Use the ruler to measure the distance from through the other five vertices. Connect the six points B to the vertex. Mark a point on the ray that is of on the rays to construct the dilated image of the that distance from B. Repeat the process for rays hexagon. through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

ANSWER: 61.

60. SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the SOLUTION: four points on the rays to construct the dilated image Draw a ray from point B through a vertex of the of the trapezoid. triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

ANSWER:

62. 61.

SOLUTION: SOLUTION: Draw a ray from point B through a vertex of the Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for times that distance from B. Repeat the process for rays through the other three vertices. Connect the rays through the other two vertices. Connect the four points on the rays to construct the dilated image three points on the rays to construct the dilated image of the trapezoid. of the triangle.

ANSWER:

ANSWER:

62.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from State whether each figure has rotational B to the vertex. Mark a point on the ray that is 3 symmetry. Write yes or no. If so, copy the times that distance from B. Repeat the process for figure, locate the center of symmetry, and state rays through the other two vertices. Connect the the order and magnitude of symmetry. three points on the rays to construct the dilated image of the triangle. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the ANSWER: magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state 65. the order and magnitude of symmetry. SOLUTION: 63. Refer to the image on page 705. The given figure cannot be mapped onto itself by any SOLUTION: rotation. So, it does not have a rotational symmetry. The figure cannot be mapped onto itself by any ANSWER: rotation. So, it does not have a rotational symmetry. No ANSWER: No 66. Refer to the image on page 705. SOLUTION: 64. Refer to the image on page 705. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. SOLUTION: The figure can be mapped onto itself by a rotation of ANSWER: 90º. So, it has a rotational symmetry. So, the No magnitude of symmetry is 90º and order of symmetry is 4. Determine the truth value of the following statement for each set of conditions. Explain ANSWER: your reasoning. Yes; 4, 90 If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

ANSWER: True; sample answer: Since the hypothesis is true 65. and the conclusion is true, then the statement is true SOLUTION: for the conditions. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. 68. You are 21 years old and do not vote. ANSWER: SOLUTION: No False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false 66. Refer to the image on page 705. for the conditions.

SOLUTION: ANSWER: The given figure cannot be mapped onto itself by any False; sample answer: Since the hypothesis is true rotation. So, it does not have a rotational symmetry. and the conclusion is false, then the statement is false for the conditions. ANSWER: No Find x. Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. 69. SOLUTION: SOLUTION: True; sample answer: Since the hypothesis is true The sum of the measures of angles around a point is and the conclusion is true, then the statement is true 360º. for the conditions.

ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true ANSWER: for the conditions. 90

68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. 70. ANSWER: SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false The angles with the measures xº and 2xº form a

for the conditions. straight line. So, the sum is 180.

Find x.

ANSWER: 60

69. SOLUTION: The sum of the measures of angles around a point is 360º. 71. SOLUTION: The angles with the measures 6xº and 3xº form a ANSWER: straight line. So, the sum is 180. 90

ANSWER: 20

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180. 72. SOLUTION: The sum of the three angles is 180. ANSWER: 60

ANSWER: 45

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? For Exercises 1–4, refer to . SOLUTION: Here, is a radius and the diameter is twice the radius.

Therefore, the diameter is 26 ft.

1. Name the circle. ANSWER: 26 ft SOLUTION: The center of the circle is N. So, the circle is The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. ANSWER: Find each measure.

2. Identify each. a. a chord b. a diameter c. a radius

SOLUTION: 5. FG a. A chord is a segment with endpoints on the circle. SOLUTION: So, here are chords. Since the diameter of B is 18 inches and A is 8 b. A diameter of a circle is a chord that passes inches, AG = 18 and AF = (8) or 4. through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is

radius. Therefore, FG = 14 in. ANSWER: a. ANSWER: b. 14 in. c. 6. FB 3. If CN = 8 centimeters, find DN. SOLUTION: SOLUTION: Since the diameter of B is 18 inches and A is 4 Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm. inches, AB = (18) or 9 and AF = (8) or 4.

ANSWER: 8 cm

4. If EN = 13 feet, what is the diameter of the circle? Therefore, FB = 5 inches.

SOLUTION: ANSWER: Here, is a radius and the diameter is twice the 5 in. radius. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the Therefore, the diameter is 26 ft. radius and circumference of the ride? Round to the nearest hundredth, if necessary. ANSWER: SOLUTION: 26 ft The radius is half the diameter. So, the radius of the The diameters of , , and are 8 circular ride is 22 feet. inches, 18 inches, and 11 inches, respectively. Find each measure.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft 5. FG SOLUTION: 8. CCSS MODELING The circumference of the Since the diameter of B is 18 inches and A is 8 circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round inches, AG = 18 and AF = (8) or 4. to the nearest hundredth.

Therefore, FG = 14 in.

ANSWER: 14 in. SOLUTION: 6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches.

ANSWER: 5 in. The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the ANSWER: radius and circumference of the ride? Round to the 17.98 ft; 8.99 ft nearest hundredth, if necessary. 9. SHORT RESPONSE The right triangle shown is SOLUTION: inscribed in . Find the exact circumference of The radius is half the diameter. So, the radius of the . circular ride is 22 feet.

Therefore, the circumference of the ride is about 138.23 feet. SOLUTION: The diameter of the circle is the hypotenuse of the ANSWER: right triangle. 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π centimeters.

ANSWER: SOLUTION:

For Exercises 10–13, refer to .

10. Name the center of the circle. SOLUTION: The diameter of the pool is about 17.98 feet and the R radius of the pool is about 8.99 feet. ANSWER: ANSWER: R 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is 11. Identify a chord that is also a diameter. inscribed in . Find the exact circumference of SOLUTION:

. Two chords are shown: and . goes through the center, R, so is a diameter.

ANSWER:

SOLUTION: 12. Is a radius? Explain. The diameter of the circle is the hypotenuse of the right triangle. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord.

ANSWER: The diameter of the circle is 4 centimeters. No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: Here, is a diameter and is a radius. The circumference of the circle is 4π centimeters.

ANSWER:

Therefore, RT = 8.1 cm. For Exercises 10–13, refer to . ANSWER: 8.1 cm

For Exercises 14–17, refer to .

10. Name the center of the circle. SOLUTION: R 14. Identify a chord that is not a diameter. ANSWER: R SOLUTION: The chords do not pass through the 11. Identify a chord that is also a diameter. center. So, they are not diameters. SOLUTION: ANSWER: Two chords are shown: and . goes through the center, R, so is a diameter. 15. If CF = 14 inches, what is the diameter of the circle?

ANSWER: SOLUTION: Here, is a radius and the diameter is twice the radius. . 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center Therefore, the diameter of the circle is 28 inches. and on the circle. But has both the end points on the circle, so it is a chord. ANSWER: 28 in. ANSWER: No; it is a chord. 16. Is ? Explain. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: SOLUTION: All radii of a circle are congruent. Since are both radii of , . Here, is a diameter and is a radius. ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF?

Therefore, RT = 8.1 cm. SOLUTION:

ANSWER: Here, is a diameter and is a radius. 8.1 cm

For Exercises 14–17, refer to .

Therefore, EF = 3.7 cm.

ANSWER: 3.7 cm

14. Identify a chord that is not a diameter. Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. The chords do not pass through the center. So, they are not diameters.

ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? 18. CK SOLUTION: SOLUTION: Here, is a radius and the diameter is twice the Since the radius of K is 8 units, BK = 8. radius. .

Therefore, CK = 2.6 units. Therefore, the diameter of the circle is 28 inches. ANSWER: ANSWER: 2.6 28 in. 19. AB 16. Is ? Explain. SOLUTION: SOLUTION: Since is a diameter of circle J and the radius is All radii of a circle are congruent. Since 10, AC = 2(10) or 20 units. are both radii of , .

ANSWER: Yes; they are both radii of . Therefore, AB = 14.6 units.

17. If DA = 7.4 centimeters, what is EF? ANSWER: 14.6 SOLUTION: Here, is a diameter and is a radius. 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Therefore, EF = 3.7 cm.

ANSWER: Since circle J has a radius of 10 units, JC = 10. 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each Therefore, JK = 12.6 units. measure. ANSWER: 12.6 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J 18. CK is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. SOLUTION: Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units. ANSWER: Therefore, AD = 30.6 units. 2.6 ANSWER: 30.6 19. AB 22. PIZZA Find the radius and circumference of the SOLUTION: pizza shown. Round to the nearest hundredth, if Since is a diameter of circle J and the radius is necessary. 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units.

ANSWER: 14.6

20. JK SOLUTION: SOLUTION: The diameter of the pizza is 16 inches. First find CK. Since circle K has a radius of 8 units, BK = 8.

. Since circle J has a radius of 10 units, JC = 10. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, JK = 12.6 units.

ANSWER: 12.6 Therefore, the circumference of the pizza is about 50.27 inches. 21. AD ANSWER: SOLUTION: 8 in.; 50.27 in. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J 23. BICYCLES A bicycle has tires with a diameter of is 10 units, so KD = 8 and AC = 2(10) or 20. Before 26 inches. Find the radius and circumference of a finding AD, we need to find CK. tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half

the diameter.

Therefore, AD = 30.6 units.

ANSWER: 30.6 So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is 22. PIZZA Find the radius and circumference of the given by pizza shown. Round to the nearest hundredth, if necessary.

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in.

Find the diameter and radius of a circle by with the given circumference. Round to the nearest SOLUTION: hundredth. The diameter of the pizza is 16 inches. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the . diameter. Then find the radius. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in. Therefore, the diameter is about 5.73 inches and the 23. BICYCLES A bicycle has tires with a diameter of radius is about 2.86 inches. 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. ANSWER: 5.73 in.; 2.86 in. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter. 25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius. So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. Therefore, the diameter is about 39.47 feet and the Find the diameter and radius of a circle by with radius is about 19.74 feet. the given circumference. Round to the nearest hundredth. ANSWER: 24. C = 18 in. 39.47 ft; 19.74 ft

SOLUTION: 26. C = 375.3 cm The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 18 in. Use the formula to find the The circumference C of a circle with diameter d is diameter. Then find the radius. given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. Therefore, the diameter is about 119.46 centimeters ANSWER: and the radius is about 59.73 centimeters. 5.73 in.; 2.86 in. ANSWER: 119.46 cm; 59.73 cm 25. C = 124 ft SOLUTION: 27. C = 2608.25 m The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 124 ft. Use the formula to find the given by diameter. Then find the radius. Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. ANSWER: 39.47 ft; 19.74 ft ANSWER: 830.23 m; 415.12 m 26. C = 375.3 cm CCSS SENSE-MAKING Find the exact SOLUTION: circumference of each circle by using the given The circumference C of a circle with diameter d is inscribed or given by circumscribed polygon. Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

28.

SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: The diameter of the circle is 17 cm. The 119.46 cm; 59.73 cm circumference C of a circle is given by

27. C = 2608.25 m

SOLUTION: The circumference C of a circle with diameter d is Therefore, the circumference of the circle is 17 given by centimeters. Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. ANSWER:

29.

SOLUTION: Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the ANSWER: diameter. 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or The diameter of the circle is 12 feet. The circumscribed polygon. circumference C of a circle is given by

Therefore, the circumference of the circle is 12 feet. 28. ANSWER: SOLUTION:

The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

30.

The diameter of the circle is 17 cm. The SOLUTION: circumference C of a circle is given by Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

Therefore, the circumference of the circle is 17 centimeters.

ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 29. inches.

SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 12 feet. The 31. circumference C of a circle is given by SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter Therefore, the circumference of the circle is 12 of the circle. feet.

ANSWER:

The diameter of the circle is 10 inches. The circumference C of a circle is given by

30. SOLUTION: Therefore, the circumference of the circle is 10 inches. Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. ANSWER: Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is inches. The 32.

circumference C of a circle is given by SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by Therefore, the circumference of the circle is inches.

ANSWER: Therefore, the circumference of the circle is 25 millimeters.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The diameter of the circle is 10 inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 14 Therefore, the circumference of the circle is 10 yards. inches. ANSWER: ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? 32. Round to the nearest tenth. SOLUTION: SOLUTION: A diameter perpendicular to a side of the square and The circumference C of a circle with diameter d is the length of each side will measure the same. So, d given by = 25 millimeters. Here, C = 66.92 cm. Use the formula to find the The circumference C of a circle is given by diameter.

Therefore, the circumference of the circle is 25 millimeters. The diameter of the disc is about 21.3 centimeters ANSWER: and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. 33. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the SOLUTION: inner circle has a circumference of approximately 25 A diameter perpendicular to a side of the square and feet, what should the radius of the circle be to the the length of each side will measure the same. So, d nearest foot? = 14 yards. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 yards. SOLUTION: ANSWER: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by

34. DISC GOLF Disc golf is similar to regular golf, Use the formula to find the circumference. except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Therefore, the circumference is about 31.42 feet.

Round to the nearest tenth. b. Here, Cinner = 25 ft. Use the formula to find the SOLUTION: radius of the inner circle. The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter.

So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft The diameter of the disc is about 21.3 centimeters b. 4 ft and the maximum weight allowed is 8.3 times the The radius, diameter, or circumference of a diameter in centimeters. circle is given. Find each missing measure to Therefore, the maximum weight is 8.3 × 21.3 or the nearest hundredth. about 176.8 grams.

36. ANSWER: 176.8 g SOLUTION: 35. PATIOS Mr. Martinez is going to build the patio The radius is half the diameter and the shown. circumference C of a circle with diameter d is given a. What is the patio’s approximate circumference? by b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

SOLUTION: ANSWER: a. The radius of the patio is 3 + 2 = 5 ft. The 4.25 in.; 26.70 in. circumference C of a circle with radius r is given by

Use the formula to find the circumference. 37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given Therefore, the circumference is about 31.42 feet. by b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the radius of the inner circle is about 4 feet. So, the diameter is 22.80 feet and the circumference ANSWER: is about 71.63 feet. a. 31.42 ft ANSWER: b. 4 ft 22.80 ft; 71.63 ft The radius, diameter, or circumference of a 38. circle is given. Find each missing measure to the nearest hundredth. SOLUTION:

36. The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 35x cm. Use the formula to find the

The radius is half the diameter and the diameter.Then find the radius. circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters Therefore, the radius is 4.25 inches and the and the radius is about 5.57x centimeters. circumference is about 26.70 inches. ANSWER: ANSWER: 11.14x cm; 5.57x cm 4.25 in.; 26.70 in.

39. 37. SOLUTION: SOLUTION: The diameter is twice the radius and the The diameter is twice the radius and the circumference C of a circle with diameter d is given circumference C of a circle with diameter d is given by by

So, the diameter is 22.80 feet and the circumference So, the diameter is 0.25x units and the circumference

is about 71.63 feet. is about 0.79x units.

ANSWER: ANSWER: 22.80 ft; 71.63 ft 0.25x; 0.79x Determine whether the circles in the figures 38. below appear to be congruent, concentric, or SOLUTION: neither. The circumference C of a circle with diameter d is 40. Refer to the photo on page 703. given by SOLUTION: Here, C = 35x cm. Use the formula to find the The circles have the same center and they are in the diameter.Then find the radius. same plane. So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: Therefore, the diameter is about 11.14x centimeters neither and the radius is about 5.57x centimeters. 42. Refer to the photo on page 703. ANSWER: 11.14x cm; 5.57x cm SOLUTION: The circles appear to have the same radius. So, they appear to be congruent. 39. ANSWER: SOLUTION: congruent The diameter is twice the radius and the circumference C of a circle with diameter d is given 43. HISTORY The Indian Shell Ring on Hilton Head by Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or SOLUTION: neither. The radius of the circle is 3 units, that is 75 feet. So, 40. Refer to the photo on page 703. the diameter is 150 ft. The circumference C of a circle with diameter d is given by SOLUTION: So, the circumference is The circles have the same center and they are in the Therefore, someone have to same plane. So, they are concentric circles. walk about 471.2 feet to go completely around the ANSWER: ring. concentric ANSWER: 471.2 ft 41. Refer to the photo on page 703. SOLUTION: 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a The circles neither have the same radius nor are they circumference of 68 feet. The outer edge of the path concentric. is going to be 4 feet from the pond all the way ANSWER: around. What is the approximate circumference of the path? Round to the nearest hundredth. neither SOLUTION: 42. Refer to the photo on page 703. The circumference C of a circle with radius r is SOLUTION: given by Here, Cpond = 68 ft. Use the The circles appear to have the same radius. So, they formula to find the radius of the pond. appear to be congruent.

ANSWER: congruent

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the So, the radius of the pond is about 10.82 feet and the coordinate grid represents 25 feet, how far would radius to the outer edge of the path will be 10.82 + 4 someone have to walk to go completely around the = 14.82 ft. Use the radius to find the circumference ring? Round to the nearest tenth. of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. SOLUTION: a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to The radius of the circle is 3 units, that is 75 feet. So, the next larger circle is 1 : 2. the diameter is 150 ft. The circumference C of a b. TABULAR Calculate the radius (to the nearest circle with diameter d is given by tenth) and circumference (to the nearest hundredth) So, the circumference is of each circle. Record your results in a table. Therefore, someone have to c. VERBAL Explain why these three circles are walk about 471.2 feet to go completely around the geometrically similar. ring. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ANSWER: ratio between their radii is 2. 471.2 ft e. ANALYTICAL The scale factor from to 44. CCSS MODELING A brick path is being installed is . Write an equation relating the around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path circumference (CA) of to the circumference is going to be 4 feet from the pond all the way (C ) of . around. What is the approximate circumference of B the path? Round to the nearest hundredth. f. NUMERICAL If the scale factor from to SOLUTION: is , and the circumference of is 12 The circumference C of a circle with radius r is inches, what is the circumference of ? given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. SOLUTION: a. Sample answer:

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference b. Sample answer. of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: c. They all have the same shape–circular. 93.13 ft d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. 45. MULTIPLE REPRESENTATIONS In this So, if the radii are in the ratio b : a, so will the problem, you will explore changing dimensions in circumference. Therefore, . circles. a. GEOMETRIC Use a compass to draw three f. Use the formula from part e. Here, the circles in which the scale factor from each circle to circumference of is 12 inches and . the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. Therefore, the circumference of is 4 inches. d. VERBAL Make a conjecture about the ratio ANSWER: between the circumferences of two circles when the ratio between their radii is 2. a. Sample answer: e. ANALYTICAL The scale factor from to is . Write an equation relating the

circumference (CA) of to the circumference 10-1( CirclesC ) of and. Circumference B f. NUMERICAL If the scale factor from to b. Sample answer. is , and the circumference of is 12 inches, what is the circumference of ? SOLUTION: a. Sample answer:

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

b. Sample answer. 46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii.

So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . a. Drop the needle onto the paper. When the needle f. Use the formula from part e. Here, the lands, record whether it touches one of the lines as a circumference of is 12 inches and . hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number Therefore, the circumference of is 4 inches. of drops to the number of hits after 25, 50, and 100 drops. ANSWER: a. Sample answer: c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. b. Sample answer.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. eSolutions Manual - Powered by Cognero Page 12 c. They all have the same shape—circular. d. The ratio of their circumferences is also 2. 47. M APS The concentric circles on the map below

e. show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center

circle? b. As the radii of the circles increases by 5 miles, by a. Drop the needle onto the paper. When the needle how much does the circumference increase? lands, record whether it touches one of the lines as a SOLUTION: hit. Record the number of hits after 25, 50, and 100 a. drops. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the

circumference of each circle.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? The difference of the circumferences of the SOLUTION: outermost circle and the center circle is 60π - 10π = a. See students’ work. 50π or about 157.1 miles. b. See students’ work. b. Let the radius of one circle be x miles and the c. Sample answer: The values are approaching 3.14, radius of the next larger circle be x + 5 miles. Find which is approximately equal to π. the circumference of each circle.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

The difference of the two circumferences is (2xπ + 47. M APS The concentric circles on the map below 10π) - (2xπ) = 10π or about 31.4 miles. show the areas that are 5, 10, 15, 20, 25, and 30 Therefore, as the radius of each circle increases by 5 miles from downtown Phoenix. miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: a. How much greater is the circumference of the Sample answer: A line and a circle may intersect in outermost circle than the circumference of the center one point, two points, or may not intersect at all. A circle? line that intersects a circle in one point can be b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. SOLUTION: A line segment with endpoints on a circle can be a. The radius of the outermost circle is 30 miles and described as a chord. If the chord passes through that of the center circle is 5 miles. Find the the center of the circle, it can be described as a circumference of each circle. diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be The difference of the circumferences of the described as a secant. A line segment with outermost circle and the center circle is 60π - 10π = endpoints on a circle can be described as a chord. 50π or about 157.1 miles. If the chord passes through the center of the circle, b. Let the radius of one circle be x miles and the it can be described as a diameter. A line segment radius of the next larger circle be x + 5 miles. Find with endpoints at the center and on the circle can be the circumference of each circle. described as a radius.

49. REASONING In the figure, a circle with radius r is The difference of the two circumferences is (2xπ + inscribed in a regular polygon and circumscribed 10π) - (2xπ) = 10π or about 31.4 miles. about another. Therefore, as the radius of each circle increases by 5 a. What are the perimeters of the circumscribed and miles, the circumference increases by about 31.4 inscribed polygons in terms of r? Explain. miles. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? ANSWER: the inscribed polygon? Write a compound inequality a. 157.1 mi comparing C to these perimeters. b. by about 31.4 mi c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. 48. WRITING IN MATH How can we describe the d. As the number of sides of both the circumscribed relationships that exist between and inscribed polygons increase, what will happen to circles and lines? the upper and lower limits of the inequality from part c, and what does this imply? SOLUTION: Sample answer: A line and a circle may intersect in SOLUTION: one point, two points, or may not intersect at all. A a. The circumscribed polygon is a square with sides line that intersects a circle in one point can be of length 2r. described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through The inscribed polygon is a hexagon with sides of the center of the circle, it can be described as a length r. (Each triangle formed using a central angle diameter. A line segment with endpoints at the will be equilateral.) center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord.

If the chord passes through the center of the circle, So, the perimeters are 8r and 6r. it can be described as a diameter. A line segment b. with endpoints at the center and on the circle can be described as a radius.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r 49. c. Since the diameter is twice the radius, 8r = 4(2r) REASONING In the figure, a circle with radius r is or 4d and 6r = 3(2r) or 3d. inscribed in a regular polygon and circumscribed The inequality is then 4d < C < 3d. Therefore, the

about another. circumference of the circle is between 3 and 4 times a. What are the perimeters of the circumscribed and its diameter. inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or d. These limits will approach a value of πd, implying less than the perimeter of the circumscribed polygon? that C = πd. the inscribed polygon? Write a compound inequality comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the ANSWER: diameter d of the circle and interpret its meaning. a. 8r and 6r ; Twice the radius of the circle, 2r is d. As the number of sides of both the circumscribed the side length of the square, so the perimeter of the and inscribed polygons increase, what will happen to square is 4(2r) or 8r. The the upper and lower limits of the inequality from part regular hexagon is made up of six equilateral c, and what does this imply? triangles with side length r, so the perimeter of the SOLUTION: hexagon is 6(r) or 6r. a. The circumscribed polygon is a square with sides b. less; greater; 6r < C < 8r of length 2r. c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

The inscribed polygon is a hexagon with sides of 50. CHALLENGE The sum of the circumferences of length r. (Each triangle formed using a central angle circles H, J, and K shown at the right is units. will be equilateral.) Find KJ.

SOLUTION: So, the perimeters are 8r and 6r. The radius of is 4x units, the radius of is x b. units, and the radius of is 2x units. Find the circumference of each circle.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying The sum of the circumferences is 56π units. that C = πd. Substitute the three circumferences and solve for x.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the So, the radius of is 4 units, the radius of is square is 4(2r) or 8r. The 2(4) or 8 units, and the radius of is 4(4) or 16 regular hexagon is made up of six equilateral units. triangles with side length r, so the perimeter of the The measure of KJ is equal to the sum of the radii hexagon is 6(r) or 6r. of and . Therefore, KJ = 16 + 8 or 24 units. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is ANSWER: between 3 and 4 times its diameter. 24 units d. These limits will approach a value of πd, implying that C = πd. 51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, 50. CHALLENGE The sum of the circumferences of always, or never less than the radius of the circle? circles H, J, and K shown at the right is units. Explain. Find KJ. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A SOLUTION: segment drawn from the center to a point inside the circle will always have a length less than the radius The radius of is 4x units, the radius of is x of the circle. units, and the radius of is 2x units. Find the circumference of each circle. 52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION:

A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. The sum of the circumferences is 56π units. For some value k, r = Substitute the three circumferences and solve for x. 2 kr1.

So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps of and . onto , the circles are Therefore, KJ = 16 + 8 or 24 units. similar. Thus, all circles are similar. ANSWER:

24 units ANSWER: 51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, A circle is a locus of points in a plane equidistant always, or never less than the radius of the circle? from a given point. For any two circles and Explain. , there exists a translation that maps center A onto center B, moving so that it is concentric SOLUTION: with . There also exists a dilation with scale A radius is a segment drawn between the center of factor k such that each point that makes up is the circle and a point on the circle. A segment drawn moved to be the same distance from center A as the from the center to a point inside the circle will always have a length less than the radius of the circle. points that make up are from center B. Therefore, is mapped onto . Since there ANSWER: exists a rigid motion followed by a scaling that maps Always; a radius is a segment drawn between the onto , the circles are similar. Thus, all center of the circle and a point on the circle. A circles are similar. segment drawn from the center to a point inside the circle will always have a length less than the radius 53. CHALLENGE In the figure, is inscribed in of the circle. equilateral triangle LMN. What is the circumference of ? 52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation SOLUTION: with scale factor k such that each point that makes up is moved to be the Draw a perpendicular to the side Also join same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

Since is equilateral triangle, MN = 8 inches

and RN = 4 inches. Also, bisects Therefore, is mapped onto . Since there Use the definition of exists a rigid motion followed by a scaling that maps tangent of an angle to find PR, which is the radius of onto , the circles are similar. Thus, all circles are similar.

ANSWER: A circle is a locus of points in a plane equidistant

from a given point. For any two circles and Therefore, the radius of is , there exists a translation that maps center A The circumference C of a circle with radius r is onto center B, moving so that it is concentric given by with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps

onto , the circles are similar. Thus, all The circumference of is inches. circles are similar. ANSWER: 53. CHALLENGE In the figure, is inscribed in

equilateral triangle LMN. What is the circumference in. of ?

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

SOLUTION: ANSWER: See students’ work. Draw a perpendicular to the side Also join 55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects SOLUTION: Use the definition of Use the Pythagorean Theorem to find the diameter

tangent of an angle to find PR, which is the radius of of the circle.

The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of is

The circumference C of a circle with radius r is given by

ANSWER: 40.8 56. What is the radius of a table with a circumference of 10 feet? A The circumference of is inches. 1.6 ft B 2.5 ft ANSWER: C 3.2 ft D 5 ft in. SOLUTION: The circumference C of a circle with radius r is 54. WRITING IN MATH Research and write about given by the history of pi and its importance to the study of Here, C = 10 ft. Use the formula to find the radius. geometry. SOLUTION: See students’ work.

ANSWER: See students’ work. The radius of the circle is about 1.6 feet, so the correct choice is A. 55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden?

F 10 SOLUTION: G 9 Use the Pythagorean Theorem to find the diameter H 8 of the circle. J 7

SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with radius r is given by given by Use the formula to find the circumference. Because 50 feet of fence can be used to border the

garden, Cmax = 50. Use the circumference formula to find the maximum radius.

ANSWER: 40.8 Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only 56. What is the radius of a table with a circumference of correct choice is J. 10 feet? ANSWER: A 1.6 ft J B 2.5 ft C 3.2 ft 58. SAT/ACT What is the radius of a circle with an D 5 ft area of square units? SOLUTION: The circumference C of a circle with radius r is A 0.4 units given by B 0.5 units Here, C = 10 ft. Use the formula to find the radius. C 2 units D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the

The radius of the circle is about 1.6 feet, so the formula Here A = square units. Use the correct choice is A. formula to find the radius.

ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable

garden with a fence around the border. If he can use Therefore, the correct choice is B. up to 50 feet of fence, what radius can he use for the garden? ANSWER: F 10 B G 9 H 8 Copy each figure and point B. Then use a ruler J 7 to draw the image of the figure under a dilation with center B and the scale factor r indicated. SOLUTION: The circumference C of a circle with radius r is 59. given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only that distance from B. Repeat the process for rays correct choice is J. through the other five vertices. Connect the six points on the rays to construct the dilated image of the ANSWER: hexagon. J

58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units D 4 units E 16 units SOLUTION: ANSWER: The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

60.

Therefore, the correct choice is B.

ANSWER: SOLUTION: B Draw a ray from point B through a vertex of the Copy each figure and point B. Then use a ruler triangle. Use the ruler to measure the distance from to draw the image of the figure under a dilation B to the vertex. Mark a point on the ray that is of with center B and the scale factor r indicated. that distance from B. Repeat the process for rays

59. through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the ANSWER: hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points 61. on the rays to construct the dilated image of the hexagon.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

60. ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from 62. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the ANSWER: three points on the rays to construct the dilated image of the triangle.

61.

SOLUTION: Draw a ray from point B through a vertex of the

trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for ANSWER: rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER: State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any 62. rotation. So, it does not have a rotational symmetry.

ANSWER: No

SOLUTION: 64. Refer to the image on page 705. Draw a ray from point B through a vertex of the SOLUTION: triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 The figure can be mapped onto itself by a rotation of times that distance from B. Repeat the process for 90º. So, it has a rotational symmetry. So, the rays through the other two vertices. Connect the magnitude of symmetry is 90º and order of symmetry

three points on the rays to construct the dilated image is 4. of the triangle. ANSWER: Yes; 4, 90

65. ANSWER: SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. State whether each figure has rotational ANSWER: symmetry. Write yes or no. If so, copy the No figure, locate the center of symmetry, and state the order and magnitude of symmetry. Determine the truth value of the following 63. Refer to the image on page 705. statement for each set of conditions. Explain your reasoning. SOLUTION: If you are over 18 years old, then you vote in all The figure cannot be mapped onto itself by any elections. rotation. So, it does not have a rotational symmetry. 67. You are 19 years old and you vote. ANSWER: SOLUTION: No True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. 64. Refer to the image on page 705. SOLUTION: ANSWER: The figure can be mapped onto itself by a rotation of True; sample answer: Since the hypothesis is true 90º. So, it has a rotational symmetry. So, the and the conclusion is true, then the statement is true magnitude of symmetry is 90º and order of symmetry for the conditions. is 4. 68. You are 21 years old and do not vote. ANSWER: Yes; 4, 90 SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x. 65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: 69. No SOLUTION: 66. Refer to the image on page 705. The sum of the measures of angles around a point is SOLUTION: 360º.

The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No ANSWER: 90 Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: 70. True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true for the conditions. The angles with the measures xº and 2xº form a straight line. So, the sum is 180. ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. ANSWER: 68. You are 21 years old and do not vote. 60 SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true 71. and the conclusion is false, then the statement is false SOLUTION: for the conditions. The angles with the measures 6xº and 3xº form a Find x. straight line. So, the sum is 180.

ANSWER: 20 69. SOLUTION: The sum of the measures of angles around a point is 360º.

72. SOLUTION: ANSWER: The sum of the three angles is 180. 90

ANSWER: 45 70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm For Exercises 1–4, refer to . 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

1. Name the circle. Therefore, the diameter is 26 ft. SOLUTION: ANSWER: The center of the circle is N. So, the circle is 26 ft

ANSWER: The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. 5. FG So, here are chords. b. A diameter of a circle is a chord that passes SOLUTION: through the center. Here, is a diameter. Since the diameter of B is 18 inches and A is 8 c. A radius is a segment with endpoints at the center inches, AG = 18 and AF = (8) or 4. and on the circle. Here, is radius.

ANSWER: a. Therefore, FG = 14 in. b. c. ANSWER: 14 in. 3. If CN = 8 centimeters, find DN. SOLUTION: 6. FB Here, are radii of the same circle. So, SOLUTION: they are equal in length. Therefore, DN = 8 cm. Since the diameter of B is 18 inches and A is 4

ANSWER: inches, AB = (18) or 9 and AF = (8) or 4. 8 cm 4. If EN = 13 feet, what is the diameter of the circle?

SOLUTION: Therefore, FB = 5 inches. Here, is a radius and the diameter is twice the radius. ANSWER: 5 in.

Therefore, the diameter is 26 ft. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the ANSWER: radius and circumference of the ride? Round to the 26 ft nearest hundredth, if necessary. SOLUTION: The diameters of , , and are 8 The radius is half the diameter. So, the radius of the inches, 18 inches, and 11 inches, respectively. circular ride is 22 feet. Find each measure.

Therefore, the circumference of the ride is about 138.23 feet.

5. FG ANSWER: SOLUTION: 22 ft; 138.23 ft Since the diameter of B is 18 inches and A is 8 8. CCSS MODELING The circumference of the inches, AG = 18 and AF = (8) or 4. circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB SOLUTION: SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning The diameter of the pool is about 17.98 feet and the of the lesson has a diameter of 44 feet. What are the radius of the pool is about 8.99 feet. radius and circumference of the ride? Round to the ANSWER: nearest hundredth, if necessary. 17.98 ft; 8.99 ft SOLUTION: The radius is half the diameter. So, the radius of the 9. SHORT RESPONSE The right triangle shown is circular ride is 22 feet. inscribed in . Find the exact circumference of .

Therefore, the circumference of the ride is about 138.23 feet. ANSWER: SOLUTION: 22 ft; 138.23 ft The diameter of the circle is the hypotenuse of the right triangle. 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π centimeters. SOLUTION: ANSWER:

For Exercises 10–13, refer to .

10. Name the center of the circle. The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. SOLUTION: R ANSWER: 17.98 ft; 8.99 ft ANSWER: R 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of 11. Identify a chord that is also a diameter. . SOLUTION: Two chords are shown: and . goes through the center, R, so is a diameter.

ANSWER: SOLUTION: The diameter of the circle is the hypotenuse of the right triangle. 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord. The diameter of the circle is 4 centimeters. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT?

SOLUTION: The circumference of the circle is 4π centimeters. Here, is a diameter and is a radius.

ANSWER:

For Exercises 10–13, refer to . Therefore, RT = 8.1 cm.

ANSWER: 8.1 cm

For Exercises 14–17, refer to .

10. Name the center of the circle. SOLUTION: R

ANSWER: R 14. Identify a chord that is not a diameter. SOLUTION: 11. Identify a chord that is also a diameter. The chords do not pass through the SOLUTION: center. So, they are not diameters.

Two chords are shown: and . goes ANSWER:

through the center, R, so is a diameter.

ANSWER: 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the 12. Is a radius? Explain. radius. . SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord. Therefore, the diameter of the circle is 28 inches.

ANSWER: ANSWER: No; it is a chord. 28 in.

13. If SU = 16.2 centimeters, what is RT? 16. Is ? Explain. SOLUTION: SOLUTION: Here, is a diameter and is a radius. All radii of a circle are congruent. Since are both radii of , .

ANSWER: Yes; they are both radii of .

Therefore, RT = 8.1 cm. 17. If DA = 7.4 centimeters, what is EF? ANSWER: SOLUTION: 8.1 cm Here, is a diameter and is a radius. For Exercises 14–17, refer to .

Therefore, EF = 3.7 cm.

ANSWER:

14. Identify a chord that is not a diameter. 3.7 cm SOLUTION: Circle J has a radius of 10 units, has a The chords do not pass through the radius of 8 units, and BC = 5.4 units. Find each center. So, they are not diameters. measure.

ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: 18. CK Here, is a radius and the diameter is twice the radius. . SOLUTION: Since the radius of K is 8 units, BK = 8.

Therefore, the diameter of the circle is 28 inches. Therefore, CK = 2.6 units. ANSWER: ANSWER: 28 in. 2.6 16. Is ? Explain. 19. AB SOLUTION: All radii of a circle are congruent. Since SOLUTION: are both radii of , . Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? Therefore, AB = 14.6 units. SOLUTION: ANSWER: Here, is a diameter and is a radius. 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8. Therefore, EF = 3.7 cm.

ANSWER:

3.7 cm Since circle J has a radius of 10 units, JC = 10. Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each

measure. Therefore, JK = 12.6 units.

ANSWER: 12.6 21. AD SOLUTION:

18. CK We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J SOLUTION: is 10 units, so KD = 8 and AC = 2(10) or 20. Before Since the radius of K is 8 units, BK = 8. finding AD, we need to find CK.

Therefore, CK = 2.6 units.

ANSWER: 2.6 Therefore, AD = 30.6 units.

19. AB ANSWER: 30.6 SOLUTION: Since is a diameter of circle J and the radius is 22. PIZZA Find the radius and circumference of the 10, AC = 2(10) or 20 units. pizza shown. Round to the nearest hundredth, if necessary.

Therefore, AB = 14.6 units.

ANSWER: 14.6 20. JK

SOLUTION: SOLUTION: First find CK. Since circle K has a radius of 8 units, The diameter of the pizza is 16 inches. BK = 8.

Since circle J has a radius of 10 units, JC = 10. . So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is Therefore, JK = 12.6 units. given by

ANSWER: 12.6

21. AD Therefore, the circumference of the pizza is about 50.27 inches. SOLUTION: We can find AD using AD = AC + CK + KD. The ANSWER: radius of circle K is 8 units and the radius of circle J 8 in.; 50.27 in. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter.

Therefore, AD = 30.6 units.

ANSWER: 30.6

22. PIZZA Find the radius and circumference of the So, the radius of the tires is 13 inches. pizza shown. Round to the nearest hundredth, if The circumference C of a circle with diameter d is necessary. given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. SOLUTION: Find the diameter and radius of a circle by with The diameter of the pizza is 16 inches. the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is . given by So, the radius of the pizza is 8 inches. Here, C = 18 in. Use the formula to find the The circumference C of a circle of diameter d is diameter. Then find the radius. given by

Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a Therefore, the diameter is about 5.73 inches and the tire. Round to the nearest hundredth, if necessary. radius is about 2.86 inches. SOLUTION: ANSWER: The diameter of a tire is 26 inches. The radius is half 5.73 in.; 2.86 in. the diameter. 25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is given by So, the radius of the tires is 13 inches. Here, C = 124 ft. Use the formula to find the The circumference C of a circle with diameter d is diameter. Then find the radius. given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest Therefore, the diameter is about 39.47 feet and the hundredth. radius is about 19.74 feet. 24. C = 18 in. ANSWER: SOLUTION: 39.47 ft; 19.74 ft The circumference C of a circle with diameter d is given by 26. C = 375.3 cm Here, C = 18 in. Use the formula to find the SOLUTION: diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: 5.73 in.; 2.86 in. Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

25. C = 124 ft ANSWER: 119.46 cm; 59.73 cm SOLUTION: The circumference C of a circle with diameter d is 27. C = 2608.25 m given by Here, C = 124 ft. Use the formula to find the SOLUTION: diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the

radius is about 19.74 feet. ANSWER: Therefore, the diameter is about 830.23 meters and

39.47 ft; 19.74 ft the radius is about 415.12 meters. ANSWER: 26. C = 375.3 cm 830.23 m; 415.12 m SOLUTION: The circumference C of a circle with diameter d is CCSS SENSE-MAKING Find the exact circumference of each circle by using the given given by inscribed or Here, C = 375.3 cm. Use the formula to find the circumscribed polygon. diameter. Then find the radius.

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: 119.46 cm; 59.73 cm The diameter of the circle is 17 cm. The 27. C = 2608.25 m circumference C of a circle is given by SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find Therefore, the circumference of the circle is 17 the diameter. Then find the radius. centimeters. ANSWER:

Therefore, the diameter is about 830.23 meters and 29. the radius is about 415.12 meters. SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the 830.23 m; 415.12 m diameter. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon. The diameter of the circle is 12 feet. The circumference C of a circle is given by

28. Therefore, the circumference of the circle is 12 feet. SOLUTION: The hypotenuse of the right triangle is a diameter of ANSWER: the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The 30. circumference C of a circle is given by SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter Therefore, the circumference of the circle is 17 of the circle. centimeters.

ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by

29. Therefore, the circumference of the circle is SOLUTION: inches. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the ANSWER: diameter.

The diameter of the circle is 12 feet. The circumference C of a circle is given by 31. SOLUTION: Each diagonal of the inscribed rectangle will pass Therefore, the circumference of the circle is 12 through the origin, so it is a diameter of the circle. feet. Use the Pythagorean Theorem to find the diameter of the circle. ANSWER:

The diameter of the circle is 10 inches. The circumference C of a circle is given by

30.

SOLUTION: Each diagonal of the inscribed rectangle will pass Therefore, the circumference of the circle is 10 through the origin, so it is a diameter of the circle. inches. Use the Pythagorean Theorem to find the diameter of the circle. ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d Therefore, the circumference of the circle is = 25 millimeters. inches. The circumference C of a circle is given by

ANSWER:

Therefore, the circumference of the circle is 25 millimeters.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d The diameter of the circle is 10 inches. The = 14 yards. circumference C of a circle is given by The circumference C of a circle is given by

Therefore, the circumference of the circle is 10 Therefore, the circumference of the circle is 14 inches. yards. ANSWER: ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in 32. centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? SOLUTION: Round to the nearest tenth. A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d SOLUTION: = 25 millimeters. The circumference C of a circle with diameter d is The circumference C of a circle is given by given by Here, C = 66.92 cm. Use the formula to find the diameter.

Therefore, the circumference of the circle is 25 millimeters.

ANSWER: The diameter of the disc is about 21.3 centimeters

and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g

33. 35. PATIOS Mr. Martinez is going to build the patio shown. SOLUTION: a. What is the patio’s approximate circumference? A diameter perpendicular to a side of the square and b. If Mr. Martinez changes the plans so that the the length of each side will measure the same. So, d inner circle has a circumference of approximately 25 = 14 yards. feet, what should the radius of the circle be to the The circumference C of a circle is given by nearest foot?

Therefore, the circumference of the circle is 14 yards.

ANSWER: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The 34. DISC GOLF Disc golf is similar to regular golf, circumference C of a circle with radius r is given by except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum Use the formula to find the circumference. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. Therefore, the circumference is about 31.42 feet. SOLUTION: b. Here, Cinner = 25 ft. Use the formula to find the The circumference C of a circle with diameter d is radius of the inner circle. given by Here, C = 66.92 cm. Use the formula to find the diameter.

So, the radius of the inner circle is about 4 feet.

ANSWER: The diameter of the disc is about 21.3 centimeters a. 31.42 ft and the maximum weight allowed is 8.3 times the b. 4 ft diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or The radius, diameter, or circumference of a about 176.8 grams. circle is given. Find each missing measure to the nearest hundredth. ANSWER:

176.8 g 36.

35. PATIOS Mr. Martinez is going to build the patio SOLUTION:

shown. The radius is half the diameter and the a. What is the patio’s approximate circumference? circumference C of a circle with diameter d is given b. If Mr. Martinez changes the plans so that the by inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the SOLUTION: circumference is about 26.70 inches. a. The radius of the patio is 3 + 2 = 5 ft. The ANSWER: circumference C of a circle with radius r is given by 4.25 in.; 26.70 in.

Use the formula to find the circumference.

37.

SOLUTION: Therefore, the circumference is about 31.42 feet. The diameter is twice the radius and the b. Here, Cinner = 25 ft. Use the formula to find the circumference C of a circle with diameter d is given radius of the inner circle. by

So, the radius of the inner circle is about 4 feet.

ANSWER: So, the diameter is 22.80 feet and the circumference a. 31.42 ft is about 71.63 feet. b. 4 ft ANSWER: The radius, diameter, or circumference of a 22.80 ft; 71.63 ft circle is given. Find each missing measure to the nearest hundredth. 38.

36. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The radius is half the diameter and the Here, C = 35x cm. Use the formula to find the circumference C of a circle with diameter d is given diameter.Then find the radius. by

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 4.25 in.; 26.70 in. ANSWER: 11.14x cm; 5.57x cm

37. 39. SOLUTION: SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given The diameter is twice the radius and the by circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet. So, the diameter is 0.25x units and the circumference

ANSWER: is about 0.79x units. 22.80 ft; 71.63 ft ANSWER: 0.25x; 0.79x 38. Determine whether the circles in the figures SOLUTION: below appear to be congruent, concentric, or The circumference C of a circle with diameter d is neither. given by 40. Refer to the photo on page 703. Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric. Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: neither ANSWER: 11.14x cm; 5.57x cm 42. Refer to the photo on page 703. SOLUTION: 39. The circles appear to have the same radius. So, they appear to be congruent. SOLUTION: ANSWER: The diameter is twice the radius and the circumference C of a circle with diameter d is given congruent by 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or neither. SOLUTION: 40. Refer to the photo on page 703. The radius of the circle is 3 units, that is 75 feet. So, SOLUTION: the diameter is 150 ft. The circumference C of a The circles have the same center and they are in the circle with diameter d is given by same plane. So, they are concentric circles. So, the circumference is Therefore, someone have to ANSWER: concentric walk about 471.2 feet to go completely around the ring.

41. Refer to the photo on page 703. ANSWER: 471.2 ft SOLUTION: The circles neither have the same radius nor are they 44. CCSS MODELING A brick path is being installed concentric. around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path ANSWER: is going to be 4 feet from the pond all the way neither around. What is the approximate circumference of the path? Round to the nearest hundredth. 42. Refer to the photo on page 703. SOLUTION: SOLUTION: The circumference C of a circle with radius r is The circles appear to have the same radius. So, they given by Here, C = 68 ft. Use the appear to be congruent. pond formula to find the radius of the pond. ANSWER: congruent

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the So, the radius of the pond is about 10.82 feet and the ring? Round to the nearest tenth. radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in SOLUTION: circles. The radius of the circle is 3 units, that is 75 feet. So, a. GEOMETRIC Use a compass to draw three the diameter is 150 ft. The circumference C of a circles in which the scale factor from each circle to circle with diameter d is given by the next larger circle is 1 : 2. So, the circumference is b. TABULAR Calculate the radius (to the nearest Therefore, someone have to tenth) and circumference (to the nearest hundredth) walk about 471.2 feet to go completely around the of each circle. Record your results in a table. ring. c. VERBAL Explain why these three circles are geometrically similar. ANSWER: d. VERBAL Make a conjecture about the ratio 471.2 ft between the circumferences of two circles when the ratio between their radii is 2. 44. CCSS MODELING A brick path is being installed e. ANALYTICAL The scale factor from to around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path is . Write an equation relating the is going to be 4 feet from the pond all the way around. What is the approximate circumference of circumference (CA) of to the circumference

the path? Round to the nearest hundredth. (CB) of . SOLUTION: f. NUMERICAL If the scale factor from to The circumference C of a circle with radius r is is , and the circumference of is 12 given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. inches, what is the circumference of ? SOLUTION: a. Sample answer:

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path. b. Sample answer.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER:

93.13 ft c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. 45. MULTIPLE REPRESENTATIONS In this e. The circumference is directly proportional to radii. problem, you will explore changing dimensions in So, if the radii are in the ratio b : a, so will the circles.

a. GEOMETRIC Use a compass to draw three circumference. Therefore, . circles in which the scale factor from each circle to the next larger circle is 1 : 2. f. Use the formula from part e. Here, the b. TABULAR Calculate the radius (to the nearest circumference of is 12 inches and . tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio Therefore, the circumference of is 4 inches. between the circumferences of two circles when the ratio between their radii is 2. ANSWER: e. ANALYTICAL The scale factor from to a. Sample answer: is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to

is , and the circumference of is 12 b. Sample answer. inches, what is the circumference of ? SOLUTION: a. Sample answer:

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e.

b. Sample answer. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . f. Use the formula from part e. Here, the a. Drop the needle onto the paper. When the needle circumference of is 12 inches and . lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

Therefore, the circumference of is 4 inches. b. Calculate the ratio of two times the total number ANSWER: of drops to the number of hits after 25, 50, and 100 a. Sample answer: drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. b. Sample answer. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, c. They all have the same shape—circular. which is approximately equal to π. d. The ratio of their circumferences is also 2.

e. 10-1 Circles and Circumference 47. M APS The concentric circles on the map below f. 4 in. show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. 46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the

outermost circle than the circumference of the center circle? a. Drop the needle onto the paper. When the needle b. As the radii of the circles increases by 5 miles, by lands, record whether it touches one of the lines as a how much does the circumference increase? hit. Record the number of hits after 25, 50, and 100 drops. SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle. b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. The difference of the circumferences of the b. See students’ work. outermost circle and the center circle is 60π - 10π = c. Sample answer: The values are approaching 3.14, 50π or about 157.1 miles. which is approximately equal to π. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below π + show the areas that are 5, 10, 15, 20, 25, and 30 The difference of the two circumferences is (2x miles from downtown Phoenix. 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi eSolutions Manual - Powered by Cognero 48. WRITING IN MATH How can we describePage the13 relationships that exist between circles and lines? a. How much greater is the circumference of the outermost circle than the circumference of the center SOLUTION: circle? Sample answer: A line and a circle may intersect in b. As the radii of the circles increases by 5 miles, by one point, two points, or may not intersect at all. A how much does the circumference increase? line that intersects a circle in one point can be SOLUTION: described as a tangent. A line that intersects a circle a. The radius of the outermost circle is 30 miles and in exactly two points can be described as a secant. that of the center circle is 5 miles. Find the A line segment with endpoints on a circle can be circumference of each circle. described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a

radius.

ANSWER: Sample answer: A line that intersects a circle in one The difference of the circumferences of the point can be described as a tangent. A line that outermost circle and the center circle is 60π - 10π = intersects a circle in exactly two points can be 50π or about 157.1 miles. described as a secant. A line segment with b. Let the radius of one circle be x miles and the endpoints on a circle can be described as a chord. radius of the next larger circle be x + 5 miles. Find If the chord passes through the center of the circle, the circumference of each circle. it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

The difference of the two circumferences is (2xπ + 49. 10π) - (2xπ) = 10π or about 31.4 miles. REASONING In the figure, a circle with radius r is Therefore, as the radius of each circle increases by 5 inscribed in a regular polygon and circumscribed miles, the circumference increases by about 31.4 about another. miles. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. ANSWER: b. Is the circumference C of the circle greater or a. 157.1 mi less than the perimeter of the circumscribed polygon? b. by about 31.4 mi the inscribed polygon? Write a compound inequality comparing C to these perimeters. 48. WRITING IN MATH How can we describe the c. Rewrite the inequality from part b in terms of the relationships that exist between diameter d of the circle and interpret its meaning. circles and lines? d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to SOLUTION: the upper and lower limits of the inequality from part Sample answer: A line and a circle may intersect in c, and what does this imply? one point, two points, or may not intersect at all. A SOLUTION: line that intersects a circle in one point can be a. The circumscribed polygon is a square with sides described as a tangent. A line that intersects a circle of length 2r. in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the The inscribed polygon is a hexagon with sides of center and on the circle can be described as a length r. (Each triangle formed using a central angle radius. will be equilateral.)

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord.

If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be So, the perimeters are 8r and 6r. described as a radius. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the 49. perimeter of the inscribed polygon. REASONING In the figure, a circle with radius r is 8r < C < 6r inscribed in a regular polygon and circumscribed c. Since the diameter is twice the radius, 8r = 4(2r) about another. or 4d and 6r = 3(2r) or 3d. a. What are the perimeters of the circumscribed and The inequality is then 4d < C < 3d. Therefore, the inscribed polygons in terms of r? Explain. circumference of the circle is between 3 and 4 times b. Is the circumference C of the circle greater or its diameter. less than the perimeter of the circumscribed polygon? d. These limits will approach a value of πd, implying the inscribed polygon? Write a compound inequality that C = πd. comparing C to these perimeters.

c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. ANSWER: d. As the number of sides of both the circumscribed a. 8r and 6r ; Twice the radius of the circle, 2r is and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part the side length of the square, so the perimeter of the c, and what does this imply? square is 4(2r) or 8r. The regular hexagon is made up of six equilateral SOLUTION: triangles with side length r, so the perimeter of the a. The circumscribed polygon is a square with sides hexagon is 6(r) or 6r. of length 2r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying The inscribed polygon is a hexagon with sides of that C = πd. length r. (Each triangle formed using a central angle will be equilateral.) 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ.

So, the perimeters are 8r and 6r. SOLUTION: b. The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle. The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x. ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The So, the radius of is 4 units, the radius of is regular hexagon is made up of six equilateral 2(4) or 8 units, and the radius of is 4(4) or 16 triangles with side length r, so the perimeter of the units. hexagon is 6(r) or 6r. The measure of KJ is equal to the sum of the radii b. less; greater; 6r < C < 8r of and . c. 3d < C < 4d; The circumference of the circle is Therefore, KJ = 16 + 8 or 24 units. between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying ANSWER: that C = πd. 24 units

50. CHALLENGE The sum of the circumferences of 51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, circles H, J, and K shown at the right is units. always, or never less than the radius of the circle? Find KJ. Explain. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the SOLUTION: center of the circle and a point on the circle. A The radius of is 4x units, the radius of is x segment drawn from the center to a point inside the units, and the radius of is 2x units. Find the circle will always have a length less than the radius circumference of each circle. of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar.

SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the The sum of the circumferences is 56π units. same distance from center A as the points that make Substitute the three circumferences and solve for x. up are from center B. For some value k, r2 = kr1.

So, the radius of is 4 units, the radius of is

2(4) or 8 units, and the radius of is 4(4) or 16

units.

The measure of KJ is equal to the sum of the radii of and . Therefore, is mapped onto . Since there Therefore, KJ = 16 + 8 or 24 units. exists a rigid motion followed by a scaling that maps onto , the circles are ANSWER: similar. Thus, all circles are similar. 24 units

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, ANSWER: always, or never less than the radius of the circle? A circle is a locus of points in a plane equidistant

Explain. from a given point. For any two circles and SOLUTION: , there exists a translation that maps center A A radius is a segment drawn between the center of onto center B, moving so that it is concentric the circle and a point on the circle. A segment drawn with . There also exists a dilation with scale from the center to a point inside the circle will always factor k such that each point that makes up is have a length less than the radius of the circle. moved to be the same distance from center A as the ANSWER: points that make up are from center B. Always; a radius is a segment drawn between the Therefore, is mapped onto . Since there center of the circle and a point on the circle. A exists a rigid motion followed by a scaling that maps segment drawn from the center to a point inside the onto , the circles are similar. Thus, all circle will always have a length less than the radius circles are similar. of the circle. 53. CHALLENGE In the figure, is inscribed in 52. PROOF Use the locus definition of a circle and equilateral triangle LMN. What is the circumference dilations to prove that all circles are similar. of ? SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the SOLUTION: same distance from center A as the points that make Draw a perpendicular to the side Also join up are from center B.

For some value k, r2 = kr1.

Since is equilateral triangle, MN = 8 inches Therefore, is mapped onto . Since there and RN = 4 inches. Also, bisects exists a rigid motion followed by a scaling that maps onto , the circles are Use the definition of similar. Thus, all circles are similar. tangent of an angle to find PR, which is the radius of

ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric Therefore, the radius of is with . There also exists a dilation with scale The circumference C of a circle with radius r is factor k such that each point that makes up is given by moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

The circumference of is inches. 53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference ANSWER: of ? in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: SOLUTION: See students’ work. Draw a perpendicular to the side Also join ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects Use the definition of SOLUTION: tangent of an angle to find PR, which is the radius of Use the Pythagorean Theorem to find the diameter of the circle.

The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of is The circumference C of a circle with radius r is given by

ANSWER: 40.8

56. What is the radius of a table with a circumference of The circumference of is inches. 10 feet? A 1.6 ft ANSWER: B 2.5 ft C 3.2 ft in. D 5 ft SOLUTION: 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of The circumference C of a circle with radius r is geometry. given by Here, C = 10 ft. Use the formula to find the radius. SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the The radius of the circle is about 1.6 feet, so the

circumference of ? Round to the nearest tenth. correct choice is A. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the SOLUTION: garden? Use the Pythagorean Theorem to find the diameter F 10 of the circle. G 9 H 8 J 7 The circumference C of a circle with diameter d is given by SOLUTION: Use the formula to find the circumference. The circumference C of a circle with radius r is given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

ANSWER: 40.8

56. What is the radius of a table with a circumference of Therefore, the radius of the garden must be less than 10 feet? or equal to about 7.96 feet. Therefore, the only correct choice is J. A 1.6 ft B 2.5 ft ANSWER: C 3.2 ft J D 5 ft 58. SAT/ACT What is the radius of a circle with an SOLUTION: The circumference C of a circle with radius r is area of square units? given by A 0.4 units Here, C = 10 ft. Use the formula to find the radius. B 0.5 units C 2 units D 4 units E 16 units SOLUTION: The radius of the circle is about 1.6 feet, so the The area of a circle of radius r is given by the correct choice is A. formula Here A = square units. Use the ANSWER: formula to find the radius. A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? Therefore, the correct choice is B. F 10 G 9 ANSWER: H 8 B J 7 Copy each figure and point B. Then use a ruler SOLUTION: to draw the image of the figure under a dilation The circumference C of a circle with radius r is with center B and the scale factor r indicated.

given by 59. Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

SOLUTION: Draw a ray from point B through a vertex of the Therefore, the radius of the garden must be less than hexagon. Use the ruler to measure the distance from or equal to about 7.96 feet. Therefore, the only B to the vertex. Mark a point on the ray that is of

correct choice is J. that distance from B. Repeat the process for rays ANSWER: through the other five vertices. Connect the six points on the rays to construct the dilated image of the J hexagon. 58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units D 4 units E 16 units SOLUTION:

The area of a circle of radius r is given by the ANSWER: formula Here A = square units. Use the formula to find the radius.

60.

Therefore, the correct choice is B.

ANSWER:

B SOLUTION: Copy each figure and point B. Then use a ruler Draw a ray from point B through a vertex of the to draw the image of the figure under a dilation triangle. Use the ruler to measure the distance from with center B and the scale factor r indicated. B to the vertex. Mark a point on the ray that is of 59. that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the 61. hexagon.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid. ANSWER:

60.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from

B to the vertex. Mark a point on the ray that is of 62. that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 ANSWER: times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. 61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image ANSWER: of the trapezoid.

ANSWER: State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. 62. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER:

No SOLUTION: Draw a ray from point B through a vertex of the Refer to the image on page 705. triangle. Use the ruler to measure the distance from 64. B to the vertex. Mark a point on the ray that is 3 SOLUTION: times that distance from B. Repeat the process for The figure can be mapped onto itself by a rotation of rays through the other two vertices. Connect the 90º. So, it has a rotational symmetry. So, the three points on the rays to construct the dilated image magnitude of symmetry is 90º and order of symmetry of the triangle. is 4.

ANSWER: Yes; 4, 90

ANSWER: 65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705.

SOLUTION: The given figure cannot be mapped onto itself by any State whether each figure has rotational rotation. So, it does not have a rotational symmetry. symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state ANSWER: the order and magnitude of symmetry. No 63. Refer to the image on page 705. Determine the truth value of the following SOLUTION: statement for each set of conditions. Explain The figure cannot be mapped onto itself by any your reasoning. rotation. So, it does not have a rotational symmetry. If you are over 18 years old, then you vote in all elections. ANSWER: 67. You are 19 years old and you vote. No SOLUTION: True; sample answer: Since the hypothesis is true 64. Refer to the image on page 705. and the conclusion is true, then the statement is true SOLUTION: for the conditions. The figure can be mapped onto itself by a rotation of ANSWER: 90º. So, it has a rotational symmetry. So, the True; sample answer: Since the hypothesis is true magnitude of symmetry is 90º and order of symmetry and the conclusion is true, then the statement is true

is 4. for the conditions. ANSWER: Yes; 4, 90 68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. 65. Find x. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No 69. 66. Refer to the image on page 705. SOLUTION: SOLUTION: The sum of the measures of angles around a point is The given figure cannot be mapped onto itself by any 360º. rotation. So, it does not have a rotational symmetry. ANSWER: No Determine the truth value of the following ANSWER: statement for each set of conditions. Explain 90 your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true

and the conclusion is true, then the statement is true 70. for the conditions. SOLUTION: ANSWER: The angles with the measures xº and 2xº form a straight line. So, the sum is 180. True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. ANSWER: SOLUTION: 60 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false 71. for the conditions. SOLUTION: Find x. The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

69. ANSWER: SOLUTION: 20 The sum of the measures of angles around a point is 360º.

ANSWER: 72. 90 SOLUTION: The sum of the three angles is 180.

ANSWER: 70. 45 SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

For Exercises 1–4, refer to . Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

5. FG 2. Identify each. SOLUTION: a. a chord Since the diameter of B is 18 inches and A is 8 b. a diameter c. a radius inches, AG = 18 and AF = (8) or 4. SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes Therefore, FG = 14 in. through the center. Here, is a diameter. ANSWER: c. A radius is a segment with endpoints at the center 14 in. and on the circle. Here, is radius. 6. FB ANSWER: SOLUTION: a. Since the diameter of B is 18 inches and A is 4 b. inches, AB = (18) or 9 and AF = (8) or 4. c.

3. If CN = 8 centimeters, find DN.

SOLUTION: Here, are radii of the same circle. So, Therefore, FB = 5 inches. they are equal in length. Therefore, DN = 8 cm. ANSWER: ANSWER: 5 in. 8 cm 7. RIDES The circular ride described at the beginning 4. If EN = 13 feet, what is the diameter of the circle? of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the SOLUTION: nearest hundredth, if necessary. Here, is a radius and the diameter is twice the SOLUTION: radius. The radius is half the diameter. So, the radius of the circular ride is 22 feet.

Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 Therefore, the circumference of the ride is about inches, 18 inches, and 11 inches, respectively. 138.23 feet. Find each measure. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

5. FG SOLUTION: Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4.

SOLUTION:

Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

ANSWER: Therefore, FB = 5 inches. 17.98 ft; 8.99 ft ANSWER: SHORT RESPONSE 5 in. 9. The right triangle shown is inscribed in . Find the exact circumference of 7. RIDES The circular ride described at the beginning . of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: The diameter of the circle is 4 centimeters. 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to .

SOLUTION:

10. Name the center of the circle. SOLUTION: R ANSWER: R

11. Identify a chord that is also a diameter. The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. SOLUTION: ANSWER: Two chords are shown: and . goes 17.98 ft; 8.99 ft through the center, R, so is a diameter.

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of ANSWER: .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord. SOLUTION: The diameter of the circle is the hypotenuse of the ANSWER: right triangle. No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: Here, is a diameter and is a radius. The diameter of the circle is 4 centimeters.

Therefore, RT = 8.1 cm. The circumference of the circle is 4π centimeters. ANSWER: 8.1 cm ANSWER: For Exercises 14–17, refer to .

For Exercises 10–13, refer to .

14. Identify a chord that is not a diameter. SOLUTION: 10. Name the center of the circle. The chords do not pass through the SOLUTION: center. So, they are not diameters. R ANSWER: ANSWER: R 15. If CF = 14 inches, what is the diameter of the circle? 11. Identify a chord that is also a diameter. SOLUTION: SOLUTION: Here, is a radius and the diameter is twice the Two chords are shown: and . goes radius. . through the center, R, so is a diameter.

ANSWER: Therefore, the diameter of the circle is 28 inches.

ANSWER: 12. Is a radius? Explain. 28 in. SOLUTION: 16. Is ? Explain. A radius is a segment with endpoints at the center and on the circle. But has both the end points on SOLUTION: the circle, so it is a chord. All radii of a circle are congruent. Since are both radii of , . ANSWER: No; it is a chord. ANSWER: Yes; they are both radii of . 13. If SU = 16.2 centimeters, what is RT? SOLUTION: 17. If DA = 7.4 centimeters, what is EF? Here, is a diameter and is a radius. SOLUTION: Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm.

ANSWER: Therefore, EF = 3.7 cm. 8.1 cm ANSWER: 3.7 cm For Exercises 14–17, refer to . Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

14. Identify a chord that is not a diameter. SOLUTION: The chords do not pass through the center. So, they are not diameters. 18. CK ANSWER: SOLUTION: Since the radius of K is 8 units, BK = 8.

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Therefore, CK = 2.6 units. Here, is a radius and the diameter is twice the ANSWER: radius. . 2.6

19. AB

Therefore, the diameter of the circle is 28 inches. SOLUTION: Since is a diameter of circle J and the radius is ANSWER: 10, AC = 2(10) or 20 units. 28 in.

16. Is ? Explain. Therefore, AB = 14.6 units. SOLUTION: All radii of a circle are congruent. Since ANSWER: are both radii of , . 14.6

ANSWER: 20. JK Yes; they are both radii of . SOLUTION: First find CK. Since circle K has a radius of 8 units, 17. If DA = 7.4 centimeters, what is EF? BK = 8. SOLUTION: Here, is a diameter and is a radius.

Since circle J has a radius of 10 units, JC = 10.

Therefore, JK = 12.6 units. Therefore, EF = 3.7 cm. ANSWER: ANSWER: 12.6 3.7 cm 21. AD Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

18. CK

SOLUTION: Therefore, AD = 30.6 units. Since the radius of K is 8 units, BK = 8. ANSWER: 30.6

Therefore, CK = 2.6 units. 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if ANSWER: necessary. 2.6

19. AB SOLUTION: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

SOLUTION: Therefore, AB = 14.6 units. The diameter of the pizza is 16 inches. ANSWER: 14.6 20. JK . SOLUTION: So, the radius of the pizza is 8 inches. First find CK. Since circle K has a radius of 8 units, The circumference C of a circle of diameter d is BK = 8. given by

Since circle J has a radius of 10 units, JC = 10. Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: Therefore, JK = 12.6 units. 8 in.; 50.27 in. ANSWER: 23. BICYCLES A bicycle has tires with a diameter of 12.6 26 inches. Find the radius and circumference of a 21. AD tire. Round to the nearest hundredth, if necessary. SOLUTION: SOLUTION: We can find AD using AD = AC + CK + KD. The The diameter of a tire is 26 inches. The radius is half radius of circle K is 8 units and the radius of circle J the diameter. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by Therefore, AD = 30.6 units.

ANSWER:

30.6 Therefore, the circumference of the bicycle tire is about 81.68 inches. 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if ANSWER:

necessary. 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is

given by SOLUTION: Here, C = 18 in. Use the formula to find the The diameter of the pizza is 16 inches. diameter. Then find the radius.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the diameter is about 5.73 inches and the Therefore, the circumference of the pizza is about radius is about 2.86 inches. 50.27 inches. ANSWER: ANSWER: 5.73 in.; 2.86 in. 8 in.; 50.27 in. 25. C = 124 ft 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a SOLUTION: tire. Round to the nearest hundredth, if necessary. The circumference C of a circle with diameter d is SOLUTION: given by The diameter of a tire is 26 inches. The radius is half Here, C = 124 ft. Use the formula to find the

the diameter. diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the diameter is about 39.47 feet and the Therefore, the circumference of the bicycle tire is radius is about 19.74 feet. about 81.68 inches. ANSWER: ANSWER: 39.47 ft; 19.74 ft 13 in.; 81.68 in. 26. C = 375.3 cm Find the diameter and radius of a circle by with SOLUTION: the given circumference. Round to the nearest hundredth. The circumference C of a circle with diameter d is 24. C = 18 in. given by Here, C = 375.3 cm. Use the formula to find the SOLUTION: diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the diameter is about 5.73 inches and the 119.46 cm; 59.73 cm radius is about 2.86 inches. 27. C = 2608.25 m ANSWER: SOLUTION: 5.73 in.; 2.86 in. The circumference C of a circle with diameter d is given by 25. C = 124 ft Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m Therefore, the diameter is about 39.47 feet and the CCSS SENSE-MAKING Find the exact

radius is about 19.74 feet. circumference of each circle by using the given ANSWER: inscribed or circumscribed polygon. 39.47 ft; 19.74 ft

26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by 28. Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the circumference of the circle is 17 119.46 cm; 59.73 cm centimeters.

27. C = 2608.25 m ANSWER:

SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. The diameter of the circle is 12 feet. The ANSWER: circumference C of a circle is given by 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or Therefore, the circumference of the circle is 12 circumscribed polygon. feet.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of 30. the circle. Use the Pythagorean Theorem to find the SOLUTION: diameter. Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

The diameter of the circle is inches. The Therefore, the circumference of the circle is 17 circumference C of a circle is given by centimeters.

ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the 31. diameter. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter The diameter of the circle is 12 feet. The of the circle. circumference C of a circle is given by

Therefore, the circumference of the circle is 12 The diameter of the circle is 10 inches. The feet. circumference C of a circle is given by

ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER:

30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The diameter of the circle is inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 25 millimeters. Therefore, the circumference of the circle is inches. ANSWER:

ANSWER:

33. 31. SOLUTION: SOLUTION: A diameter perpendicular to a side of the square and Each diagonal of the inscribed rectangle will pass the length of each side will measure the same. So, d through the origin, so it is a diameter of the circle. = 14 yards. Use the Pythagorean Theorem to find the diameter The circumference C of a circle is given by of the circle.

Therefore, the circumference of the circle is 14

The diameter of the circle is 10 inches. The yards.

circumference C of a circle is given by ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, Therefore, the circumference of the circle is 10 except that a flying disc is used instead of a ball and inches. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in ANSWER: centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the 32. diameter. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters.

The circumference C of a circle is given by The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or Therefore, the circumference of the circle is 25 about 176.8 grams. millimeters. ANSWER: ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards.

The circumference C of a circle is given by SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by

Therefore, the circumference of the circle is 14 Use the formula to find the circumference. yards.

ANSWER:

Therefore, the circumference is about 31.42 feet.

34. DISC GOLF Disc golf is similar to regular golf, b. Here, Cinner = 25 ft. Use the formula to find the except that a flying disc is used instead of a ball and radius of the inner circle. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth.

SOLUTION: So, the radius of the inner circle is about 4 feet. The circumference C of a circle with diameter d is given by ANSWER: Here, C = 66.92 cm. Use the formula to find the a. 31.42 ft diameter. b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

36. The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the SOLUTION:

diameter in centimeters. The radius is half the diameter and the Therefore, the maximum weight is 8.3 × 21.3 or circumference C of a circle with diameter d is given about 176.8 grams. by ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25

feet, what should the radius of the circle be to the nearest foot? Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

37.

SOLUTION: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The The diameter is twice the radius and the circumference C of a circle with radius r is given by circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle. So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft

So, the radius of the inner circle is about 4 feet. 38. ANSWER: SOLUTION: a. 31.42 ft The circumference C of a circle with diameter d is b. 4 ft given by The radius, diameter, or circumference of a Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. circle is given. Find each missing measure to the nearest hundredth.

36.

SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters.

ANSWER: 11.14x cm; 5.57x cm

Therefore, the radius is 4.25 inches and the 39. circumference is about 26.70 inches. SOLUTION: ANSWER: The diameter is twice the radius and the 4.25 in.; 26.70 in. circumference C of a circle with diameter d is given by

37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures So, the diameter is 22.80 feet and the circumference below appear to be congruent, concentric, or is about 71.63 feet. neither. ANSWER: 40. Refer to the photo on page 703. 22.80 ft; 71.63 ft SOLUTION: The circles have the same center and they are in the 38. same plane. So, they are concentric circles.

SOLUTION: ANSWER: The circumference C of a circle with diameter d is concentric given by Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. 41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they Therefore, the diameter is about 11.14x centimeters appear to be congruent. and the radius is about 5.57x centimeters. ANSWER: ANSWER: congruent 11.14x cm; 5.57x cm 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the 39. coordinate grid represents 25 feet, how far would someone have to walk to go completely around the SOLUTION: ring? Round to the nearest tenth. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, ANSWER: the diameter is 150 ft. The circumference C of a 0.25x; 0.79x circle with diameter d is given by So, the circumference is Determine whether the circles in the figures Therefore, someone have to below appear to be congruent, concentric, or neither. walk about 471.2 feet to go completely around the ring. 40. Refer to the photo on page 703. SOLUTION: ANSWER: The circles have the same center and they are in the 471.2 ft same plane. So, they are concentric circles. 44. CCSS MODELING A brick path is being installed ANSWER: around a circular pond. The pond has a concentric circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way around. What is the approximate circumference of 41. Refer to the photo on page 703. the path? Round to the nearest hundredth. SOLUTION: SOLUTION: The circles neither have the same radius nor are they The circumference C of a circle with radius r is concentric. given by Here, Cpond = 68 ft. Use the ANSWER: formula to find the radius of the pond. neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they appear to be congruent. So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 ANSWER: = 14.82 ft. Use the radius to find the circumference congruent of the brick path.

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would Therefore, the circumference of the path is about someone have to walk to go completely around the 93.13 feet. ring? Round to the nearest tenth. ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. SOLUTION: c. VERBAL Explain why these three circles are

The radius of the circle is 3 units, that is 75 feet. So, geometrically similar. the diameter is 150 ft. The circumference C of a d. VERBAL Make a conjecture about the ratio circle with diameter d is given by between the circumferences of two circles when the

So, the circumference is ratio between their radii is 2. e. ANALYTICAL The scale factor from to Therefore, someone have to walk about 471.2 feet to go completely around the is . Write an equation relating the ring. circumference (CA) of to the circumference ANSWER: (CB) of . 471.2 ft f. NUMERICAL If the scale factor from to CCSS MODELING 44. A brick path is being installed is , and the circumference of is 12 around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path inches, what is the circumference of ? is going to be 4 feet from the pond all the way around. What is the approximate circumference of SOLUTION: the path? Round to the nearest hundredth. a. Sample answer: SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

b. Sample answer.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path. c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. Therefore, the circumference of the path is about So, if the radii are in the ratio b : a, so will the

93.13 feet. circumference. Therefore, . ANSWER: f. Use the formula from part e. Here, the 93.13 ft circumference of is 12 inches and . 45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to Therefore, the circumference of is 4 inches. the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest ANSWER: tenth) and circumference (to the nearest hundredth) a. Sample answer: of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2.

e. ANALYTICAL The scale factor from to b. Sample answer. is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to

is , and the circumference of is 12 c. They all have the same shape—circular. inches, what is the circumference of ? d. The ratio of their circumferences is also 2.

SOLUTION: e. a. Sample answer: f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

b. Sample answer.

a. Drop the needle onto the paper. When the needle c. They all have the same shape–circular. lands, record whether it touches one of the lines as a d. T he ratio of their circumferences is also 2. hit. Record the number of hits after 25, 50, and 100 e. The circumference is directly proportional to radii. drops. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, .

f. Use the formula from part e. Here, the b. Calculate the ratio of two times the total number circumference of is 12 inches and . of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π?

Therefore, the circumference of is 4 inches. SOLUTION: a. See students’ work. ANSWER: b. See students’ work. a. Sample answer: c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, b. Sample answer. which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a a. sheet of plain white paper. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? SOLUTION: a. The radius of the outermost circle is 30 miles and

that of the center circle is 5 miles. Find the circumference of each circle. a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number The difference of the circumferences of the of drops to the number of hits after 25, 50, and 100 outermost circle and the center circle is 60π - 10π = drops. 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the c. How are the values you found in part b related to radius of the next larger circle be x + 5 miles. Find π? the circumference of each circle. SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: The difference of the two circumferences is (2xπ + a. See students’ work. 10π) - (2xπ) = 10π or about 31.4 miles. b. See students’ work. Therefore, as the radius of each circle increases by 5 c. Sample answer: The values are approaching 3.14, miles, the circumference increases by about 31.4 which is approximately equal to π. miles.

ANSWER: 47. M APS The concentric circles on the map below a. 157.1 mi show the areas that are 5, 10, 15, 20, 25, and 30 b. by about 31.4 mi miles from downtown Phoenix. 48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. a. How much greater is the circumference of the A line segment with endpoints on a circle can be outermost circle than the circumference of the center described as a chord. If the chord passes through circle? the center of the circle, it can be described as a b. As the radii of the circles increases by 5 miles, by diameter. A line segment with endpoints at the how much does the circumference increase? center and on the circle can be described as a SOLUTION: radius. a. The radius of the outermost circle is 30 miles and ANSWER: that of the center circle is 5 miles. Find the Sample answer: A line that intersects a circle in one circumference of each circle. point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be The difference of the circumferences of the described as a radius. outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. 49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? The difference of the two circumferences is (2xπ + the inscribed polygon? Write a compound inequality 10π) - (2xπ) = 10π or about 31.4 miles. comparing C to these perimeters. Therefore, as the radius of each circle increases by 5 c. Rewrite the inequality from part b in terms of the miles, the circumference increases by about 31.4 diameter d of the circle and interpret its meaning. miles. d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to ANSWER: the upper and lower limits of the inequality from part 10-1a. Circles 157.1 mi and Circumference c, and what does this imply? b. by about 31.4 mi SOLUTION: 48. WRITING IN MATH How can we describe the a. The circumscribed polygon is a square with sides relationships that exist between of length 2r. circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A The inscribed polygon is a hexagon with sides of line that intersects a circle in one point can be length r. (Each triangle formed using a central angle will be equilateral.) described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius. ANSWER: Sample answer: A line that intersects a circle in one So, the perimeters are 8r and 6r. point can be described as a tangent. A line that b. intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. The circumference is less than the perimeter of the If the chord passes through the center of the circle, circumscribed polygon and greater than the it can be described as a diameter. A line segment perimeter of the inscribed polygon. with endpoints at the center and on the circle can be 8r < C < 6r described as a radius. c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. 49. d. These limits will approach a value of πd, implying REASONING In the figure, a circle with radius r is that C = πd. inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and ANSWER: inscribed polygons in terms of r? Explain. a. 8r and 6r ; Twice the radius of the circle, 2r is b. Is the circumference C of the circle greater or the side length of the square, so the perimeter of the less than the perimeter of the circumscribed polygon? square is 4(2r) or 8r. The the inscribed polygon? Write a compound inequality regular hexagon is made up of six equilateral comparing C to these perimeters. triangles with side length r, so the perimeter of the c. Rewrite the inequality from part b in terms of the hexagon is 6(r) or 6r. diameter d of the circle and interpret its meaning. b. less; greater; 6r < C < 8r d. As the number of sides of both the circumscribed c. 3d < C < 4d; The circumference of the circle is and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part between 3 and 4 times its diameter. c, and what does this imply? d. These limits will approach a value of πd, implying that C = πd. SOLUTION: eSolutionsa. TheManual circumscribed- Powered by Cogneropolygon is a square with sides 50. CHALLENGE The sum of the circumferencesPage of 14 of length 2r. circles H, J, and K shown at the right is units. Find KJ.

The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r The sum of the circumferences is 56π units. c. Since the diameter is twice the radius, 8r = 4(2r) Substitute the three circumferences and solve for x. or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying So, the radius of is 4 units, the radius of is that C = πd. 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii ANSWER: of and . a. 8r and 6r ; Twice the radius of the circle, 2r is Therefore, KJ = 16 + 8 or 24 units. the side length of the square, so the perimeter of the square is 4(2r) or 8r. The ANSWER: regular hexagon is made up of six equilateral 24 units triangles with side length r, so the perimeter of the 51. REASONING Is the distance from the center of a hexagon is 6(r) or 6r. circle to a point in the interior of a circle sometimes, b. less; greater; 6r < C < 8r always, or never less than the radius of the circle? c. 3d < C < 4d; The circumference of the circle is Explain. between 3 and 4 times its diameter. SOLUTION: d. These limits will approach a value of πd, implying that C = πd. A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always 50. CHALLENGE The sum of the circumferences of have a length less than the radius of the circle. circles H, J, and K shown at the right is units. Find KJ. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: SOLUTION: The radius of is 4x units, the radius of is x A circle is a locus of points in a plane equidistant units, and the radius of is 2x units. Find the from a given point. For any two circles and circumference of each circle. , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r = 2 kr1.

The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are So, the radius of is 4 units, the radius of is similar. Thus, all circles are similar. 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii ANSWER: of and . A circle is a locus of points in a plane equidistant

Therefore, KJ = 16 + 8 or 24 units. from a given point. For any two circles and ANSWER: , there exists a translation that maps center A 24 units onto center B, moving so that it is concentric with . There also exists a dilation with scale 51. REASONING Is the distance from the center of a factor k such that each point that makes up is circle to a point in the interior of a circle sometimes, moved to be the same distance from center A as the always, or never less than the radius of the circle? points that make up are from center B.

Explain. Therefore, is mapped onto . Since there SOLUTION: exists a rigid motion followed by a scaling that maps A radius is a segment drawn between the center of onto , the circles are similar. Thus, all the circle and a point on the circle. A segment drawn circles are similar. from the center to a point inside the circle will always have a length less than the radius of the circle. 53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference ANSWER: of ? Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: SOLUTION: A circle is a locus of points in a plane equidistant Draw a perpendicular to the side Also join from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = Since is equilateral triangle, MN = 8 inches kr1. and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. Therefore, the radius of is

The circumference C of a circle with radius r is given by ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric

with . There also exists a dilation with scale factor k such that each point that makes up is The circumference of is inches. moved to be the same distance from center A as the points that make up are from center B. ANSWER: Therefore, is mapped onto . Since there in. exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of 53. CHALLENGE In the figure, is inscribed in geometry. equilateral triangle LMN. What is the circumference SOLUTION: of ? See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

SOLUTION: Draw a perpendicular to the side Also join

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

The circumference C of a circle with diameter d is Since is equilateral triangle, MN = 8 inches given by and RN = 4 inches. Also, bisects Use the formula to find the circumference. Use the definition of tangent of an angle to find PR, which is the radius of

ANSWER:

Therefore, the radius of is 40.8 The circumference C of a circle with radius r is 56. What is the radius of a table with a circumference of given by 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft SOLUTION:

The circumference of is inches. The circumference C of a circle with radius r is given by

ANSWER: Here, C = 10 ft. Use the formula to find the radius.

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of The radius of the circle is about 1.6 feet, so the geometry. correct choice is A. SOLUTION: ANSWER: See students’ work. A ANSWER: See students’ work. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use 55. GRIDDED RESPONSE What is the up to 50 feet of fence, what radius can he use for the

circumference of ? Round to the nearest tenth. garden? F 10 G 9 H 8 J 7 SOLUTION:

The circumference C of a circle with radius r is SOLUTION: given by Use the Pythagorean Theorem to find the diameter Because 50 feet of fence can be used to border the of the circle. garden, Cmax = 50. Use the circumference formula to find the maximum radius. The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J. ANSWER: ANSWER: J 40.8 58. SAT/ACT What is the radius of a circle with an

56. What is the radius of a table with a circumference of area of square units? 10 feet? A 1.6 ft A 0.4 units B 2.5 ft B 0.5 units C 3.2 ft C 2 units D 5 ft D 4 units E 16 units SOLUTION: The circumference C of a circle with radius r is SOLUTION: given by The area of a circle of radius r is given by the

Here, C = 10 ft. Use the formula to find the radius. formula Here A = square units. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the correct choice is A. Therefore, the correct choice is B.

ANSWER: ANSWER: A B 57. ALGEBRA Bill is planning a circular vegetable Copy each figure and point B. Then use a ruler garden with a fence around the border. If he can use to draw the image of the figure under a dilation up to 50 feet of fence, what radius can he use for the with center B and the scale factor r indicated. garden? F 10 59. G 9 H 8 J 7 SOLUTION: The circumference C of a circle with radius r is given by SOLUTION: Because 50 feet of fence can be used to border the Draw a ray from point B through a vertex of the garden, Cmax = 50. Use the circumference formula to hexagon. Use the ruler to measure the distance from find the maximum radius. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units ANSWER: B 0.5 units C 2 units D 4 units E 16 units

SOLUTION: 60. The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from

Therefore, the correct choice is B. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays ANSWER: through the other two vertices. Connect the three B points on the rays to construct the dilated image of Copy each figure and point B. Then use a ruler the triangle. to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

61. SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other five vertices. Connect the six points Draw a ray from point B through a vertex of the on the rays to construct the dilated image of the trapezoid. Use the ruler to measure the distance from hexagon. B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER: ANSWER:

60.

62.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is of Draw a ray from point B through a vertex of the that distance from B. Repeat the process for rays triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 through the other two vertices. Connect the three times that distance from B. Repeat the process for points on the rays to construct the dilated image of rays through the other two vertices. Connect the the triangle. three points on the rays to construct the dilated image of the triangle.

ANSWER:

61.

ANSWER: SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. ANSWER: SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

62. 64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. Draw a ray from point B through a vertex of the ANSWER: triangle. Use the ruler to measure the distance from Yes; 4, 90 B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No ANSWER: 66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. State whether each figure has rotational If you are over 18 years old, then you vote in all symmetry. Write yes or no. If so, copy the elections. figure, locate the center of symmetry, and state 67. You are 19 years old and you vote. the order and magnitude of symmetry. SOLUTION: 63. Refer to the image on page 705. True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true The figure cannot be mapped onto itself by any for the conditions. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: True; sample answer: Since the hypothesis is true No and the conclusion is true, then the statement is true for the conditions. 64. Refer to the image on page 705. 68. You are 21 years old and do not vote. SOLUTION: SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the False; sample answer: Since the hypothesis is true magnitude of symmetry is 90º and order of symmetry and the conclusion is false, then the statement is false is 4. for the conditions.

ANSWER: ANSWER: Yes; 4, 90 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. 69. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any The sum of the measures of angles around a point is rotation. So, it does not have a rotational symmetry. 360º.

ANSWER: No

66. Refer to the image on page 705. ANSWER: SOLUTION: 90 The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following

statement for each set of conditions. Explain 70. your reasoning. SOLUTION: If you are over 18 years old, then you vote in all The angles with the measures xº and 2xº form a elections. straight line. So, the sum is 180. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. ANSWER: 60 ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: 71. False; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is false, then the statement is false The angles with the measures 6xº and 3xº form a for the conditions. straight line. So, the sum is 180. ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. ANSWER: Find x. 20

69. 72. SOLUTION: The sum of the measures of angles around a point is SOLUTION: 360º. The sum of the three angles is 180.

ANSWER: ANSWER: 90 45

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm For Exercises 1–4, refer to . 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

1. Name the circle. Therefore, the diameter is 26 ft.

SOLUTION: ANSWER: The center of the circle is N. So, the circle is 26 ft

ANSWER: The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. 5. FG b. A diameter of a circle is a chord that passes SOLUTION: through the center. Here, is a diameter. Since the diameter of B is 18 inches and A is 8 c. A radius is a segment with endpoints at the center inches, AG = 18 and AF = (8) or 4. and on the circle. Here, is radius.

ANSWER:

a. Therefore, FG = 14 in. b. c. ANSWER: 14 in. 3. If CN = 8 centimeters, find DN. SOLUTION: 6. FB Here, are radii of the same circle. So, SOLUTION:

they are equal in length. Therefore, DN = 8 cm. Since the diameter of B is 18 inches and A is 4 ANSWER: inches, AB = (18) or 9 and AF = (8) or 4. 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Therefore, FB = 5 inches. Here, is a radius and the diameter is twice the radius. ANSWER: 5 in.

Therefore, the diameter is 26 ft. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the ANSWER: radius and circumference of the ride? Round to the 26 ft nearest hundredth, if necessary.

The diameters of , , and are 8 SOLUTION: inches, 18 inches, and 11 inches, respectively. The radius is half the diameter. So, the radius of the Find each measure. circular ride is 22 feet.

Therefore, the circumference of the ride is about 138.23 feet.

5. FG ANSWER: SOLUTION: 22 ft; 138.23 ft Since the diameter of B is 18 inches and A is 8 8. CCSS MODELING The circumference of the inches, AG = 18 and AF = (8) or 4. circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB SOLUTION: SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning The diameter of the pool is about 17.98 feet and the of the lesson has a diameter of 44 feet. What are the radius of the pool is about 8.99 feet. radius and circumference of the ride? Round to the nearest hundredth, if necessary. ANSWER: 17.98 ft; 8.99 ft SOLUTION: The radius is half the diameter. So, the radius of the 9. SHORT RESPONSE The right triangle shown is circular ride is 22 feet. inscribed in . Find the exact circumference of .

Therefore, the circumference of the ride is about 138.23 feet. ANSWER: SOLUTION: 22 ft; 138.23 ft The diameter of the circle is the hypotenuse of the right triangle. 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π SOLUTION: centimeters. ANSWER:

For Exercises 10–13, refer to .

10. Name the center of the circle. The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet. SOLUTION: R ANSWER: 17.98 ft; 8.99 ft ANSWER: R 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of 11. Identify a chord that is also a diameter. . SOLUTION: Two chords are shown: and . goes through the center, R, so is a diameter.

ANSWER: SOLUTION: The diameter of the circle is the hypotenuse of the right triangle. 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on

the circle, so it is a chord. The diameter of the circle is 4 centimeters. ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT?

The circumference of the circle is 4π SOLUTION: centimeters. Here, is a diameter and is a radius. ANSWER:

For Exercises 10–13, refer to . Therefore, RT = 8.1 cm.

ANSWER: 8.1 cm

For Exercises 14–17, refer to .

10. Name the center of the circle. SOLUTION: R ANSWER:

R 14. Identify a chord that is not a diameter.

11. Identify a chord that is also a diameter. SOLUTION: The chords do not pass through the SOLUTION: center. So, they are not diameters. Two chords are shown: and . goes ANSWER: through the center, R, so is a diameter.

ANSWER: 15. If CF = 14 inches, what is the diameter of the circle?

SOLUTION: 12. Is a radius? Explain. Here, is a radius and the diameter is twice the radius. . SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord. Therefore, the diameter of the circle is 28 inches.

ANSWER: ANSWER: No; it is a chord. 28 in. 13. If SU = 16.2 centimeters, what is RT? 16. Is ? Explain. SOLUTION: SOLUTION: Here, is a diameter and is a radius. All radii of a circle are congruent. Since are both radii of , .

ANSWER: Yes; they are both radii of .

Therefore, RT = 8.1 cm. 17. If DA = 7.4 centimeters, what is EF? ANSWER: SOLUTION: 8.1 cm Here, is a diameter and is a radius. For Exercises 14–17, refer to .

Therefore, EF = 3.7 cm.

ANSWER: 14. Identify a chord that is not a diameter. 3.7 cm

SOLUTION: Circle J has a radius of 10 units, has a The chords do not pass through the radius of 8 units, and BC = 5.4 units. Find each center. So, they are not diameters. measure. ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the 18. CK radius. . SOLUTION: Since the radius of K is 8 units, BK = 8.

Therefore, the diameter of the circle is 28 inches. Therefore, CK = 2.6 units. ANSWER: 28 in. ANSWER: 2.6 16. Is ? Explain. SOLUTION: 19. AB All radii of a circle are congruent. Since SOLUTION: are both radii of , . Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? Therefore, AB = 14.6 units.

SOLUTION: ANSWER: Here, is a diameter and is a radius. 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8. Therefore, EF = 3.7 cm.

ANSWER: 3.7 cm Since circle J has a radius of 10 units, JC = 10. Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure. Therefore, JK = 12.6 units.

ANSWER: 12.6 21. AD SOLUTION: 18. CK We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J SOLUTION: is 10 units, so KD = 8 and AC = 2(10) or 20. Before Since the radius of K is 8 units, BK = 8. finding AD, we need to find CK.

Therefore, CK = 2.6 units.

ANSWER:

2.6 Therefore, AD = 30.6 units.

19. AB ANSWER: 30.6 SOLUTION: Since is a diameter of circle J and the radius is 22. PIZZA Find the radius and circumference of the 10, AC = 2(10) or 20 units. pizza shown. Round to the nearest hundredth, if necessary.

Therefore, AB = 14.6 units.

ANSWER: 14.6 20. JK

SOLUTION: First find CK. Since circle K has a radius of 8 units, SOLUTION: BK = 8. The diameter of the pizza is 16 inches.

Since circle J has a radius of 10 units, JC = 10. . So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is Therefore, JK = 12.6 units. given by ANSWER: 12.6

21. AD Therefore, the circumference of the pizza is about 50.27 inches. SOLUTION: We can find AD using AD = AC + CK + KD. The ANSWER: radius of circle K is 8 units and the radius of circle J 8 in.; 50.27 in. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary.

SOLUTION: The diameter of a tire is 26 inches. The radius is half

the diameter. Therefore, AD = 30.6 units.

ANSWER: 30.6

22. PIZZA Find the radius and circumference of the So, the radius of the tires is 13 inches. pizza shown. Round to the nearest hundredth, if The circumference C of a circle with diameter d is necessary. given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. SOLUTION: Find the diameter and radius of a circle by with The diameter of the pizza is 16 inches. the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is .

So, the radius of the pizza is 8 inches. given by The circumference C of a circle of diameter d is Here, C = 18 in. Use the formula to find the diameter. Then find the radius. given by

Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a Therefore, the diameter is about 5.73 inches and the tire. Round to the nearest hundredth, if necessary. radius is about 2.86 inches.

SOLUTION: ANSWER: The diameter of a tire is 26 inches. The radius is half 5.73 in.; 2.86 in. the diameter.

25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is given by So, the radius of the tires is 13 inches. Here, C = 124 ft. Use the formula to find the The circumference C of a circle with diameter d is diameter. Then find the radius. given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in.

Find the diameter and radius of a circle by with the given circumference. Round to the nearest Therefore, the diameter is about 39.47 feet and the hundredth. radius is about 19.74 feet. 24. C = 18 in. ANSWER: SOLUTION: 39.47 ft; 19.74 ft The circumference C of a circle with diameter d is given by 26. C = 375.3 cm Here, C = 18 in. Use the formula to find the diameter. Then find the radius. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: 5.73 in.; 2.86 in. Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

25. C = 124 ft ANSWER: SOLUTION: 119.46 cm; 59.73 cm The circumference C of a circle with diameter d is 27. C = 2608.25 m given by Here, C = 124 ft. Use the formula to find the SOLUTION: diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet.

ANSWER: Therefore, the diameter is about 830.23 meters and 39.47 ft; 19.74 ft the radius is about 415.12 meters.

26. C = 375.3 cm ANSWER: 830.23 m; 415.12 m SOLUTION: The circumference C of a circle with diameter d is CCSS SENSE-MAKING Find the exact given by circumference of each circle by using the given Here, C = 375.3 cm. Use the formula to find the inscribed or diameter. Then find the radius. circumscribed polygon.

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the

diameter. Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: 119.46 cm; 59.73 cm The diameter of the circle is 17 cm. The 27. C = 2608.25 m circumference C of a circle is given by SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find Therefore, the circumference of the circle is 17 the diameter. Then find the radius. centimeters.

ANSWER:

Therefore, the diameter is about 830.23 meters and 29. the radius is about 415.12 meters. SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of 830.23 m; 415.12 m the circle. Use the Pythagorean Theorem to find the diameter. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon. The diameter of the circle is 12 feet. The circumference C of a circle is given by

28. Therefore, the circumference of the circle is 12 SOLUTION: feet. The hypotenuse of the right triangle is a diameter of ANSWER: the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The 30. circumference C of a circle is given by SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter Therefore, the circumference of the circle is 17 of the circle. centimeters.

ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by

29. Therefore, the circumference of the circle is SOLUTION: inches. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the ANSWER:

diameter.

The diameter of the circle is 12 feet. The circumference C of a circle is given by 31. SOLUTION: Each diagonal of the inscribed rectangle will pass Therefore, the circumference of the circle is 12 through the origin, so it is a diameter of the circle. feet. Use the Pythagorean Theorem to find the diameter of the circle. ANSWER:

The diameter of the circle is 10 inches. The circumference C of a circle is given by 30. SOLUTION: Each diagonal of the inscribed rectangle will pass Therefore, the circumference of the circle is 10 through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter inches. of the circle. ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d Therefore, the circumference of the circle is = 25 millimeters. inches. The circumference C of a circle is given by

ANSWER:

Therefore, the circumference of the circle is 25 millimeters.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 33. SOLUTION: A diameter perpendicular to a side of the square and The diameter of the circle is 10 inches. The the length of each side will measure the same. So, d circumference C of a circle is given by = 14 yards. The circumference C of a circle is given by

Therefore, the circumference of the circle is 10 inches. Therefore, the circumference of the circle is 14 yards. ANSWER: ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in 32. centimeters. What is the maximum allowable weight SOLUTION: for a disc with circumference 66.92 centimeters? Round to the nearest tenth. A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d SOLUTION: = 25 millimeters. The circumference C of a circle with diameter d is The circumference C of a circle is given by given by Here, C = 66.92 cm. Use the formula to find the diameter.

Therefore, the circumference of the circle is 25 millimeters.

ANSWER: The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g

33. 35. PATIOS Mr. Martinez is going to build the patio shown. SOLUTION: a. What is the patio’s approximate circumference? A diameter perpendicular to a side of the square and b. If Mr. Martinez changes the plans so that the the length of each side will measure the same. So, d inner circle has a circumference of approximately 25 = 14 yards. feet, what should the radius of the circle be to the The circumference C of a circle is given by nearest foot?

Therefore, the circumference of the circle is 14 yards.

ANSWER: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The 34. DISC GOLF Disc golf is similar to regular golf, circumference C of a circle with radius r is given by except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum Use the formula to find the circumference. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. Therefore, the circumference is about 31.42 feet. SOLUTION: b. Here, C = 25 ft. Use the formula to find the The circumference C of a circle with diameter d is inner given by radius of the inner circle. Here, C = 66.92 cm. Use the formula to find the diameter.

So, the radius of the inner circle is about 4 feet.

ANSWER: The diameter of the disc is about 21.3 centimeters a. and the maximum weight allowed is 8.3 times the 31.42 ft diameter in centimeters. b. 4 ft Therefore, the maximum weight is 8.3 × 21.3 or The radius, diameter, or circumference of a about 176.8 grams. circle is given. Find each missing measure to ANSWER: the nearest hundredth.

176.8 g 36. 35. PATIOS Mr. Martinez is going to build the patio shown. SOLUTION: a. What is the patio’s approximate circumference? The radius is half the diameter and the circumference C of a circle with diameter d is given b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 by feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the SOLUTION: circumference is about 26.70 inches. a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by ANSWER: 4.25 in.; 26.70 in. Use the formula to find the circumference.

37.

SOLUTION: Therefore, the circumference is about 31.42 feet. The diameter is twice the radius and the b. Here, Cinner = 25 ft. Use the formula to find the circumference C of a circle with diameter d is given radius of the inner circle. by

So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft So, the diameter is 22.80 feet and the circumference

b. 4 ft is about 71.63 feet. ANSWER: The radius, diameter, or circumference of a circle is given. Find each missing measure to 22.80 ft; 71.63 ft the nearest hundredth.

38. 36. SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The radius is half the diameter and the given by circumference C of a circle with diameter d is given Here, C = 35x cm. Use the formula to find the

by diameter.Then find the radius.

Therefore, the radius is 4.25 inches and the

circumference is about 26.70 inches. Therefore, the diameter is about 11.14x centimeters ANSWER: and the radius is about 5.57x centimeters. 4.25 in.; 26.70 in. ANSWER: 11.14x cm; 5.57x cm

37. 39. SOLUTION: The diameter is twice the radius and the SOLUTION: circumference C of a circle with diameter d is given The diameter is twice the radius and the by circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet. So, the diameter is 0.25x units and the circumference ANSWER: is about 0.79x units. 22.80 ft; 71.63 ft ANSWER: 38. 0.25x; 0.79x

SOLUTION: Determine whether the circles in the figures below appear to be congruent, concentric, or The circumference C of a circle with diameter d is neither. given by Here, C = 35x cm. Use the formula to find the 40. Refer to the photo on page 703. diameter.Then find the radius. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric. Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: neither ANSWER: 11.14x cm; 5.57x cm 42. Refer to the photo on page 703. SOLUTION: 39. The circles appear to have the same radius. So, they appear to be congruent. SOLUTION: The diameter is twice the radius and the ANSWER: circumference C of a circle with diameter d is given congruent by 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or neither. 40. Refer to the photo on page 703. SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, SOLUTION: the diameter is 150 ft. The circumference C of a The circles have the same center and they are in the circle with diameter d is given by

same plane. So, they are concentric circles. So, the circumference is ANSWER: Therefore, someone have to concentric walk about 471.2 feet to go completely around the ring.

41. Refer to the photo on page 703. ANSWER: SOLUTION: 471.2 ft The circles neither have the same radius nor are they CCSS MODELING concentric. 44. A brick path is being installed around a circular pond. The pond has a ANSWER: circumference of 68 feet. The outer edge of the path neither is going to be 4 feet from the pond all the way around. What is the approximate circumference of 42. Refer to the photo on page 703. the path? Round to the nearest hundredth. SOLUTION: SOLUTION: The circles appear to have the same radius. So, they The circumference C of a circle with radius r is appear to be congruent. given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. ANSWER: congruent

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the So, the radius of the pond is about 10.82 feet and the ring? Round to the nearest tenth. radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will explore changing dimensions in circles. The radius of the circle is 3 units, that is 75 feet. So, a. GEOMETRIC the diameter is 150 ft. The circumference C of a Use a compass to draw three circles in which the scale factor from each circle to circle with diameter d is given by the next larger circle is 1 : 2. So, the circumference is b. TABULAR Calculate the radius (to the nearest Therefore, someone have to tenth) and circumference (to the nearest hundredth) walk about 471.2 feet to go completely around the of each circle. Record your results in a table. ring. c. VERBAL Explain why these three circles are geometrically similar. ANSWER: d. VERBAL Make a conjecture about the ratio 471.2 ft between the circumferences of two circles when the ratio between their radii is 2. 44. CCSS MODELING A brick path is being installed e. ANALYTICAL The scale factor from to around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path is . Write an equation relating the is going to be 4 feet from the pond all the way around. What is the approximate circumference of circumference (CA) of to the circumference the path? Round to the nearest hundredth. (CB) of . SOLUTION: f. NUMERICAL If the scale factor from to The circumference C of a circle with radius r is is , and the circumference of is 12 given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. inches, what is the circumference of ? SOLUTION: a. Sample answer:

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path. b. Sample answer.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft c. They all have the same shape–circular. 45. MULTIPLE REPRESENTATIONS In this d. The ratio of their circumferences is also 2. problem, you will explore changing dimensions in e. The circumference is directly proportional to radii. circles. So, if the radii are in the ratio b : a, so will the a. GEOMETRIC Use a compass to draw three circumference. Therefore, . circles in which the scale factor from each circle to the next larger circle is 1 : 2. f. Use the formula from part e. Here, the b. TABULAR Calculate the radius (to the nearest circumference of is 12 inches and . tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the Therefore, the circumference of is 4 inches. ratio between their radii is 2. ANSWER: e. ANALYTICAL The scale factor from to a. Sample answer: is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to

is , and the circumference of is 12 b. Sample answer. inches, what is the circumference of ? SOLUTION: a. Sample answer:

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. b. Sample answer. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, .

f. Use the formula from part e. Here, the circumference of is 12 inches and . a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

Therefore, the circumference of is 4 inches. b. Calculate the ratio of two times the total number ANSWER: of drops to the number of hits after 25, 50, and 100 a. Sample answer: drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. b. Sample answer. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. They all have the same shape—circular. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. d. The ratio of their circumferences is also 2.

e. 47. M APS The concentric circles on the map below f. 4 in. show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. 46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center a. Drop the needle onto the paper. When the needle circle? lands, record whether it touches one of the lines as a b. As the radii of the circles increases by 5 miles, by hit. Record the number of hits after 25, 50, and 100 how much does the circumference increase? drops. SOLUTION:

a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle. b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π?

SOLUTION: a. See students’ work. The difference of the circumferences of the b. See students’ work. outermost circle and the center circle is 60π - 10π = c. Sample answer: The values are approaching 3.14, 50π or about 157.1 miles. which is approximately equal to π. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find

ANSWER: the circumference of each circle. a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 The difference of the two circumferences is (2xπ + miles from downtown Phoenix. 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between

a. How much greater is the circumference of the circles and lines? outermost circle than the circumference of the center SOLUTION: circle? Sample answer: A line and a circle may intersect in b. As the radii of the circles increases by 5 miles, by one point, two points, or may not intersect at all. A how much does the circumference increase? line that intersects a circle in one point can be SOLUTION: described as a tangent. A line that intersects a circle a. The radius of the outermost circle is 30 miles and in exactly two points can be described as a secant. that of the center circle is 5 miles. Find the A line segment with endpoints on a circle can be circumference of each circle. described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one The difference of the circumferences of the point can be described as a tangent. A line that outermost circle and the center circle is 60π - 10π = intersects a circle in exactly two points can be 50π or about 157.1 miles. described as a secant. A line segment with b. Let the radius of one circle be x miles and the endpoints on a circle can be described as a chord. radius of the next larger circle be x + 5 miles. Find If the chord passes through the center of the circle, the circumference of each circle. it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

The difference of the two circumferences is (2xπ + 49. 10π) - (2xπ) = 10π or about 31.4 miles. REASONING In the figure, a circle with radius r is Therefore, as the radius of each circle increases by 5 inscribed in a regular polygon and circumscribed miles, the circumference increases by about 31.4 about another. miles. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. ANSWER: b. Is the circumference C of the circle greater or a. 157.1 mi less than the perimeter of the circumscribed polygon? b. by about 31.4 mi the inscribed polygon? Write a compound inequality comparing C to these perimeters. 48. WRITING IN MATH How can we describe the c. Rewrite the inequality from part b in terms of the relationships that exist between diameter d of the circle and interpret its meaning. circles and lines? d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to SOLUTION: the upper and lower limits of the inequality from part Sample answer: A line and a circle may intersect in c, and what does this imply? one point, two points, or may not intersect at all. A line that intersects a circle in one point can be SOLUTION: described as a tangent. A line that intersects a circle a. The circumscribed polygon is a square with sides in exactly two points can be described as a secant. of length 2r. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the The inscribed polygon is a hexagon with sides of center and on the circle can be described as a length r. (Each triangle formed using a central angle radius. will be equilateral.)

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment

with endpoints at the center and on the circle can be So, the perimeters are 8r and 6r. described as a radius. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the 49. perimeter of the inscribed polygon. REASONING In the figure, a circle with radius r is 8r < C < 6r inscribed in a regular polygon and circumscribed c. Since the diameter is twice the radius, 8r = 4(2r) about another. or 4d and 6r = 3(2r) or 3d. a. What are the perimeters of the circumscribed and The inequality is then 4d < C < 3d. Therefore, the inscribed polygons in terms of r? Explain. circumference of the circle is between 3 and 4 times b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? its diameter. the inscribed polygon? Write a compound inequality d. These limits will approach a value of πd, implying comparing C to these perimeters. that C = πd. c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed ANSWER: and inscribed polygons increase, what will happen to a. 8r and 6r ; Twice the radius of the circle, 2r is the upper and lower limits of the inequality from part the side length of the square, so the perimeter of the c, and what does this imply? square is 4(2r) or 8r. The SOLUTION: regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the a. The circumscribed polygon is a square with sides hexagon is 6(r) or 6r. of length 2r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying The inscribed polygon is a hexagon with sides of that C = πd. length r. (Each triangle formed using a central angle will be equilateral.) 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ.

So, the perimeters are 8r and 6r. b. SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle. The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x. ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral So, the radius of is 4 units, the radius of is triangles with side length r, so the perimeter of the 2(4) or 8 units, and the radius of is 4(4) or 16 units. hexagon is 6(r) or 6r. The measure of KJ is equal to the sum of the radii b. less; greater; 6r < C < 8r of and . c. 3d < C < 4d; The circumference of the circle is Therefore, KJ = 16 + 8 or 24 units. between 3 and 4 times its diameter. 10-1d. Circles These andlimits Circumference will approach a value of πd, implying ANSWER: that C = πd. 24 units

50. CHALLENGE The sum of the circumferences of 51. REASONING Is the distance from the center of a circles H, J, and K shown at the right is units. circle to a point in the interior of a circle sometimes, Find KJ. always, or never less than the radius of the circle? Explain. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: SOLUTION: Always; a radius is a segment drawn between the The radius of is 4x units, the radius of is x center of the circle and a point on the circle. A units, and the radius of is 2x units. Find the segment drawn from the center to a point inside the circumference of each circle. circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such The sum of the circumferences is 56π units. that each point that makes up is moved to be the Substitute the three circumferences and solve for x. same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii of and . Therefore, KJ = 16 + 8 or 24 units. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps ANSWER: onto , the circles are 24 units similar. Thus, all circles are similar.

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, ANSWER: always, or never less than the radius of the circle? Explain. A circle is a locus of points in a plane equidistant from a given point. For any two circles and SOLUTION: , there exists a translation that maps center A A radius is a segment drawn between the center of onto center B, moving so that it is concentric the circle and a point on the circle. A segment drawn with . There also exists a dilation with scale from the center to a point inside the circle will always eSolutions Manual - Powered by Cognero factor k such that each point that makes up Page is 15 have a length less than the radius of the circle. moved to be the same distance from center A as the ANSWER: points that make up are from center B. Always; a radius is a segment drawn between the Therefore, is mapped onto . Since there center of the circle and a point on the circle. A exists a rigid motion followed by a scaling that maps segment drawn from the center to a point inside the onto , the circles are similar. Thus, all circle will always have a length less than the radius circles are similar. of the circle. 53. CHALLENGE In the figure, is inscribed in PROOF Use the locus definition of a circle and 52. equilateral triangle LMN. What is the circumference dilations to prove that all circles are similar. of ? SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the SOLUTION: same distance from center A as the points that make up are from center B. Draw a perpendicular to the side Also join

For some value k, r2 = kr1.

Since is equilateral triangle, MN = 8 inches Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps and RN = 4 inches. Also, bisects onto , the circles are Use the definition of similar. Thus, all circles are similar. tangent of an angle to find PR, which is the radius of

ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric Therefore, the radius of is with . There also exists a dilation with scale The circumference C of a circle with radius r is factor k such that each point that makes up is given by moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

The circumference of is inches. 53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference ANSWER: of ?

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: SOLUTION: See students’ work.

Draw a perpendicular to the side Also join ANSWER:

See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects Use the definition of SOLUTION: tangent of an angle to find PR, which is the radius of Use the Pythagorean Theorem to find the diameter of the circle.

The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of is The circumference C of a circle with radius r is given by

ANSWER: 40.8

The circumference of is inches. 56. What is the radius of a table with a circumference of 10 feet? A ANSWER: 1.6 ft B 2.5 ft

in. C 3.2 ft D 5 ft 54. WRITING IN MATH Research and write about SOLUTION: the history of pi and its importance to the study of The circumference C of a circle with radius r is geometry. given by Here, C = 10 ft. Use the formula to find the radius. SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the The radius of the circle is about 1.6 feet, so the circumference of ? Round to the nearest tenth. correct choice is A. ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use

up to 50 feet of fence, what radius can he use for the SOLUTION: garden? Use the Pythagorean Theorem to find the diameter F 10 of the circle. G 9 H 8 The circumference C of a circle with diameter d is J 7 given by SOLUTION: Use the formula to find the circumference. The circumference C of a circle with radius r is

given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

ANSWER: 40.8

56. What is the radius of a table with a circumference of Therefore, the radius of the garden must be less than 10 feet? or equal to about 7.96 feet. Therefore, the only A 1.6 ft correct choice is J. B 2.5 ft ANSWER: C 3.2 ft J D 5 ft SOLUTION: 58. SAT/ACT What is the radius of a circle with an

The circumference C of a circle with radius r is area of square units? given by Here, C = 10 ft. Use the formula to find the radius. A 0.4 units B 0.5 units C 2 units D 4 units E 16 units SOLUTION: The radius of the circle is about 1.6 feet, so the The area of a circle of radius r is given by the correct choice is A. formula Here A = square units. Use the ANSWER: formula to find the radius. A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? Therefore, the correct choice is B. F 10 G 9 ANSWER: H 8 B J 7 Copy each figure and point B. Then use a ruler SOLUTION: to draw the image of the figure under a dilation The circumference C of a circle with radius r is with center B and the scale factor r indicated. given by Because 50 feet of fence can be used to border the 59. garden, Cmax = 50. Use the circumference formula to find the maximum radius.

SOLUTION: Draw a ray from point B through a vertex of the Therefore, the radius of the garden must be less than hexagon. Use the ruler to measure the distance from or equal to about 7.96 feet. Therefore, the only B to the vertex. Mark a point on the ray that is of correct choice is J. that distance from B. Repeat the process for rays ANSWER: through the other five vertices. Connect the six points J on the rays to construct the dilated image of the hexagon. 58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the ANSWER: formula to find the radius.

60.

Therefore, the correct choice is B.

ANSWER: B SOLUTION: Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation Draw a ray from point B through a vertex of the with center B and the scale factor r indicated. triangle. Use the ruler to measure the distance from

59. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon. 61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid. ANSWER:

60.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from

B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays 62. through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 ANSWER: times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. 61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for

rays through the other three vertices. Connect the four points on the rays to construct the dilated image ANSWER: of the trapezoid.

ANSWER:

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. 62. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: SOLUTION: No Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from 64. Refer to the image on page 705. B to the vertex. Mark a point on the ray that is 3 SOLUTION: times that distance from B. Repeat the process for rays through the other two vertices. Connect the The figure can be mapped onto itself by a rotation of three points on the rays to construct the dilated image 90º. So, it has a rotational symmetry. So, the of the triangle. magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

ANSWER: 65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any State whether each figure has rotational rotation. So, it does not have a rotational symmetry. symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state ANSWER: the order and magnitude of symmetry. No 63. Refer to the image on page 705. Determine the truth value of the following SOLUTION: statement for each set of conditions. Explain The figure cannot be mapped onto itself by any your reasoning. rotation. So, it does not have a rotational symmetry. If you are over 18 years old, then you vote in all elections. ANSWER: 67. You are 19 years old and you vote. No SOLUTION: 64. Refer to the image on page 705. True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true SOLUTION: for the conditions. The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the ANSWER: magnitude of symmetry is 90º and order of symmetry True; sample answer: Since the hypothesis is true is 4. and the conclusion is true, then the statement is true for the conditions. ANSWER: Yes; 4, 90 68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false 65. for the conditions. SOLUTION: Find x. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705. 69. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any The sum of the measures of angles around a point is

rotation. So, it does not have a rotational symmetry. 360º.

ANSWER: No

Determine the truth value of the following ANSWER: statement for each set of conditions. Explain 90 your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true 70. for the conditions. SOLUTION: ANSWER: The angles with the measures xº and 2xº form a True; sample answer: Since the hypothesis is true straight line. So, the sum is 180. and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. ANSWER: SOLUTION: 60 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. 71.

Find x. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

69. ANSWER: SOLUTION: 20 The sum of the measures of angles around a point is 360º.

ANSWER: 72. 90 SOLUTION: The sum of the three angles is 180.

ANSWER: 70. 45 SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: For Exercises 1–4, refer to . 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

1. Name the circle. Therefore, the diameter is 26 ft. SOLUTION: The center of the circle is N. So, the circle is ANSWER: 26 ft ANSWER: The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle.

So, here are chords. 5. FG b. A diameter of a circle is a chord that passes SOLUTION: through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center Since the diameter of B is 18 inches and A is 8

and on the circle. Here, is inches, AG = 18 and AF = (8) or 4. radius.

ANSWER: a. b. Therefore, FG = 14 in. c. ANSWER: 3. If CN = 8 centimeters, find DN. 14 in. SOLUTION: 6. FB Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm. SOLUTION: Since the diameter of B is 18 inches and A is 4 ANSWER: inches, AB = (18) or 9 and AF = (8) or 4. 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION:

Here, is a radius and the diameter is twice the Therefore, FB = 5 inches. radius. ANSWER: 5 in. Therefore, the diameter is 26 ft. 7. RIDES The circular ride described at the beginning ANSWER: of the lesson has a diameter of 44 feet. What are the 26 ft radius and circumference of the ride? Round to the nearest hundredth, if necessary. The diameters of , , and are 8 SOLUTION: inches, 18 inches, and 11 inches, respectively. The radius is half the diameter. So, the radius of the Find each measure. circular ride is 22 feet.

Therefore, the circumference of the ride is about 138.23 feet. 5. FG ANSWER: SOLUTION: 22 ft; 138.23 ft Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4. 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB SOLUTION: SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the The diameter of the pool is about 17.98 feet and the radius and circumference of the ride? Round to the radius of the pool is about 8.99 feet.

nearest hundredth, if necessary. ANSWER: SOLUTION: 17.98 ft; 8.99 ft The radius is half the diameter. So, the radius of the circular ride is 22 feet. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of .

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: SOLUTION: 22 ft; 138.23 ft The diameter of the circle is the hypotenuse of the 8. CCSS MODELING The circumference of the right triangle. circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π SOLUTION: centimeters.

ANSWER:

For Exercises 10–13, refer to .

The diameter of the pool is about 17.98 feet and the 10. Name the center of the circle. radius of the pool is about 8.99 feet. SOLUTION: ANSWER: R 17.98 ft; 8.99 ft ANSWER: 9. SHORT RESPONSE The right triangle shown is R inscribed in . Find the exact circumference of . 11. Identify a chord that is also a diameter. SOLUTION: Two chords are shown: and . goes through the center, R, so is a diameter.

SOLUTION: ANSWER: The diameter of the circle is the hypotenuse of the right triangle. 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on The diameter of the circle is 4 centimeters. the circle, so it is a chord.

ANSWER: No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? The circumference of the circle is 4π SOLUTION: centimeters. Here, is a diameter and is a radius. ANSWER:

For Exercises 10–13, refer to .

Therefore, RT = 8.1 cm.

ANSWER: 8.1 cm

For Exercises 14–17, refer to . 10. Name the center of the circle. SOLUTION: R ANSWER: R 14. Identify a chord that is not a diameter. 11. Identify a chord that is also a diameter. SOLUTION: SOLUTION: The chords do not pass through the Two chords are shown: and . goes center. So, they are not diameters. through the center, R, so is a diameter. ANSWER:

ANSWER: 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: 12. Is a radius? Explain. Here, is a radius and the diameter is twice the SOLUTION: radius. . A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord. Therefore, the diameter of the circle is 28 inches. ANSWER: No; it is a chord. ANSWER: 28 in. 13. If SU = 16.2 centimeters, what is RT?

SOLUTION: 16. Is ? Explain. Here, is a diameter and is a radius. SOLUTION: All radii of a circle are congruent. Since are both radii of , .

ANSWER: Yes; they are both radii of . Therefore, RT = 8.1 cm. 17. If DA = 7.4 centimeters, what is EF? ANSWER: 8.1 cm SOLUTION: Here, is a diameter and is a radius. For Exercises 14–17, refer to .

Therefore, EF = 3.7 cm.

ANSWER: 14. Identify a chord that is not a diameter. 3.7 cm SOLUTION: Circle J has a radius of 10 units, has a The chords do not pass through the center. So, they are not diameters. radius of 8 units, and BC = 5.4 units. Find each measure. ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION:

Here, is a radius and the diameter is twice the 18. CK radius. . SOLUTION: Since the radius of K is 8 units, BK = 8.

Therefore, the diameter of the circle is 28 inches.

ANSWER: Therefore, CK = 2.6 units. 28 in. ANSWER: 2.6 16. Is ? Explain.

SOLUTION: 19. AB All radii of a circle are congruent. Since SOLUTION: are both radii of , . Since is a diameter of circle J and the radius is ANSWER: 10, AC = 2(10) or 20 units. Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? Therefore, AB = 14.6 units. SOLUTION: ANSWER: Here, is a diameter and is a radius. 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, Therefore, EF = 3.7 cm. BK = 8. ANSWER: 3.7 cm Since circle J has a radius of 10 units, JC = 10. Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

Therefore, JK = 12.6 units.

ANSWER: 12.6 21. AD SOLUTION: 18. CK We can find AD using AD = AC + CK + KD. The SOLUTION: radius of circle K is 8 units and the radius of circle J Since the radius of K is 8 units, BK = 8. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

Therefore, CK = 2.6 units. ANSWER: 2.6 Therefore, AD = 30.6 units. 19. AB ANSWER: SOLUTION: 30.6 Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary.

Therefore, AB = 14.6 units.

ANSWER: 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, SOLUTION: BK = 8. The diameter of the pizza is 16 inches.

Since circle J has a radius of 10 units, JC = 10.

. So, the radius of the pizza is 8 inches. Therefore, JK = 12.6 units. The circumference C of a circle of diameter d is given by ANSWER: 12.6 21. AD Therefore, the circumference of the pizza is about SOLUTION: 50.27 inches. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J ANSWER: is 10 units, so KD = 8 and AC = 2(10) or 20. Before 8 in.; 50.27 in. finding AD, we need to find CK. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary.

SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter. Therefore, AD = 30.6 units.

ANSWER: 30.6

22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if So, the radius of the tires is 13 inches. necessary. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER:

13 in.; 81.68 in. SOLUTION: The diameter of the pizza is 16 inches. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: . The circumference C of a circle with diameter d is So, the radius of the pizza is 8 inches. given by The circumference C of a circle of diameter d is Here, C = 18 in. Use the formula to find the given by diameter. Then find the radius.

Therefore, the circumference of the pizza is about 50.27 inches. ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. SOLUTION: The diameter of a tire is 26 inches. The radius is half ANSWER: the diameter. 5.73 in.; 2.86 in.

25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is So, the radius of the tires is 13 inches. given by The circumference C of a circle with diameter d is Here, C = 124 ft. Use the formula to find the given by diameter. Then find the radius.

Therefore, the circumference of the bicycle tire is about 81.68 inches. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. Therefore, the diameter is about 39.47 feet and the 24. C = 18 in. radius is about 19.74 feet. SOLUTION: ANSWER: The circumference C of a circle with diameter d is 39.47 ft; 19.74 ft given by Here, C = 18 in. Use the formula to find the 26. C = 375.3 cm

diameter. Then find the radius. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: 5.73 in.; 2.86 in. Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. 25. C = 124 ft ANSWER: SOLUTION: 119.46 cm; 59.73 cm The circumference C of a circle with diameter d is given by 27. C = 2608.25 m Here, C = 124 ft. Use the formula to find the diameter. Then find the radius. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet.

ANSWER: 39.47 ft; 19.74 ft Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

26. C = 375.3 cm ANSWER: SOLUTION: 830.23 m; 415.12 m The circumference C of a circle with diameter d is CCSS SENSE-MAKING Find the exact given by circumference of each circle by using the given Here, C = 375.3 cm. Use the formula to find the inscribed or diameter. Then find the radius. circumscribed polygon.

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the Therefore, the diameter is about 119.46 centimeters diameter. and the radius is about 59.73 centimeters.

ANSWER: 119.46 cm; 59.73 cm

27. C = 2608.25 m The diameter of the circle is 17 cm. The circumference C of a circle is given by SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. Therefore, the circumference of the circle is 17 centimeters.

ANSWER:

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. 29. SOLUTION: ANSWER: The hypotenuse of the right triangle is a diameter of 830.23 m; 415.12 m the circle. Use the Pythagorean Theorem to find the CCSS SENSE-MAKING Find the exact diameter. circumference of each circle by using the given inscribed or circumscribed polygon.

The diameter of the circle is 12 feet. The circumference C of a circle is given by

28. Therefore, the circumference of the circle is 12 SOLUTION: feet. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the ANSWER: diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Therefore, the circumference of the circle is 17 Use the Pythagorean Theorem to find the diameter centimeters. of the circle.

ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by

29.

SOLUTION: Therefore, the circumference of the circle is The hypotenuse of the right triangle is a diameter of inches. the circle. Use the Pythagorean Theorem to find the diameter. ANSWER:

The diameter of the circle is 12 feet. The circumference C of a circle is given by

31. SOLUTION: Therefore, the circumference of the circle is 12 Each diagonal of the inscribed rectangle will pass feet. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter ANSWER: of the circle.

The diameter of the circle is 10 inches. The circumference C of a circle is given by 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Therefore, the circumference of the circle is 10 Use the Pythagorean Theorem to find the diameter inches. of the circle. ANSWER:

The diameter of the circle is inches. The circumference C of a circle is given by 32. SOLUTION: A diameter perpendicular to a side of the square and Therefore, the circumference of the circle is the length of each side will measure the same. So, d inches. = 25 millimeters. The circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 25 millimeters.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

33. SOLUTION: A diameter perpendicular to a side of the square and The diameter of the circle is 10 inches. The the length of each side will measure the same. So, d

circumference C of a circle is given by = 14 yards. The circumference C of a circle is given by

Therefore, the circumference of the circle is 10

inches. Therefore, the circumference of the circle is 14 ANSWER: yards. ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum 32. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight SOLUTION: for a disc with circumference 66.92 centimeters? A diameter perpendicular to a side of the square and Round to the nearest tenth. the length of each side will measure the same. So, d = 25 millimeters. SOLUTION: The circumference C of a circle is given by The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter.

Therefore, the circumference of the circle is 25 millimeters.

ANSWER:

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g 33. 35. PATIOS Mr. Martinez is going to build the patio SOLUTION: shown. A diameter perpendicular to a side of the square and a. What is the patio’s approximate circumference? the length of each side will measure the same. So, d b. If Mr. Martinez changes the plans so that the = 14 yards. inner circle has a circumference of approximately 25 The circumference C of a circle is given by feet, what should the radius of the circle be to the nearest foot?

Therefore, the circumference of the circle is 14 yards.

ANSWER:

SOLUTION: 34. DISC GOLF Disc golf is similar to regular golf, a. The radius of the patio is 3 + 2 = 5 ft. The except that a flying disc is used instead of a ball and circumference C of a circle with radius r is given by clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in Use the formula to find the circumference. centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth.

SOLUTION: Therefore, the circumference is about 31.42 feet.

The circumference C of a circle with diameter d is b. Here, Cinner = 25 ft. Use the formula to find the given by radius of the inner circle. Here, C = 66.92 cm. Use the formula to find the diameter.

So, the radius of the inner circle is about 4 feet.

The diameter of the disc is about 21.3 centimeters ANSWER: and the maximum weight allowed is 8.3 times the a. 31.42 ft diameter in centimeters. b. 4 ft Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. The radius, diameter, or circumference of a circle is given. Find each missing measure to ANSWER: the nearest hundredth. 176.8 g 36. 35. PATIOS Mr. Martinez is going to build the patio shown. SOLUTION: a. What is the patio’s approximate circumference? The radius is half the diameter and the b. If Mr. Martinez changes the plans so that the circumference C of a circle with diameter d is given inner circle has a circumference of approximately 25 by feet, what should the radius of the circle be to the nearest foot?

SOLUTION: Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by ANSWER: 4.25 in.; 26.70 in. Use the formula to find the circumference.

37.

Therefore, the circumference is about 31.42 feet. SOLUTION: b. Here, Cinner = 25 ft. Use the formula to find the The diameter is twice the radius and the radius of the inner circle. circumference C of a circle with diameter d is given by

So, the radius of the inner circle is about 4 feet.

ANSWER:

a. 31.42 ft So, the diameter is 22.80 feet and the circumference b. 4 ft is about 71.63 feet.

The radius, diameter, or circumference of a ANSWER: circle is given. Find each missing measure to 22.80 ft; 71.63 ft the nearest hundredth. 38. 36. SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The radius is half the diameter and the given by circumference C of a circle with diameter d is given Here, C = 35x cm. Use the formula to find the by diameter.Then find the radius.

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. Therefore, the diameter is about 11.14x centimeters ANSWER: and the radius is about 5.57x centimeters. 4.25 in.; 26.70 in. ANSWER: 11.14x cm; 5.57x cm 37.

39. SOLUTION: The diameter is twice the radius and the SOLUTION: circumference C of a circle with diameter d is given by The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet. ANSWER: So, the diameter is 0.25x units and the circumference is about 0.79x units. 22.80 ft; 71.63 ft ANSWER:

38. 0.25x; 0.79x SOLUTION: Determine whether the circles in the figures The circumference C of a circle with diameter d is below appear to be congruent, concentric, or given by neither. Here, C = 35x cm. Use the formula to find the 40. Refer to the photo on page 703. diameter.Then find the radius. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they Therefore, the diameter is about 11.14x centimeters concentric. and the radius is about 5.57x centimeters. ANSWER: ANSWER: neither 11.14x cm; 5.57x cm 42. Refer to the photo on page 703.

39. SOLUTION: The circles appear to have the same radius. So, they SOLUTION: appear to be congruent. The diameter is twice the radius and the ANSWER: circumference C of a circle with diameter d is given by congruent 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or neither. 40. Refer to the photo on page 703. SOLUTION: SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, The circles have the same center and they are in the the diameter is 150 ft. The circumference C of a same plane. So, they are concentric circles. circle with diameter d is given by So, the circumference is ANSWER: Therefore, someone have to concentric walk about 471.2 feet to go completely around the ring. 41. Refer to the photo on page 703. ANSWER: SOLUTION: 471.2 ft The circles neither have the same radius nor are they concentric. 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a ANSWER: circumference of 68 feet. The outer edge of the path neither is going to be 4 feet from the pond all the way around. What is the approximate circumference of 42. Refer to the photo on page 703. the path? Round to the nearest hundredth. SOLUTION: SOLUTION: The circles appear to have the same radius. So, they The circumference C of a circle with radius r is appear to be congruent. given by Here, Cpond = 68 ft. Use the ANSWER: formula to find the radius of the pond. congruent

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ring? Round to the nearest tenth. So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will explore changing dimensions in The radius of the circle is 3 units, that is 75 feet. So, circles. the diameter is 150 ft. The circumference C of a a. GEOMETRIC Use a compass to draw three circle with diameter d is given by circles in which the scale factor from each circle to So, the circumference is the next larger circle is 1 : 2. Therefore, someone have to b. TABULAR Calculate the radius (to the nearest walk about 471.2 feet to go completely around the tenth) and circumference (to the nearest hundredth) ring. of each circle. Record your results in a table. c. VERBAL Explain why these three circles are ANSWER: geometrically similar. 471.2 ft d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the 44. CCSS MODELING A brick path is being installed ratio between their radii is 2. around a circular pond. The pond has a e. ANALYTICAL The scale factor from to circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way is . Write an equation relating the around. What is the approximate circumference of the path? Round to the nearest hundredth. circumference (CA) of to the circumference (C ) of . SOLUTION: B The circumference C of a circle with radius r is f. NUMERICAL If the scale factor from to given by Here, Cpond = 68 ft. Use the is , and the circumference of is 12 formula to find the radius of the pond. inches, what is the circumference of ? SOLUTION: a. Sample answer:

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

b. Sample answer.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft c. They all have the same shape–circular. 45. MULTIPLE REPRESENTATIONS In this d. The ratio of their circumferences is also 2. problem, you will explore changing dimensions in e. The circumference is directly proportional to radii. circles. So, if the radii are in the ratio b : a, so will the a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to circumference. Therefore, . the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest f. Use the formula from part e. Here, the tenth) and circumference (to the nearest hundredth) circumference of is 12 inches and . of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the Therefore, the circumference of is 4 inches. ratio between their radii is 2. e. ANALYTICAL The scale factor from to ANSWER: is . Write an equation relating the a. Sample answer:

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to

is , and the circumference of is 12 b. Sample answer. inches, what is the circumference of ? SOLUTION: a. Sample answer:

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

b. Sample answer. e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, .

f. Use the formula from part e. Here, the circumference of is 12 inches and . a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

Therefore, the circumference of is 4 inches.

ANSWER: b. Calculate the ratio of two times the total number a. Sample answer: of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. Sample answer. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. They all have the same shape—circular. c. Sample answer: The values are approaching 3.14, d. The ratio of their circumferences is also 2. which is approximately equal to π.

e. f. 4 in. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 46. BUFFON’S NEEDLE Measure the length of a miles from downtown Phoenix. needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center a. Drop the needle onto the paper. When the needle circle? lands, record whether it touches one of the lines as a b. As the radii of the circles increases by 5 miles, by hit. Record the number of hits after 25, 50, and 100 how much does the circumference increase? drops. SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the b. Calculate the ratio of two times the total number circumference of each circle. of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. The difference of the circumferences of the c. Sample answer: The values are approaching 3.14, outermost circle and the center circle is 60π - 10π = which is approximately equal to π. 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find ANSWER: the circumference of each circle. a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 π + miles from downtown Phoenix. The difference of the two circumferences is (2x 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between a. How much greater is the circumference of the circles and lines? outermost circle than the circumference of the center circle? SOLUTION: b. As the radii of the circles increases by 5 miles, by Sample answer: A line and a circle may intersect in how much does the circumference increase? one point, two points, or may not intersect at all. A SOLUTION: line that intersects a circle in one point can be described as a tangent. A line that intersects a circle a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the in exactly two points can be described as a secant. circumference of each circle. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER: The difference of the circumferences of the Sample answer: A line that intersects a circle in one outermost circle and the center circle is 60π - 10π = point can be described as a tangent. A line that 50π or about 157.1 miles. intersects a circle in exactly two points can be b. Let the radius of one circle be x miles and the described as a secant. A line segment with radius of the next larger circle be x + 5 miles. Find endpoints on a circle can be described as a chord. the circumference of each circle. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

The difference of the two circumferences is (2xπ + 10π) - (2xπ) = 10π or about 31.4 miles. 49. Therefore, as the radius of each circle increases by 5 REASONING In the figure, a circle with radius r is miles, the circumference increases by about 31.4 inscribed in a regular polygon and circumscribed miles. about another. a. What are the perimeters of the circumscribed and ANSWER: inscribed polygons in terms of r? Explain. a. 157.1 mi b. Is the circumference C of the circle greater or b. by about 31.4 mi less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality 48. WRITING IN MATH How can we describe the comparing C to these perimeters. relationships that exist between c. Rewrite the inequality from part b in terms of the circles and lines? diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed SOLUTION: and inscribed polygons increase, what will happen to Sample answer: A line and a circle may intersect in the upper and lower limits of the inequality from part one point, two points, or may not intersect at all. A c, and what does this imply? line that intersects a circle in one point can be SOLUTION: described as a tangent. A line that intersects a circle a. The circumscribed polygon is a square with sides in exactly two points can be described as a secant. of length 2r. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the The inscribed polygon is a hexagon with sides of center and on the circle can be described as a length r. (Each triangle formed using a central angle radius. will be equilateral.) ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius. So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the 49. circumscribed polygon and greater than the REASONING In the figure, a circle with radius r is perimeter of the inscribed polygon. inscribed in a regular polygon and circumscribed 8r < C < 6r about another. c. Since the diameter is twice the radius, 8r = 4(2r) a. What are the perimeters of the circumscribed and or 4d and 6r = 3(2r) or 3d. inscribed polygons in terms of r? Explain. The inequality is then 4d < C < 3d. Therefore, the b. Is the circumference C of the circle greater or circumference of the circle is between 3 and 4 times less than the perimeter of the circumscribed polygon? its diameter. the inscribed polygon? Write a compound inequality d. These limits will approach a value of πd, implying comparing C to these perimeters. that C = πd. c. Rewrite the inequality from part b in terms of the

diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed ANSWER: and inscribed polygons increase, what will happen to a. 8r and 6r ; Twice the radius of the circle, 2r is the upper and lower limits of the inequality from part c, and what does this imply? the side length of the square, so the perimeter of the square is 4(2r) or 8r. The SOLUTION: regular hexagon is made up of six equilateral a. The circumscribed polygon is a square with sides triangles with side length r, so the perimeter of the of length 2r. hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. The inscribed polygon is a hexagon with sides of d. These limits will approach a value of πd, implying length r. (Each triangle formed using a central angle that C = πd. will be equilateral.) 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ.

So, the perimeters are 8r and 6r. b. SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the The circumference is less than the perimeter of the circumference of each circle. circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd. The sum of the circumferences is 56π units. ANSWER: Substitute the three circumferences and solve for x. a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The

regular hexagon is made up of six equilateral So, the radius of is 4 units, the radius of is triangles with side length r, so the perimeter of the 2(4) or 8 units, and the radius of is 4(4) or 16 hexagon is 6(r) or 6r. units. b. less; greater; 6r < C < 8r The measure of KJ is equal to the sum of the radii c. 3d < C < 4d; The circumference of the circle is of and . between 3 and 4 times its diameter. Therefore, KJ = 16 + 8 or 24 units. d. These limits will approach a value of πd, implying that C = πd. ANSWER: 24 units 50. CHALLENGE The sum of the circumferences of 51. REASONING Is the distance from the center of a circles H, J, and K shown at the right is units. circle to a point in the interior of a circle sometimes, Find KJ. always, or never less than the radius of the circle? Explain. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: SOLUTION: Always; a radius is a segment drawn between the The radius of is 4x units, the radius of is x center of the circle and a point on the circle. A units, and the radius of is 2x units. Find the segment drawn from the center to a point inside the circumference of each circle. circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such The sum of the circumferences is 56π units. that each point that makes up is moved to be the Substitute the three circumferences and solve for x. same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

So, the radius of is 4 units, the radius of is

2(4) or 8 units, and the radius of is 4(4) or 16

units.

The measure of KJ is equal to the sum of the radii of and . Therefore, KJ = 16 + 8 or 24 units. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps ANSWER: onto , the circles are 24 units similar. Thus, all circles are similar.

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? ANSWER: Explain. A circle is a locus of points in a plane equidistant SOLUTION: from a given point. For any two circles and A radius is a segment drawn between the center of , there exists a translation that maps center A the circle and a point on the circle. A segment drawn onto center B, moving so that it is concentric from the center to a point inside the circle will always with . There also exists a dilation with scale have a length less than the radius of the circle. factor k such that each point that makes up is moved to be the same distance from center A as the ANSWER: points that make up are from center B. Always; a radius is a segment drawn between the center of the circle and a point on the circle. A Therefore, is mapped onto . Since there segment drawn from the center to a point inside the exists a rigid motion followed by a scaling that maps 10-1circle Circles will and always Circumference have a length less than the radius onto , the circles are similar. Thus, all of the circle. circles are similar.

52. PROOF Use the locus definition of a circle and 53. CHALLENGE In the figure, is inscribed in dilations to prove that all circles are similar. equilateral triangle LMN. What is the circumference of ? SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make SOLUTION: up are from center B. Draw a perpendicular to the side Also join For some value k, r = 2 kr1.

Therefore, is mapped onto . Since there Since is equilateral triangle, MN = 8 inches exists a rigid motion followed by a scaling that maps and RN = 4 inches. Also, bisects onto , the circles are similar. Thus, all circles are similar. Use the definition of tangent of an angle to find PR, which is the radius of

ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale Therefore, the radius of is factor k such that each point that makes up is The circumference C of a circle with radius r is moved to be the same distance from center A as the given by points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

53. CHALLENGE In the figure, is inscribed in The circumference of is inches. equilateral triangle LMN. What is the circumference of ? ANSWER:

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of eSolutions Manual - Powered by Cognero geometry. Page 16

SOLUTION: SOLUTION: See students’ work. Draw a perpendicular to the side Also join ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of is The circumference C of a circle with radius r is given by

ANSWER: 40.8

The circumference of is inches. 56. What is the radius of a table with a circumference of 10 feet? ANSWER: A 1.6 ft B 2.5 ft in. C 3.2 ft D 5 ft 54. WRITING IN MATH Research and write about SOLUTION: the history of pi and its importance to the study of geometry. The circumference C of a circle with radius r is given by SOLUTION: Here, C = 10 ft. Use the formula to find the radius. See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use SOLUTION: up to 50 feet of fence, what radius can he use for the garden? Use the Pythagorean Theorem to find the diameter of the circle. F 10 G 9 H 8 The circumference C of a circle with diameter d is J 7 given by Use the formula to find the circumference. SOLUTION: The circumference C of a circle with radius r is given by Because 50 feet of fence can be used to border the garden, C = 50. Use the circumference formula to max find the maximum radius.

ANSWER: 40.8 56. What is the radius of a table with a circumference of 10 feet? Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only A 1.6 ft correct choice is J. B 2.5 ft C 3.2 ft ANSWER: D 5 ft J

SOLUTION: 58. SAT/ACT What is the radius of a circle with an The circumference C of a circle with radius r is given by area of square units? Here, C = 10 ft. Use the formula to find the radius. A 0.4 units B 0.5 units C 2 units D 4 units E 16 units

The radius of the circle is about 1.6 feet, so the SOLUTION: correct choice is A. The area of a circle of radius r is given by the ANSWER: formula Here A = square units. Use the A formula to find the radius.

57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? F 10 Therefore, the correct choice is B. G 9 H 8 ANSWER: J 7 B SOLUTION: Copy each figure and point B. Then use a ruler The circumference C of a circle with radius r is to draw the image of the figure under a dilation given by with center B and the scale factor r indicated.

Because 50 feet of fence can be used to border the 59. garden, Cmax = 50. Use the circumference formula to find the maximum radius.

SOLUTION:

Therefore, the radius of the garden must be less than Draw a ray from point B through a vertex of the or equal to about 7.96 feet. Therefore, the only hexagon. Use the ruler to measure the distance from correct choice is J. B to the vertex. Mark a point on the ray that is of ANSWER: that distance from B. Repeat the process for rays through the other five vertices. Connect the six points J on the rays to construct the dilated image of the hexagon. 58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the ANSWER: formula to find the radius.

60. Therefore, the correct choice is B.

ANSWER: B

Copy each figure and point B. Then use a ruler SOLUTION: to draw the image of the figure under a dilation Draw a ray from point B through a vertex of the with center B and the scale factor r indicated. triangle. Use the ruler to measure the distance from

59. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of ANSWER: that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon. 61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image ANSWER: of the trapezoid.

60.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays 62. through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. 61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid. ANSWER:

ANSWER:

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. 62. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: SOLUTION: No Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from Refer to the image on page 705. B to the vertex. Mark a point on the ray that is 3 64. times that distance from B. Repeat the process for SOLUTION: rays through the other two vertices. Connect the The figure can be mapped onto itself by a rotation of three points on the rays to construct the dilated image 90º. So, it has a rotational symmetry. So, the of the triangle. magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

ANSWER: 65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705.

SOLUTION: State whether each figure has rotational The given figure cannot be mapped onto itself by any symmetry. Write yes or no. If so, copy the rotation. So, it does not have a rotational symmetry. figure, locate the center of symmetry, and state the order and magnitude of symmetry. ANSWER: 63. Refer to the image on page 705. No SOLUTION: Determine the truth value of the following The figure cannot be mapped onto itself by any statement for each set of conditions. Explain rotation. So, it does not have a rotational symmetry. your reasoning. If you are over 18 years old, then you vote in all ANSWER: elections. No 67. You are 19 years old and you vote. SOLUTION: 64. Refer to the image on page 705. True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true for the conditions. The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the ANSWER: magnitude of symmetry is 90º and order of symmetry True; sample answer: Since the hypothesis is true

is 4. and the conclusion is true, then the statement is true ANSWER: for the conditions. Yes; 4, 90 68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false 65. for the conditions. SOLUTION: Find x. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705. 69. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any The sum of the measures of angles around a point is rotation. So, it does not have a rotational symmetry. 360º.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain ANSWER: your reasoning. 90 If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true

for the conditions. 70. SOLUTION: ANSWER: The angles with the measures xº and 2xº form a True; sample answer: Since the hypothesis is true straight line. So, the sum is 180. and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: ANSWER: False; sample answer: Since the hypothesis is true 60 and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. 71. Find x. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

69. ANSWER: SOLUTION: 20 The sum of the measures of angles around a point is 360º.

ANSWER: 72. 90 SOLUTION: The sum of the three angles is 180.

70. ANSWER: SOLUTION: 45 The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

For Exercises 1–4, refer to . 3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm

1. Name the circle. 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: SOLUTION: The center of the circle is N. So, the circle is Here, is a radius and the diameter is twice the radius. ANSWER:

Therefore, the diameter is 26 ft. 2. Identify each. ANSWER: a. a chord 26 ft b. a diameter c. a radius The diameters of , , and are 8 SOLUTION: inches, 18 inches, and 11 inches, respectively. Find each measure. a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius. 5. FG ANSWER: a. SOLUTION: b. Since the diameter of B is 18 inches and A is 8 c. inches, AG = 18 and AF = (8) or 4.

3. If CN = 8 centimeters, find DN. SOLUTION:

Here, are radii of the same circle. So, Therefore, FG = 14 in. they are equal in length. Therefore, DN = 8 cm. ANSWER: ANSWER: 14 in. 8 cm 4. If EN = 13 feet, what is the diameter of the circle? 6. FB SOLUTION: SOLUTION: Here, is a radius and the diameter is twice the Since the diameter of B is 18 inches and A is 4 radius. inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, the diameter is 26 ft.

ANSWER: Therefore, FB = 5 inches. 26 ft ANSWER: The diameters of , , and are 8 5 in. inches, 18 inches, and 11 inches, respectively. Find each measure. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet.

5. FG SOLUTION: Since the diameter of B is 18 inches and A is 8

inches, AG = 18 and AF = (8) or 4. Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft Therefore, FG = 14 in. 8. CCSS MODELING The circumference of the ANSWER: circular swimming pool shown is about 56.5 feet. 14 in. What are the diameter and radius of the pool? Round to the nearest hundredth. 6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

SOLUTION: Therefore, FB = 5 inches.

ANSWER: 5 in.

7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

ANSWER: 17.98 ft; 8.99 ft

Therefore, the circumference of the ride is about 9. SHORT RESPONSE The right triangle shown is 138.23 feet. inscribed in . Find the exact circumference of . ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

SOLUTION: The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π centimeters. ANSWER:

For Exercises 10–13, refer to . The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

ANSWER: 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of 10. Name the center of the circle. . SOLUTION: R ANSWER: R

11. Identify a chord that is also a diameter. SOLUTION: SOLUTION: The diameter of the circle is the hypotenuse of the right triangle. Two chords are shown: and . goes through the center, R, so is a diameter.

ANSWER:

The diameter of the circle is 4 centimeters. 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center

The circumference of the circle is 4π and on the circle. But has both the end points on centimeters. the circle, so it is a chord. ANSWER: ANSWER: No; it is a chord.

13. If SU = 16.2 centimeters, what is RT? For Exercises 10–13, refer to . SOLUTION: Here, is a diameter and is a radius.

10. Name the center of the circle. Therefore, RT = 8.1 cm. SOLUTION: R ANSWER: 8.1 cm ANSWER: R For Exercises 14–17, refer to .

11. Identify a chord that is also a diameter. SOLUTION: Two chords are shown: and . goes through the center, R, so is a diameter. 14. Identify a chord that is not a diameter. SOLUTION: ANSWER: The chords do not pass through the center. So, they are not diameters.

12. Is a radius? Explain. ANSWER: SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on 15. If CF = 14 inches, what is the diameter of the circle?

the circle, so it is a chord. SOLUTION: ANSWER: Here, is a radius and the diameter is twice the No; it is a chord. radius. . 13. If SU = 16.2 centimeters, what is RT? SOLUTION:

Here, is a diameter and is a radius. Therefore, the diameter of the circle is 28 inches. ANSWER: 28 in.

16. Is ? Explain.

Therefore, RT = 8.1 cm. SOLUTION: All radii of a circle are congruent. Since ANSWER:

8.1 cm are both radii of , . ANSWER: For Exercises 14–17, refer to . Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? SOLUTION: Here, is a diameter and is a radius.

14. Identify a chord that is not a diameter. SOLUTION: The chords do not pass through the center. So, they are not diameters. Therefore, EF = 3.7 cm.

ANSWER: ANSWER: 3.7 cm

Circle J has a radius of 10 units, has a 15. If CF = 14 inches, what is the diameter of the circle? radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. Here, is a radius and the diameter is twice the radius. .

Therefore, the diameter of the circle is 28 inches. 18. CK ANSWER: 28 in. SOLUTION: Since the radius of K is 8 units, BK = 8. 16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since Therefore, CK = 2.6 units. are both radii of , . ANSWER: ANSWER: 2.6 Yes; they are both radii of . 19. AB 17. If DA = 7.4 centimeters, what is EF? SOLUTION: SOLUTION: Since is a diameter of circle J and the radius is Here, is a diameter and is a radius. 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units.

Therefore, EF = 3.7 cm. ANSWER: 14.6 ANSWER: 3.7 cm 20. JK

Circle J has a radius of 10 units, has a SOLUTION: radius of 8 units, and BC = 5.4 units. Find each First find CK. Since circle K has a radius of 8 units,

measure. BK = 8.

Since circle J has a radius of 10 units, JC = 10.

18. CK Therefore, JK = 12.6 units. SOLUTION: ANSWER: Since the radius of K is 8 units, BK = 8. 12.6 21. AD SOLUTION: Therefore, CK = 2.6 units. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J ANSWER: is 10 units, so KD = 8 and AC = 2(10) or 20. Before 2.6 finding AD, we need to find CK.

19. AB SOLUTION: Since is a diameter of circle J and the radius is

10, AC = 2(10) or 20 units. Therefore, AD = 30.6 units.

ANSWER: Therefore, AB = 14.6 units. 30.6

ANSWER: 22. PIZZA Find the radius and circumference of the 14.6 pizza shown. Round to the nearest hundredth, if necessary. 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Since circle J has a radius of 10 units, JC = 10. SOLUTION: The diameter of the pizza is 16 inches. Therefore, JK = 12.6 units.

ANSWER: 12.6 . 21. AD So, the radius of the pizza is 8 inches. SOLUTION: The circumference C of a circle of diameter d is We can find AD using AD = AC + CK + KD. The given by radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in.

Therefore, AD = 30.6 units. 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a ANSWER: tire. Round to the nearest hundredth, if necessary. 30.6 SOLUTION: 22. PIZZA Find the radius and circumference of the The diameter of a tire is 26 inches. The radius is half pizza shown. Round to the nearest hundredth, if the diameter. necessary.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

SOLUTION: The diameter of the pizza is 16 inches. Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. . So, the radius of the pizza is 8 inches. Find the diameter and radius of a circle by with The circumference C of a circle of diameter d is the given circumference. Round to the nearest given by hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is Therefore, the circumference of the pizza is about given by 50.27 inches. Here, C = 18 in. Use the formula to find the diameter. Then find the radius. ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter.

Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: So, the radius of the tires is 13 inches. 5.73 in.; 2.86 in. The circumference C of a circle with diameter d is given by 25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is Therefore, the circumference of the bicycle tire is given by about 81.68 inches. Here, C = 124 ft. Use the formula to find the diameter. Then find the radius. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the Therefore, the diameter is about 39.47 feet and the

diameter. Then find the radius. radius is about 19.74 feet.

ANSWER: 39.47 ft; 19.74 ft

26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches.

ANSWER: 5.73 in.; 2.86 in.

25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the Therefore, the diameter is about 119.46 centimeters diameter. Then find the radius. and the radius is about 59.73 centimeters. ANSWER: 119.46 cm; 59.73 cm

27. C = 2608.25 m

SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet.

ANSWER: 39.47 ft; 19.74 ft

26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by Therefore, the diameter is about 830.23 meters and Here, C = 375.3 cm. Use the formula to find the the radius is about 415.12 meters. diameter. Then find the radius. ANSWER: 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or circumscribed polygon.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. 28. ANSWER: SOLUTION: 119.46 cm; 59.73 cm The hypotenuse of the right triangle is a diameter of 27. C = 2608.25 m the circle. Use the Pythagorean Theorem to find the diameter. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find

the diameter. Then find the radius. The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the circumference of the circle is 17 centimeters.

ANSWER:

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given 29. inscribed or circumscribed polygon. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

28.

SOLUTION: The diameter of the circle is 12 feet. The The hypotenuse of the right triangle is a diameter of circumference C of a circle is given by the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the circumference of the circle is 12 feet.

The diameter of the circle is 17 cm. The ANSWER: circumference C of a circle is given by

Therefore, the circumference of the circle is 17 centimeters. 30. ANSWER: SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

29. SOLUTION: The diameter of the circle is inches. The The hypotenuse of the right triangle is a diameter of circumference C of a circle is given by the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the circumference of the circle is inches. The diameter of the circle is 12 feet. The circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 12 feet.

ANSWER: 31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter The diameter of the circle is 10 inches. The of the circle. circumference C of a circle is given by

Therefore, the circumference of the circle is 10 inches. The diameter of the circle is inches. The circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is inches. 32. ANSWER: SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by

31.

SOLUTION: Therefore, the circumference of the circle is 25 Each diagonal of the inscribed rectangle will pass millimeters. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter ANSWER: of the circle.

The diameter of the circle is 10 inches. The circumference C of a circle is given by

33. SOLUTION: Therefore, the circumference of the circle is 10 A diameter perpendicular to a side of the square and inches. the length of each side will measure the same. So, d = 14 yards. ANSWER: The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 yards.

32. ANSWER: SOLUTION: A diameter perpendicular to a side of the square and 34. DISC GOLF Disc golf is similar to regular golf, the length of each side will measure the same. So, d except that a flying disc is used instead of a ball and = 25 millimeters. clubs. For professional competitions, the maximum

The circumference C of a circle is given by weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters?

Round to the nearest tenth. Therefore, the circumference of the circle is 25 SOLUTION: millimeters. The circumference C of a circle with diameter d is

ANSWER: given by Here, C = 66.92 cm. Use the formula to find the diameter.

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the 33. diameter in centimeters. SOLUTION: Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d ANSWER:

= 14 yards. 176.8 g The circumference C of a circle is given by 35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference?

Therefore, the circumference of the circle is 14 b. If Mr. Martinez changes the plans so that the yards. inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the ANSWER: nearest foot?

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? SOLUTION:

Round to the nearest tenth. a. The radius of the patio is 3 + 2 = 5 ft. The SOLUTION: circumference C of a circle with radius r is given by The circumference C of a circle with diameter d is given by Use the formula to find the circumference. Here, C = 66.92 cm. Use the formula to find the diameter.

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. So, the radius of the inner circle is about 4 feet. ANSWER: ANSWER: 176.8 g a. 31.42 ft 35. PATIOS Mr. Martinez is going to build the patio b. 4 ft shown. a. What is the patio’s approximate circumference? The radius, diameter, or circumference of a b. If Mr. Martinez changes the plans so that the circle is given. Find each missing measure to inner circle has a circumference of approximately 25 the nearest hundredth.

feet, what should the radius of the circle be to the 36. nearest foot? SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by

Use the formula to find the circumference.

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

Therefore, the circumference is about 31.42 feet. ANSWER: b. Here, Cinner = 25 ft. Use the formula to find the 4.25 in.; 26.70 in. radius of the inner circle.

37.

SOLUTION: The diameter is twice the radius and the

So, the radius of the inner circle is about 4 feet. circumference C of a circle with diameter d is given by ANSWER: a. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

36. So, the diameter is 22.80 feet and the circumference is about 71.63 feet. SOLUTION: ANSWER: The radius is half the diameter and the circumference C of a circle with diameter d is given 22.80 ft; 71.63 ft by 38.

SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 35x cm. Use the formula to find the diameter.Then find the radius.

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

37. Therefore, the diameter is about 11.14x centimeters SOLUTION: and the radius is about 5.57x centimeters. The diameter is twice the radius and the circumference C of a circle with diameter d is given ANSWER: by 11.14x cm; 5.57x cm

39.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft 38.

SOLUTION: The circumference C of a circle with diameter d is So, the diameter is 0.25x units and the circumference given by is about 0.79x units. Here, C = 35x cm. Use the formula to find the ANSWER: diameter.Then find the radius. 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or neither. 40. Refer to the photo on page 703. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles.

ANSWER: Therefore, the diameter is about 11.14x centimeters concentric and the radius is about 5.57x centimeters.

ANSWER: 41. Refer to the photo on page 703. 11.14x cm; 5.57x cm SOLUTION: The circles neither have the same radius nor are they 39. concentric. ANSWER: SOLUTION: neither The diameter is twice the radius and the circumference C of a circle with diameter d is given 42. Refer to the photo on page 703. by SOLUTION: The circles appear to have the same radius. So, they appear to be congruent.

ANSWER: congruent

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the So, the diameter is 0.25x units and the circumference coordinate grid represents 25 feet, how far would is about 0.79x units. someone have to walk to go completely around the ring? Round to the nearest tenth. ANSWER: 0.25x; 0.79x Determine whether the circles in the figures below appear to be congruent, concentric, or neither. 40. Refer to the photo on page 703. SOLUTION: The circles have the same center and they are in the same plane. So, they are concentric circles. ANSWER: concentric SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, 41. Refer to the photo on page 703. the diameter is 150 ft. The circumference C of a circle with diameter d is given by SOLUTION: So, the circumference is The circles neither have the same radius nor are they Therefore, someone have to concentric. walk about 471.2 feet to go completely around the ANSWER: ring. neither ANSWER: 42. Refer to the photo on page 703. 471.2 ft

SOLUTION: 44. CCSS MODELING A brick path is being installed The circles appear to have the same radius. So, they around a circular pond. The pond has a appear to be congruent. circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way ANSWER: around. What is the approximate circumference of congruent the path? Round to the nearest hundredth.

43. HISTORY The Indian Shell Ring on Hilton Head SOLUTION: Island approximates a circle. If each unit on the The circumference C of a circle with radius r is coordinate grid represents 25 feet, how far would given by Here, Cpond = 68 ft. Use the someone have to walk to go completely around the formula to find the radius of the pond. ring? Round to the nearest tenth.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

SOLUTION: Therefore, the circumference of the path is about The radius of the circle is 3 units, that is 75 feet. So, 93.13 feet. the diameter is 150 ft. The circumference C of a ANSWER: circle with diameter d is given by 93.13 ft So, the circumference is Therefore, someone have to 45. MULTIPLE REPRESENTATIONS In this walk about 471.2 feet to go completely around the problem, you will explore changing dimensions in ring. circles. a. GEOMETRIC Use a compass to draw three ANSWER: circles in which the scale factor from each circle to 471.2 ft the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest 44. CCSS MODELING A brick path is being installed tenth) and circumference (to the nearest hundredth) around a circular pond. The pond has a of each circle. Record your results in a table. circumference of 68 feet. The outer edge of the path c. VERBAL Explain why these three circles are is going to be 4 feet from the pond all the way geometrically similar. around. What is the approximate circumference of d. VERBAL Make a conjecture about the ratio the path? Round to the nearest hundredth. between the circumferences of two circles when the SOLUTION: ratio between their radii is 2. The circumference C of a circle with radius r is e. ANALYTICAL The scale factor from to given by Here, C = 68 ft. Use the pond is . Write an equation relating the formula to find the radius of the pond. circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to

is , and the circumference of is 12 So, the radius of the pond is about 10.82 feet and the inches, what is the circumference of ? radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference SOLUTION: of the brick path. a. Sample answer:

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft b. Sample answer.

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. c. b. TABULAR Calculate the radius (to the nearest They all have the same shape–circular. tenth) and circumference (to the nearest hundredth) d. The ratio of their circumferences is also 2. of each circle. Record your results in a table. e. The circumference is directly proportional to radii. c. VERBAL Explain why these three circles are So, if the radii are in the ratio b : a, so will the

geometrically similar. circumference. Therefore, . d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the f. Use the formula from part e. Here, the ratio between their radii is 2. circumference of is 12 inches and . e. ANALYTICAL The scale factor from to is . Write an equation relating the circumference (C ) of to the circumference A (CB) of . Therefore, the circumference of is 4 inches. f. NUMERICAL If the scale factor from to ANSWER: is , and the circumference of is 12 a. Sample answer: inches, what is the circumference of ? SOLUTION: a. Sample answer:

b. Sample answer.

b. Sample answer.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e.

c. They all have the same shape–circular. f. 4 in. d. The ratio of their circumferences is also 2. 46. BUFFON’S NEEDLE Measure the length of a e. The circumference is directly proportional to radii. needle (or toothpick) in centimeters. Next, draw a set So, if the radii are in the ratio b : a, so will the of horizontal lines that are centimeters apart on a circumference. Therefore, . sheet of plain white paper. f. Use the formula from part e. Here, the circumference of is 12 inches and .

Therefore, the circumference of is 4 inches. a. Drop the needle onto the paper. When the needle ANSWER: lands, record whether it touches one of the lines as a a. Sample answer: hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 b. Sample answer. drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work.

c. Sample answer: The values are approaching 3.14, c. They all have the same shape—circular. which is approximately equal to π. d. T he ratio of their circumferences is also 2.

e. ANSWER: f. 4 in. a. See students’ work. b. See students’ work. 46. BUFFON’S NEEDLE Measure the length of a c. Sample answer: The values are approaching 3.14, needle (or toothpick) in centimeters. Next, draw a set which is approximately equal to π. of horizontal lines that are centimeters apart on a sheet of plain white paper. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

a. How much greater is the circumference of the outermost circle than the circumference of the center b. Calculate the ratio of two times the total number circle? of drops to the number of hits after 25, 50, and 100 b. As the radii of the circles increases by 5 miles, by drops. how much does the circumference increase? SOLUTION: c. How are the values you found in part b related to a. π? The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the SOLUTION: circumference of each circle. a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. The difference of the circumferences of the b. See students’ work. outermost circle and the center circle is 60π - 10π = c. Sample answer: The values are approaching 3.14, 50π or about 157.1 miles. which is approximately equal to π. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

The difference of the two circumferences is (2xπ + 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

a. How much greater is the circumference of the ANSWER: outermost circle than the circumference of the center a. 157.1 mi circle? b. by about 31.4 mi b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? 48. WRITING IN MATH How can we describe the SOLUTION: relationships that exist between a. The radius of the outermost circle is 30 miles and circles and lines? that of the center circle is 5 miles. Find the SOLUTION: circumference of each circle. Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through The difference of the circumferences of the the center of the circle, it can be described as a outermost circle and the center circle is 60π - 10π = diameter. A line segment with endpoints at the 50π or about 157.1 miles. center and on the circle can be described as a b. Let the radius of one circle be x miles and the radius. radius of the next larger circle be x + 5 miles. Find the circumference of each circle. ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment The difference of the two circumferences is (2xπ + with endpoints at the center and on the circle can be 10π) - (2xπ) = 10π or about 31.4 miles. described as a radius. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi 49. b. by about 31.4 mi REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed 48. WRITING IN MATH How can we describe the about another. relationships that exist between a. What are the perimeters of the circumscribed and circles and lines? inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or SOLUTION: less than the perimeter of the circumscribed polygon? Sample answer: A line and a circle may intersect in the inscribed polygon? Write a compound inequality one point, two points, or may not intersect at all. A comparing C to these perimeters. line that intersects a circle in one point can be c. Rewrite the inequality from part b in terms of the described as a tangent. A line that intersects a circle diameter d of the circle and interpret its meaning. in exactly two points can be described as a secant. d. As the number of sides of both the circumscribed A line segment with endpoints on a circle can be and inscribed polygons increase, what will happen to described as a chord. If the chord passes through the upper and lower limits of the inequality from part the center of the circle, it can be described as a c, and what does this imply? diameter. A line segment with endpoints at the SOLUTION: center and on the circle can be described as a a. The circumscribed polygon is a square with sides radius. of length 2r. ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be The inscribed polygon is a hexagon with sides of described as a secant. A line segment with length r. (Each triangle formed using a central angle will be equilateral.) endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

49. So, the perimeters are 8r and 6r. REASONING In the figure, a circle with radius r is b. inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. The circumference is less than the perimeter of the b. Is the circumference C of the circle greater or circumscribed polygon and greater than the less than the perimeter of the circumscribed polygon? perimeter of the inscribed polygon. the inscribed polygon? Write a compound inequality 8r < C < 6r comparing C to these perimeters. c. Since the diameter is twice the radius, 8r = 4(2r) c. Rewrite the inequality from part b in terms of the or 4d and 6r = 3(2r) or 3d. diameter d of the circle and interpret its meaning. The inequality is then 4d < C < 3d. Therefore, the d. As the number of sides of both the circumscribed circumference of the circle is between 3 and 4 times and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part its diameter. c, and what does this imply? d. These limits will approach a value of πd, implying that C = πd. SOLUTION: a. The circumscribed polygon is a square with sides of length 2r. ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The

The inscribed polygon is a hexagon with sides of regular hexagon is made up of six equilateral length r. (Each triangle formed using a central angle triangles with side length r, so the perimeter of the will be equilateral.) hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the

perimeter of the inscribed polygon. 8r < C < 6r SOLUTION: c. Since the diameter is twice the radius, 8r = 4(2r) The radius of is 4x units, the radius of is x or 4d and 6r = 3(2r) or 3d. units, and the radius of is 2x units. Find the

The inequality is then 4d < C < 3d. Therefore, the circumference of each circle. circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral The sum of the circumferences is 56π units. triangles with side length r, so the perimeter of the Substitute the three circumferences and solve for x. hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying So, the radius of is 4 units, the radius of is that C = πd. 2(4) or 8 units, and the radius of is 4(4) or 16 units. 50. CHALLENGE The sum of the circumferences of The measure of KJ is equal to the sum of the radii circles H, J, and K shown at the right is units. of and . Find KJ. Therefore, KJ = 16 + 8 or 24 units.

ANSWER: 24 units

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Explain. SOLUTION: SOLUTION: The radius of is 4x units, the radius of is x A radius is a segment drawn between the center of units, and the radius of is 2x units. Find the the circle and a point on the circle. A segment drawn circumference of each circle. from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar.

SOLUTION: The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x. A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such So, the radius of is 4 units, the radius of is that each point that makes up is moved to be the 2(4) or 8 units, and the radius of is 4(4) or 16 same distance from center A as the points that make units. up are from center B. The measure of KJ is equal to the sum of the radii For some value k, r2 = of and . kr . Therefore, KJ = 16 + 8 or 24 units. 1

ANSWER: 24 units

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Therefore, is mapped onto . Since there Explain. exists a rigid motion followed by a scaling that maps onto , the circles are SOLUTION: similar. Thus, all circles are similar. A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle. ANSWER: A circle is a locus of points in a plane equidistant ANSWER: from a given point. For any two circles and Always; a radius is a segment drawn between the , there exists a translation that maps center A center of the circle and a point on the circle. A onto center B, moving so that it is concentric segment drawn from the center to a point inside the circle will always have a length less than the radius with . There also exists a dilation with scale of the circle. factor k such that each point that makes up is moved to be the same distance from center A as the 52. PROOF Use the locus definition of a circle and points that make up are from center B. dilations to prove that all circles are similar. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps SOLUTION: onto , the circles are similar. Thus, all A circle is a locus of points in a plane equidistant from a given point. For any two circles and circles are similar. , there exists a translation that CHALLENGE maps center A onto center B, moving so that it 53. In the figure, is inscribed in is concentric with . There also exists a dilation equilateral triangle LMN. What is the circumference with scale factor k such of ? that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

SOLUTION: Draw a perpendicular to the side Also join

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

ANSWER: Since is equilateral triangle, MN = 8 inches A circle is a locus of points in a plane equidistant and RN = 4 inches. Also, bisects from a given point. For any two circles and Use the definition of , there exists a translation that maps center A tangent of an angle to find PR, which is the radius of onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps Therefore, the radius of is onto , the circles are similar. Thus, all circles are similar. The circumference C of a circle with radius r is given by 53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference of ?

The circumference of is inches.

ANSWER:

SOLUTION: in. Draw a perpendicular to the side Also join 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

ANSWER: See students’ work. Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects 55. GRIDDED RESPONSE What is the Use the definition of circumference of ? Round to the nearest tenth. tangent of an angle to find PR, which is the radius of

SOLUTION: Use the Pythagorean Theorem to find the diameter Therefore, the radius of is of the circle. The circumference C of a circle with radius r is given by The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

The circumference of is inches.

ANSWER:

10-1 Circles and in.Circumference ANSWER: 40.8

54. WRITING IN MATH Research and write about 56. What is the radius of a table with a circumference of the history of pi and its importance to the study of 10 feet? geometry. A 1.6 ft B 2.5 ft SOLUTION: C 3.2 ft See students’ work. D 5 ft ANSWER: SOLUTION: See students’ work. The circumference C of a circle with radius r is

55. GRIDDED RESPONSE What is the given by Here, C = 10 ft. Use the formula to find the radius. circumference of ? Round to the nearest tenth.

The radius of the circle is about 1.6 feet, so the correct choice is A. SOLUTION: Use the Pythagorean Theorem to find the diameter ANSWER: of the circle. A

57. ALGEBRA Bill is planning a circular vegetable The circumference C of a circle with diameter d is garden with a fence around the border. If he can use

given by up to 50 feet of fence, what radius can he use for the

Use the formula to find the circumference. garden?

F 10 G 9 H 8 J 7

SOLUTION:

The circumference C of a circle with radius r is ANSWER: given by 40.8 Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to 56. What is the radius of a table with a circumference of find the maximum radius. 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft

SOLUTION: Therefore, the radius of the garden must be less than The circumference C of a circle with radius r is or equal to about 7.96 feet. Therefore, the only given by correct choice is J. Here, C = 10 ft. Use the formula to find the radius. ANSWER: J

58. SAT/ACT What is the radius of a circle with an eSolutions Manual - Powered by Cognero Page 17 area of square units? The radius of the circle is about 1.6 feet, so the A 0.4 units correct choice is A. B 0.5 units ANSWER: C 2 units A D 4 units E 16 units 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use SOLUTION: up to 50 feet of fence, what radius can he use for the The area of a circle of radius r is given by the garden? formula Here A = square units. Use the F 10 formula to find the radius. G 9 H 8 J 7 SOLUTION:

The circumference C of a circle with radius r is Therefore, the correct choice is B. given by Because 50 feet of fence can be used to border the ANSWER: garden, Cmax = 50. Use the circumference formula to B find the maximum radius. Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J SOLUTION: Draw a ray from point B through a vertex of the SAT/ACT 58. What is the radius of a circle with an hexagon. Use the ruler to measure the distance from

area of square units? B to the vertex. Mark a point on the ray that is of A 0.4 units that distance from B. Repeat the process for rays through the other five vertices. Connect the six points B 0.5 units on the rays to construct the dilated image of the C 2 units hexagon. D 4 units E 16 units SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

Therefore, the correct choice is B. ANSWER:

ANSWER: B

Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation 60. with center B and the scale factor r indicated.

59.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from

B to the vertex. Mark a point on the ray that is of SOLUTION: that distance from B. Repeat the process for rays Draw a ray from point B through a vertex of the through the other two vertices. Connect the three hexagon. Use the ruler to measure the distance from points on the rays to construct the dilated image of B to the vertex. Mark a point on the ray that is of the triangle. that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

ANSWER:

61.

SOLUTION: ANSWER: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image 60. of the trapezoid.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of ANSWER: that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

62.

ANSWER:

SOLUTION: 61. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image SOLUTION: of the triangle. Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

ANSWER:

62. State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. SOLUTION: 63. Refer to the image on page 705. Draw a ray from point B through a vertex of the SOLUTION: triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 The figure cannot be mapped onto itself by any

times that distance from B. Repeat the process for rotation. So, it does not have a rotational symmetry. rays through the other two vertices. Connect the ANSWER: three points on the rays to construct the dilated image No of the triangle.

64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

ANSWER:

65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. State whether each figure has rotational symmetry. Write yes or no. If so, copy the ANSWER: figure, locate the center of symmetry, and state No the order and magnitude of symmetry. 66. Refer to the image on page 705. 63. Refer to the image on page 705. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: No No Determine the truth value of the following 64. Refer to the image on page 705. statement for each set of conditions. Explain your reasoning. SOLUTION: If you are over 18 years old, then you vote in all The figure can be mapped onto itself by a rotation of elections. 90º. So, it has a rotational symmetry. So, the 67. You are 19 years old and you vote. magnitude of symmetry is 90º and order of symmetry is 4. SOLUTION: True; sample answer: Since the hypothesis is true ANSWER: and the conclusion is true, then the statement is true Yes; 4, 90 for the conditions. ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: 65. False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false SOLUTION: for the conditions. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. ANSWER: False; sample answer: Since the hypothesis is true ANSWER: and the conclusion is false, then the statement is false No for the conditions.

66. Refer to the image on page 705. Find x. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No 69. Determine the truth value of the following SOLUTION: statement for each set of conditions. Explain The sum of the measures of angles around a point is your reasoning. 360º. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true 90 for the conditions.

ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. 70. 68. You are 21 years old and do not vote. SOLUTION: SOLUTION: The angles with the measures xº and 2xº form a False; sample answer: Since the hypothesis is true straight line. So, the sum is 180. and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true ANSWER: and the conclusion is false, then the statement is false 60 for the conditions.

Find x.

71. 69. SOLUTION: The angles with the measures 6xº and 3xº form a SOLUTION: straight line. So, the sum is 180. The sum of the measures of angles around a point is 360º.

ANSWER: 20 ANSWER: 90

72. 70. SOLUTION: SOLUTION: The sum of the three angles is 180.

The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 45 ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. For Exercises 1–4, refer to . So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: 1. Name the circle. a. SOLUTION: b. c. The center of the circle is N. So, the circle is 3. If CN = 8 centimeters, find DN. ANSWER: SOLUTION: Here, are radii of the same circle. So, 2. Identify each. they are equal in length. Therefore, DN = 8 cm. a. a chord ANSWER: b. a diameter 8 cm c. a radius SOLUTION: 4. If EN = 13 feet, what is the diameter of the circle? a. A chord is a segment with endpoints on the circle. SOLUTION: So, here are chords. Here, is a radius and the diameter is twice the b. A diameter of a circle is a chord that passes radius. through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is Therefore, the diameter is 26 ft. radius. ANSWER: ANSWER: 26 ft a. The diameters of , , and are 8 b. inches, 18 inches, and 11 inches, respectively. c. Find each measure. 3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 5. FG 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Since the diameter of B is 18 inches and A is 8 SOLUTION: inches, AG = 18 and AF = (8) or 4. Here, is a radius and the diameter is twice the radius.

Therefore, the diameter is 26 ft. Therefore, FG = 14 in.

ANSWER: ANSWER: 26 ft 14 in. The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. 6. FB Find each measure. SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

5. FG Therefore, FB = 5 inches. SOLUTION: ANSWER: Since the diameter of B is 18 inches and A is 8 5 in. inches, AG = 18 and AF = (8) or 4. 7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary.

Therefore, FG = 14 in. SOLUTION: The radius is half the diameter. So, the radius of the ANSWER: circular ride is 22 feet. 14 in.

6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 Therefore, the circumference of the ride is about inches, AB = (18) or 9 and AF = (8) or 4. 138.23 feet. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the Therefore, FB = 5 inches. circular swimming pool shown is about 56.5 feet. ANSWER: What are the diameter and radius of the pool? Round to the nearest hundredth. 5 in.

7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. SOLUTION:

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. The diameter of the pool is about 17.98 feet and the What are the diameter and radius of the pool? Round radius of the pool is about 8.99 feet. to the nearest hundredth. ANSWER: 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of .

SOLUTION:

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

The diameter of the pool is about 17.98 feet and the The diameter of the circle is 4 centimeters. radius of the pool is about 8.99 feet.

ANSWER: 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is The circumference of the circle is 4π inscribed in . Find the exact circumference of centimeters. . ANSWER:

For Exercises 10–13, refer to .

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

10. Name the center of the circle. SOLUTION: R

The diameter of the circle is 4 centimeters. ANSWER: R

11. Identify a chord that is also a diameter. SOLUTION: The circumference of the circle is 4π centimeters. Two chords are shown: and . goes through the center, R, so is a diameter. ANSWER:

ANSWER: For Exercises 10–13, refer to .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on

10. Name the center of the circle. the circle, so it is a chord. SOLUTION: ANSWER: R No; it is a chord. ANSWER: 13. If SU = 16.2 centimeters, what is RT? R SOLUTION: Here, is a diameter and is a radius. 11. Identify a chord that is also a diameter. SOLUTION: Two chords are shown: and . goes

through the center, R, so is a diameter. Therefore, RT = 8.1 cm.

ANSWER: ANSWER: 8.1 cm

For Exercises 14–17, refer to . 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord.

ANSWER: 14. Identify a chord that is not a diameter. No; it is a chord. SOLUTION: 13. If SU = 16.2 centimeters, what is RT? The chords do not pass through the SOLUTION: center. So, they are not diameters. Here, is a diameter and is a radius. ANSWER:

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Therefore, RT = 8.1 cm. Here, is a radius and the diameter is twice the ANSWER: radius. . 8.1 cm

For Exercises 14–17, refer to .

Therefore, the diameter of the circle is 28 inches.

ANSWER: 28 in.

16. Is ? Explain. 14. Identify a chord that is not a diameter. SOLUTION: SOLUTION: All radii of a circle are congruent. Since The chords do not pass through the are both radii of , . center. So, they are not diameters. ANSWER: ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Here, is a diameter and is a radius. Here, is a radius and the diameter is twice the radius. .

Therefore, EF = 3.7 cm. Therefore, the diameter of the circle is 28 inches. ANSWER: ANSWER: 3.7 cm 28 in. Circle J has a radius of 10 units, has a 16. Is ? Explain. radius of 8 units, and BC = 5.4 units. Find each measure. SOLUTION: All radii of a circle are congruent. Since are both radii of , .

ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? 18. CK SOLUTION: SOLUTION: Here, is a diameter and is a radius. Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units.

ANSWER: Therefore, EF = 3.7 cm. 2.6 ANSWER: 3.7 cm 19. AB Circle J has a radius of 10 units, has a SOLUTION: radius of 8 units, and BC = 5.4 units. Find each Since is a diameter of circle J and the radius is measure. 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units.

ANSWER: 14.6 18. CK 20. JK SOLUTION: SOLUTION: Since the radius of K is 8 units, BK = 8. First find CK. Since circle K has a radius of 8 units, BK = 8.

Therefore, CK = 2.6 units.

ANSWER: Since circle J has a radius of 10 units, JC = 10. 2.6

19. AB Therefore, JK = 12.6 units. SOLUTION: ANSWER: Since is a diameter of circle J and the radius is 12.6 10, AC = 2(10) or 20 units. 21. AD SOLUTION: Therefore, AB = 14.6 units. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J ANSWER: is 10 units, so KD = 8 and AC = 2(10) or 20. Before 14.6 finding AD, we need to find CK. 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Therefore, AD = 30.6 units.

ANSWER: Since circle J has a radius of 10 units, JC = 10. 30.6

22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if Therefore, JK = 12.6 units. necessary. ANSWER: 12.6 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J

is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. SOLUTION: The diameter of the pizza is 16 inches.

. Therefore, AD = 30.6 units. So, the radius of the pizza is 8 inches. ANSWER: The circumference C of a circle of diameter d is given by 30.6

22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary. Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary.

SOLUTION: SOLUTION: The diameter of a tire is 26 inches. The radius is half The diameter of the pizza is 16 inches. the diameter.

. So, the radius of the pizza is 8 inches. So, the radius of the tires is 13 inches. The circumference C of a circle of diameter d is The circumference C of a circle with diameter d is given by given by

Therefore, the circumference of the pizza is about Therefore, the circumference of the bicycle tire is 50.27 inches. about 81.68 inches.

ANSWER: ANSWER: 8 in.; 50.27 in. 13 in.; 81.68 in. 23. BICYCLES A bicycle has tires with a diameter of Find the diameter and radius of a circle by with 26 inches. Find the radius and circumference of a the given circumference. Round to the nearest tire. Round to the nearest hundredth, if necessary. hundredth. SOLUTION: 24. C = 18 in. The diameter of a tire is 26 inches. The radius is half SOLUTION: the diameter. The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is

about 81.68 inches. ANSWER: Therefore, the diameter is about 5.73 inches and the

13 in.; 81.68 in. radius is about 2.86 inches. ANSWER: Find the diameter and radius of a circle by with the given circumference. Round to the nearest 5.73 in.; 2.86 in. hundredth. 24. C = 18 in. 25. C = 124 ft SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is given by given by Here, C = 18 in. Use the formula to find the Here, C = 124 ft. Use the formula to find the diameter. Then find the radius. diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the Therefore, the diameter is about 39.47 feet and the radius is about 2.86 inches. radius is about 19.74 feet.

ANSWER: ANSWER: 5.73 in.; 2.86 in. 39.47 ft; 19.74 ft

25. C = 124 ft 26. C = 375.3 cm SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is

given by given by Here, C = 124 ft. Use the formula to find the Here, C = 375.3 cm. Use the formula to find the

diameter. Then find the radius. diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters Therefore, the diameter is about 39.47 feet and the and the radius is about 59.73 centimeters. radius is about 19.74 feet. ANSWER: ANSWER: 119.46 cm; 59.73 cm 39.47 ft; 19.74 ft 27. C = 2608.25 m 26. C = 375.3 cm SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is given by given by Here, C = 2608.25 meters. Use the formula to find Here, C = 375.3 cm. Use the formula to find the the diameter. Then find the radius. diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and Therefore, the diameter is about 119.46 centimeters the radius is about 415.12 meters. and the radius is about 59.73 centimeters. ANSWER: ANSWER: 830.23 m; 415.12 m 119.46 cm; 59.73 cm CCSS SENSE-MAKING Find the exact 27. C = 2608.25 m circumference of each circle by using the given inscribed or SOLUTION: circumscribed polygon. The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

28. SOLUTION:

The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. The diameter of the circle is 17 cm. The ANSWER: circumference C of a circle is given by 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or Therefore, the circumference of the circle is 17 circumscribed polygon. centimeters. ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. The diameter of the circle is 17 cm. The circumference C of a circle is given by

The diameter of the circle is 12 feet. The circumference C of a circle is given by Therefore, the circumference of the circle is 17 centimeters.

ANSWER: Therefore, the circumference of the circle is 12 feet.

ANSWER:

29. SOLUTION: 30. The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the SOLUTION: diameter. Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is 12 feet. The circumference C of a circle is given by

The diameter of the circle is inches. The

Therefore, the circumference of the circle is 12 circumference C of a circle is given by feet.

ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. The diameter of the circle is inches. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The

Therefore, the circumference of the circle is circumference C of a circle is given by inches.

ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. 32. Use the Pythagorean Theorem to find the diameter of the circle. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 25 millimeters.

Therefore, the circumference of the circle is 10 ANSWER: inches. ANSWER:

33. 32. SOLUTION: SOLUTION: A diameter perpendicular to a side of the square and A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d the length of each side will measure the same. So, d = 14 yards. = 25 millimeters. The circumference C of a circle is given by The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 Therefore, the circumference of the circle is 25 yards. millimeters. ANSWER: ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. 33. SOLUTION: SOLUTION: The circumference C of a circle with diameter d is A diameter perpendicular to a side of the square and given by the length of each side will measure the same. So, d Here, C = 66.92 cm. Use the formula to find the = 14 yards. diameter. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 The diameter of the disc is about 21.3 centimeters yards. and the maximum weight allowed is 8.3 times the diameter in centimeters. ANSWER: Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

34. DISC GOLF Disc golf is similar to regular golf, ANSWER: except that a flying disc is used instead of a ball and 176.8 g clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in 35. PATIOS Mr. Martinez is going to build the patio centimeters. What is the maximum allowable weight shown. for a disc with circumference 66.92 centimeters? a. What is the patio’s approximate circumference? Round to the nearest tenth. b. If Mr. Martinez changes the plans so that the SOLUTION: inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the The circumference C of a circle with diameter d is nearest foot? given by Here, C = 66.92 cm. Use the formula to find the diameter.

The diameter of the disc is about 21.3 centimeters SOLUTION: and the maximum weight allowed is 8.3 times the a. The radius of the patio is 3 + 2 = 5 ft. The diameter in centimeters. circumference C of a circle with radius r is given by Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. Use the formula to find the circumference.

ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio Therefore, the circumference is about 31.42 feet. shown. b. Here, Cinner = 25 ft. Use the formula to find the a. What is the patio’s approximate circumference? radius of the inner circle. b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft b. 4 ft

SOLUTION: The radius, diameter, or circumference of a circle is given. Find each missing measure to a. The radius of the patio is 3 + 2 = 5 ft. The the nearest hundredth. circumference C of a circle with radius r is given by

36. Use the formula to find the circumference. SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given Therefore, the circumference is about 31.42 feet. by

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the radius of the inner circle is about 4 feet. Therefore, the radius is 4.25 inches and the ANSWER: circumference is about 26.70 inches. a. 31.42 ft b. 4 ft ANSWER: 4.25 in.; 26.70 in. The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth. 37.

36. SOLUTION: SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given The radius is half the diameter and the by circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

Therefore, the radius is 4.25 inches and the ANSWER: circumference is about 26.70 inches. 22.80 ft; 71.63 ft

ANSWER: 38. 4.25 in.; 26.70 in. SOLUTION: The circumference C of a circle with diameter d is 37. given by Here, C = 35x cm. Use the formula to find the SOLUTION: diameter.Then find the radius. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference Therefore, the diameter is about 11.14x centimeters is about 71.63 feet. and the radius is about 5.57x centimeters.

ANSWER: ANSWER: 22.80 ft; 71.63 ft 11.14x cm; 5.57x cm

38. 39. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The diameter is twice the radius and the Here, C = 35x cm. Use the formula to find the circumference C of a circle with diameter d is given diameter.Then find the radius. by

So, the diameter is 0.25x units and the circumference is about 0.79x units. Therefore, the diameter is about 11.14x centimeters ANSWER: and the radius is about 5.57x centimeters. 0.25x; 0.79x ANSWER: Determine whether the circles in the figures 11.14x cm; 5.57x cm below appear to be congruent, concentric, or neither.

39. 40. Refer to the photo on page 703. SOLUTION: SOLUTION: The circles have the same center and they are in the The diameter is twice the radius and the same plane. So, they are concentric circles. circumference C of a circle with diameter d is given by ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: So, the diameter is 0.25x units and the circumference neither is about 0.79x units.

ANSWER: 42. Refer to the photo on page 703. 0.25x; 0.79x SOLUTION: The circles appear to have the same radius. So, they Determine whether the circles in the figures appear to be congruent. below appear to be congruent, concentric, or neither. ANSWER: 40. Refer to the photo on page 703. congruent SOLUTION: 43. HISTORY The Indian Shell Ring on Hilton Head The circles have the same center and they are in the Island approximates a circle. If each unit on the same plane. So, they are concentric circles. coordinate grid represents 25 feet, how far would ANSWER: someone have to walk to go completely around the ring? Round to the nearest tenth. concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703.

SOLUTION: SOLUTION: The circles appear to have the same radius. So, they The radius of the circle is 3 units, that is 75 feet. So, appear to be congruent. the diameter is 150 ft. The circumference C of a ANSWER: circle with diameter d is given by congruent So, the circumference is Therefore, someone have to 43. HISTORY The Indian Shell Ring on Hilton Head walk about 471.2 feet to go completely around the Island approximates a circle. If each unit on the ring. coordinate grid represents 25 feet, how far would someone have to walk to go completely around the ANSWER: ring? Round to the nearest tenth. 471.2 ft

44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond. SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a circle with diameter d is given by So, the circumference is Therefore, someone have to So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 walk about 471.2 feet to go completely around the = 14.82 ft. Use the radius to find the circumference ring. of the brick path. ANSWER: 471.2 ft

44. CCSS MODELING A brick path is being installed Therefore, the circumference of the path is about around a circular pond. The pond has a 93.13 feet. circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way ANSWER: around. What is the approximate circumference of 93.13 ft the path? Round to the nearest hundredth. 45. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will explore changing dimensions in The circumference C of a circle with radius r is circles. given by Here, Cpond = 68 ft. Use the a. GEOMETRIC Use a compass to draw three formula to find the radius of the pond. circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. So, the radius of the pond is about 10.82 feet and the d. VERBAL Make a conjecture about the ratio radius to the outer edge of the path will be 10.82 + 4 between the circumferences of two circles when the = 14.82 ft. Use the radius to find the circumference ratio between their radii is 2. of the brick path. e. ANALYTICAL The scale factor from to is . Write an equation relating the

Therefore, the circumference of the path is about circumference (CA) of to the circumference 93.13 feet. (CB) of . ANSWER: f. NUMERICAL If the scale factor from to 93.13 ft is , and the circumference of is 12 45. MULTIPLE REPRESENTATIONS In this inches, what is the circumference of ? problem, you will explore changing dimensions in circles. SOLUTION: a. GEOMETRIC Use a compass to draw three a. Sample answer: circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. b. Sample answer. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to is . Write an equation relating the circumference (C ) of to the circumference A c. They all have the same shape–circular. (CB) of . d. The ratio of their circumferences is also 2. f. NUMERICAL If the scale factor from to e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the is , and the circumference of is 12 circumference. Therefore, . inches, what is the circumference of ? f. Use the formula from part e. Here, the SOLUTION: circumference of is 12 inches and . a. Sample answer:

Therefore, the circumference of is 4 inches.

ANSWER: b. Sample answer. a. Sample answer:

c. They all have the same shape–circular. b. Sample answer. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . f. Use the formula from part e. Here, the

circumference of is 12 inches and . c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in. Therefore, the circumference of is 4 inches. 46. BUFFON’S NEEDLE Measure the length of a ANSWER: needle (or toothpick) in centimeters. Next, draw a set a. Sample answer: of horizontal lines that are centimeters apart on a sheet of plain white paper.

b. Sample answer.

a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 f. 4 in. drops.

46. BUFFON’S NEEDLE Measure the length of a c. How are the values you found in part b related to needle (or toothpick) in centimeters. Next, draw a set π? of horizontal lines that are centimeters apart on a SOLUTION: sheet of plain white paper. a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. a. Drop the needle onto the paper. When the needle c. Sample answer: The values are approaching 3.14, lands, record whether it touches one of the lines as a which is approximately equal to π. hit. Record the number of hits after 25, 50, and 100 drops. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, a. which is approximately equal to π. How much greater is the circumference of the outermost circle than the circumference of the center circle? ANSWER: b. As the radii of the circles increases by 5 miles, by

a. See students’ work. how much does the circumference increase? b. See students’ work. SOLUTION: c. Sample answer: The values are approaching 3.14, a. The radius of the outermost circle is 30 miles and which is approximately equal to π. that of the center circle is 5 miles. Find the circumference of each circle. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

The difference of the circumferences of the outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase?

SOLUTION: a. The radius of the outermost circle is 30 miles and The difference of the two circumferences is (2xπ + that of the center circle is 5 miles. Find the 10π) - (2xπ) = 10π or about 31.4 miles.

circumference of each circle. Therefore, as the radius of each circle increases by 5 miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi

The difference of the circumferences of the 48. WRITING IN MATH How can we describe the outermost circle and the center circle is 60π - 10π = relationships that exist between 50π or about 157.1 miles. circles and lines? b. Let the radius of one circle be x miles and the SOLUTION: radius of the next larger circle be x + 5 miles. Find the circumference of each circle. Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a The difference of the two circumferences is (2xπ + diameter. A line segment with endpoints at the 10π) - (2xπ) = 10π or about 31.4 miles. center and on the circle can be described as a Therefore, as the radius of each circle increases by 5 radius. miles, the circumference increases by about 31.4 miles. ANSWER: Sample answer: A line that intersects a circle in one ANSWER: point can be described as a tangent. A line that a. 157.1 mi intersects a circle in exactly two points can be b. by about 31.4 mi described as a secant. A line segment with endpoints on a circle can be described as a chord. WRITING IN MATH How can we describe the 48. If the chord passes through the center of the circle, relationships that exist between it can be described as a diameter. A line segment circles and lines? with endpoints at the center and on the circle can be SOLUTION: described as a radius. Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. 49. A line segment with endpoints on a circle can be REASONING In the figure, a circle with radius r is described as a chord. If the chord passes through inscribed in a regular polygon and circumscribed the center of the circle, it can be described as a about another. diameter. A line segment with endpoints at the a. What are the perimeters of the circumscribed and center and on the circle can be described as a inscribed polygons in terms of r? Explain. radius. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? ANSWER: the inscribed polygon? Write a compound inequality Sample answer: A line that intersects a circle in one comparing C to these perimeters. point can be described as a tangent. A line that c. Rewrite the inequality from part b in terms of the intersects a circle in exactly two points can be diameter d of the circle and interpret its meaning. described as a secant. A line segment with d. As the number of sides of both the circumscribed endpoints on a circle can be described as a chord. and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part If the chord passes through the center of the circle, c, and what does this imply? it can be described as a diameter. A line segment with endpoints at the center and on the circle can be SOLUTION: described as a radius. a. The circumscribed polygon is a square with sides of length 2r.

The inscribed polygon is a hexagon with sides of 49. REASONING In the figure, a circle with radius r is length r. (Each triangle formed using a central angle inscribed in a regular polygon and circumscribed will be equilateral.) about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality comparing C to these perimeters. c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. d. As the number of sides of both the circumscribed and inscribed polygons increase, what will happen to So, the perimeters are 8r and 6r. the upper and lower limits of the inequality from part b. c, and what does this imply? SOLUTION: a. The circumscribed polygon is a square with sides The circumference is less than the perimeter of the of length 2r. circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inscribed polygon is a hexagon with sides of The inequality is then 4d < C < 3d. Therefore, the length r. (Each triangle formed using a central angle circumference of the circle is between 3 and 4 times will be equilateral.) its diameter. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral So, the perimeters are 8r and 6r. triangles with side length r, so the perimeter of the b. hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is The circumference is less than the perimeter of the between 3 and 4 times its diameter. circumscribed polygon and greater than the d. These limits will approach a value of πd, implying perimeter of the inscribed polygon. that C = πd. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) 50. CHALLENGE The sum of the circumferences of or 4d and 6r = 3(2r) or 3d. circles H, J, and K shown at the right is units.

The inequality is then 4d < C < 3d. Therefore, the Find KJ. circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the SOLUTION: square is 4(2r) or 8r. The The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the regular hexagon is made up of six equilateral circumference of each circle. triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying π that C = d. 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ. The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

So, the radius of is 4 units, the radius of is

2(4) or 8 units, and the radius of is 4(4) or 16 SOLUTION: units. The radius of is 4x units, the radius of is x The measure of KJ is equal to the sum of the radii units, and the radius of is 2x units. Find the of and . circumference of each circle. Therefore, KJ = 16 + 8 or 24 units.

ANSWER: 24 units

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Explain. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn The sum of the circumferences is 56π units. from the center to a point inside the circle will always Substitute the three circumferences and solve for x. have a length less than the radius of the circle. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the So, the radius of is 4 units, the radius of is circle will always have a length less than the radius 2(4) or 8 units, and the radius of is 4(4) or 16 of the circle. units. The measure of KJ is equal to the sum of the radii 52. PROOF Use the locus definition of a circle and of and . dilations to prove that all circles are similar. Therefore, KJ = 16 + 8 or 24 units. SOLUTION: ANSWER: A circle is a locus of points in a plane equidistant 24 units from a given point. For any two circles and , there exists a translation that 51. REASONING Is the distance from the center of a maps center A onto center B, moving so that it circle to a point in the interior of a circle sometimes, is concentric with . There also exists a dilation always, or never less than the radius of the circle? with scale factor k such Explain. that each point that makes up is moved to be the same distance from center A as the points that make SOLUTION: up are from center B. A radius is a segment drawn between the center of For some value k, r = the circle and a point on the circle. A segment drawn 2 from the center to a point inside the circle will always kr1. have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius Therefore, is mapped onto . Since there of the circle. exists a rigid motion followed by a scaling that maps onto , the circles are 52. PROOF Use the locus definition of a circle and similar. Thus, all circles are similar. dilations to prove that all circles are similar.

SOLUTION: A circle is a locus of points in a plane equidistant ANSWER: from a given point. For any two circles and A circle is a locus of points in a plane equidistant , there exists a translation that from a given point. For any two circles and maps center A onto center B, moving so that it , there exists a translation that maps center A is concentric with . There also exists a dilation with scale factor k such onto center B, moving so that it is concentric that each point that makes up is moved to be the with . There also exists a dilation with scale same distance from center A as the points that make factor k such that each point that makes up is up are from center B. moved to be the same distance from center A as the For some value k, r2 = points that make up are from center B. Therefore, is mapped onto . Since there kr1. exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference Therefore, is mapped onto . Since there of ? exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

ANSWER:

A circle is a locus of points in a plane equidistant from a given point. For any two circles and SOLUTION: , there exists a translation that maps center A Draw a perpendicular to the side Also join onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps

onto , the circles are similar. Thus, all Since is equilateral triangle, MN = 8 inches circles are similar. and RN = 4 inches. Also, bisects 53. CHALLENGE In the figure, is inscribed in Use the definition of equilateral triangle LMN. What is the circumference tangent of an angle to find PR, which is the radius of of ?

Therefore, the radius of is SOLUTION: The circumference C of a circle with radius r is given by Draw a perpendicular to the side Also join

The circumference of is inches.

Since is equilateral triangle, MN = 8 inches ANSWER: and RN = 4 inches. Also, bisects in. Use the definition of tangent of an angle to find PR, which is the radius of 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

ANSWER: Therefore, the radius of is See students’ work. The circumference C of a circle with radius r is given by 55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

The circumference of is inches. SOLUTION: ANSWER: Use the Pythagorean Theorem to find the diameter of the circle. in. The circumference C of a circle with diameter d is 54. WRITING IN MATH Research and write about given by the history of pi and its importance to the study of Use the formula to find the circumference. geometry. SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the ANSWER: circumference of ? Round to the nearest tenth. 40.8 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft SOLUTION: D 5 ft Use the Pythagorean Theorem to find the diameter SOLUTION: of the circle. The circumference C of a circle with radius r is given by The circumference C of a circle with diameter d is Here, C = 10 ft. Use the formula to find the radius. given by Use the formula to find the circumference.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: A ANSWER: 40.8 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use 56. What is the radius of a table with a circumference of up to 50 feet of fence, what radius can he use for the

10 feet? garden? A 1.6 ft F 10 B 2.5 ft G 9 C 3.2 ft H 8 D 5 ft J 7 SOLUTION: SOLUTION: The circumference C of a circle with radius r is The circumference C of a circle with radius r is

given by given by

Here, C = 10 ft. Use the formula to find the radius. Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: Therefore, the radius of the garden must be less than A or equal to about 7.96 feet. Therefore, the only correct choice is J. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use ANSWER: up to 50 feet of fence, what radius can he use for the J garden? F 10 58. SAT/ACT What is the radius of a circle with an G 9 area of square units? H 8 J 7 A 0.4 units B 0.5 units SOLUTION: C 2 units The circumference C of a circle with radius r is D 4 units given by E Because 50 feet of fence can be used to border the 16 units garden, Cmax = 50. Use the circumference formula to SOLUTION: find the maximum radius. The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J. Therefore, the correct choice is B.

10-1ANSWER: Circles and Circumference ANSWER: J B

58. SAT/ACT What is the radius of a circle with an Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation area of square units? with center B and the scale factor r indicated.

A 0.4 units 59. B 0.5 units C 2 units D 4 units E 16 units

SOLUTION: The area of a circle of radius r is given by the SOLUTION: formula Here A = square units. Use the Draw a ray from point B through a vertex of the formula to find the radius. hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the correct choice is B. ANSWER: B Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the

hexagon. Use the ruler to measure the distance from 60. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon. SOLUTION:

Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of eSolutions Manual - Powered by Cognero the triangle. Page 18

ANSWER:

ANSWER:

60. 61.

SOLUTION: SOLUTION: Draw a ray from point B through a vertex of the Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 B to the vertex. Mark a point on the ray that is of times that distance from B. Repeat the process for that distance from B. Repeat the process for rays rays through the other three vertices. Connect the through the other two vertices. Connect the three four points on the rays to construct the dilated image points on the rays to construct the dilated image of of the trapezoid. the triangle.

ANSWER:

ANSWER:

61.

SOLUTION: 62. Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image SOLUTION: of the trapezoid. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

62. ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No ANSWER:

64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. 65. ANSWER: SOLUTION: No The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

64. Refer to the image on page 705. ANSWER: SOLUTION: No The figure can be mapped onto itself by a rotation of 66. Refer to the image on page 705. 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. ANSWER: Yes; 4, 90 ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: 65. True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true The given figure cannot be mapped onto itself by any for the conditions. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: True; sample answer: Since the hypothesis is true No and the conclusion is true, then the statement is true for the conditions. 66. Refer to the image on page 705. SOLUTION: 68. You are 21 years old and do not vote. The given figure cannot be mapped onto itself by any SOLUTION: rotation. So, it does not have a rotational symmetry. False; sample answer: Since the hypothesis is true ANSWER: and the conclusion is false, then the statement is false for the conditions. No ANSWER: Determine the truth value of the following False; sample answer: Since the hypothesis is true statement for each set of conditions. Explain and the conclusion is false, then the statement is false your reasoning. for the conditions. If you are over 18 years old, then you vote in all elections. Find x. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. 69. ANSWER: True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true The sum of the measures of angles around a point is for the conditions. 360º.

68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true ANSWER: and the conclusion is false, then the statement is false 90 for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x. 70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

69. SOLUTION: The sum of the measures of angles around a point is ANSWER: 360º. 60

ANSWER: 90 71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

70. SOLUTION: The angles with the measures xº and 2xº form a ANSWER: straight line. So, the sum is 180. 20

ANSWER: 60 72. SOLUTION: The sum of the three angles is 180.

71. SOLUTION: ANSWER: The angles with the measures 6xº and 3xº form a 45 straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes For Exercises 1–4, refer to . through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a.

1. Name the circle. b. c. SOLUTION: The center of the circle is N. So, the circle is 3. If CN = 8 centimeters, find DN.

ANSWER: SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

2. Identify each. ANSWER: a. a chord 8 cm b. a diameter c. a radius 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: SOLUTION: a. A chord is a segment with endpoints on the circle. Here, is a radius and the diameter is twice the So, here are chords. radius. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. Therefore, the diameter is 26 ft. c. A radius is a segment with endpoints at the center and on the circle. Here, is ANSWER: radius. 26 ft

ANSWER: The diameters of , , and are 8 a. inches, 18 inches, and 11 inches, respectively. b. Find each measure. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 5. FG 8 cm SOLUTION: Since the diameter of B is 18 inches and A is 8 4. If EN = 13 feet, what is the diameter of the circle? inches, AG = 18 and AF = (8) or 4. SOLUTION: Here, is a radius and the diameter is twice the radius.

Therefore, FG = 14 in. Therefore, the diameter is 26 ft. ANSWER: ANSWER: 14 in. 26 ft 6. FB The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. SOLUTION: Find each measure. Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches.

5. FG ANSWER: SOLUTION: 5 in. Since the diameter of B is 18 inches and A is 8 7. RIDES The circular ride described at the beginning inches, AG = 18 and AF = (8) or 4. of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION:

Therefore, FG = 14 in. The radius is half the diameter. So, the radius of the circular ride is 22 feet. ANSWER: 14 in.

6. FB SOLUTION: Therefore, the circumference of the ride is about Since the diameter of B is 18 inches and A is 4 138.23 feet. inches, AB = (18) or 9 and AF = (8) or 4. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. Therefore, FB = 5 inches. What are the diameter and radius of the pool? Round to the nearest hundredth. ANSWER: 5 in.

7. RIDES The circular ride described at the beginning of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary.

SOLUTION: The radius is half the diameter. So, the radius of the SOLUTION: circular ride is 22 feet.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the The diameter of the pool is about 17.98 feet and the circular swimming pool shown is about 56.5 feet. radius of the pool is about 8.99 feet. What are the diameter and radius of the pool? Round ANSWER: to the nearest hundredth. 17.98 ft; 8.99 ft

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of .

SOLUTION:

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

The diameter of the circle is 4 centimeters.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

ANSWER: 17.98 ft; 8.99 ft The circumference of the circle is 4π centimeters. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of ANSWER: .

For Exercises 10–13, refer to .

SOLUTION: The diameter of the circle is the hypotenuse of the right triangle. 10. Name the center of the circle. SOLUTION: R

ANSWER: The diameter of the circle is 4 centimeters. R

11. Identify a chord that is also a diameter. SOLUTION:

Two chords are shown: and . goes π The circumference of the circle is 4 centimeters. through the center, R, so is a diameter.

ANSWER: ANSWER:

For Exercises 10–13, refer to . 12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord.

10. Name the center of the circle. ANSWER: No; it is a chord. SOLUTION: R 13. If SU = 16.2 centimeters, what is RT? ANSWER: SOLUTION: R Here, is a diameter and is a radius.

11. Identify a chord that is also a diameter. SOLUTION: Two chords are shown: and . goes through the center, R, so is a diameter. Therefore, RT = 8.1 cm. ANSWER: ANSWER: 8.1 cm For Exercises 14–17, refer to .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on the circle, so it is a chord. 14. Identify a chord that is not a diameter. ANSWER: SOLUTION: No; it is a chord. The chords do not pass through the 13. If SU = 16.2 centimeters, what is RT? center. So, they are not diameters. SOLUTION: ANSWER: Here, is a diameter and is a radius.

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the

Therefore, RT = 8.1 cm. radius. . ANSWER: 8.1 cm

For Exercises 14–17, refer to . Therefore, the diameter of the circle is 28 inches.

ANSWER: 28 in.

16. Is ? Explain. SOLUTION: 14. Identify a chord that is not a diameter. All radii of a circle are congruent. Since SOLUTION: are both radii of , . The chords do not pass through the ANSWER: center. So, they are not diameters. Yes; they are both radii of . ANSWER: 17. If DA = 7.4 centimeters, what is EF?

SOLUTION: 15. If CF = 14 inches, what is the diameter of the circle? Here, is a diameter and is a radius. SOLUTION: Here, is a radius and the diameter is twice the radius. .

Therefore, EF = 3.7 cm.

ANSWER: Therefore, the diameter of the circle is 28 inches. 3.7 cm ANSWER: Circle J has a radius of 10 units, has a 28 in. radius of 8 units, and BC = 5.4 units. Find each measure. 16. Is ? Explain. SOLUTION: All radii of a circle are congruent. Since are both radii of , .

ANSWER: Yes; they are both radii of . 18. CK 17. If DA = 7.4 centimeters, what is EF? SOLUTION: SOLUTION: Since the radius of K is 8 units, BK = 8. Here, is a diameter and is a radius.

Therefore, CK = 2.6 units.

ANSWER: 2.6 Therefore, EF = 3.7 cm.

ANSWER: 19. AB 3.7 cm SOLUTION: Circle J has a radius of 10 units, has a Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. radius of 8 units, and BC = 5.4 units. Find each measure.

Therefore, AB = 14.6 units.

ANSWER: 14.6 20. JK 18. CK SOLUTION: SOLUTION: First find CK. Since circle K has a radius of 8 units, Since the radius of K is 8 units, BK = 8. BK = 8.

Therefore, CK = 2.6 units. Since circle J has a radius of 10 units, JC = 10. ANSWER: 2.6

Therefore, JK = 12.6 units. 19. AB ANSWER: SOLUTION: 12.6 Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units. 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The

Therefore, AB = 14.6 units. radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before ANSWER: finding AD, we need to find CK. 14.6 20. JK SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8. Therefore, AD = 30.6 units.

ANSWER: 30.6 Since circle J has a radius of 10 units, JC = 10. 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary. Therefore, JK = 12.6 units.

ANSWER: 12.6 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J SOLUTION: is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. The diameter of the pizza is 16 inches.

. So, the radius of the pizza is 8 inches. Therefore, AD = 30.6 units. The circumference C of a circle of diameter d is given by ANSWER: 30.6

22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if Therefore, the circumference of the pizza is about necessary. 50.27 inches. ANSWER: 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a tire. Round to the nearest hundredth, if necessary. SOLUTION:

The diameter of a tire is 26 inches. The radius is half SOLUTION: the diameter. The diameter of the pizza is 16 inches.

. So, the radius of the tires is 13 inches. So, the radius of the pizza is 8 inches. The circumference C of a circle with diameter d is The circumference C of a circle of diameter d is given by given by

Therefore, the circumference of the bicycle tire is Therefore, the circumference of the pizza is about about 81.68 inches. 50.27 inches. ANSWER: ANSWER: 13 in.; 81.68 in. 8 in.; 50.27 in. Find the diameter and radius of a circle by with 23. BICYCLES A bicycle has tires with a diameter of the given circumference. Round to the nearest 26 inches. Find the radius and circumference of a hundredth. tire. Round to the nearest hundredth, if necessary. 24. C = 18 in. SOLUTION: SOLUTION: The diameter of a tire is 26 inches. The radius is half The circumference C of a circle with diameter d is the diameter. given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches. Therefore, the diameter is about 5.73 inches and the radius is about 2.86 inches. ANSWER: 13 in.; 81.68 in. ANSWER: 5.73 in.; 2.86 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 25. C = 124 ft 24. C = 18 in. SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is given by given by Here, C = 124 ft. Use the formula to find the Here, C = 18 in. Use the formula to find the diameter. Then find the radius. diameter. Then find the radius.

Therefore, the diameter is about 39.47 feet and the Therefore, the diameter is about 5.73 inches and the radius is about 19.74 feet. radius is about 2.86 inches. ANSWER: ANSWER: 39.47 ft; 19.74 ft 5.73 in.; 2.86 in. 26. C = 375.3 cm

25. C = 124 ft SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 375.3 cm. Use the formula to find the given by diameter. Then find the radius. Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. ANSWER: 119.46 cm; 59.73 cm ANSWER: 39.47 ft; 19.74 ft 27. C = 2608.25 m

26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The circumference C of a circle with diameter d is Here, C = 2608.25 meters. Use the formula to find given by the diameter. Then find the radius. Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. Therefore, the diameter is about 119.46 centimeters ANSWER: and the radius is about 59.73 centimeters. 830.23 m; 415.12 m ANSWER: CCSS SENSE-MAKING Find the exact 119.46 cm; 59.73 cm circumference of each circle by using the given inscribed or 27. C = 2608.25 m circumscribed polygon. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius. 28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and The diameter of the circle is 17 cm. The the radius is about 415.12 meters. circumference C of a circle is given by ANSWER: 830.23 m; 415.12 m

CCSS SENSE-MAKING Find the exact Therefore, the circumference of the circle is 17 circumference of each circle by using the given centimeters. inscribed or circumscribed polygon. ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of 29. the circle. Use the Pythagorean Theorem to find the diameter. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

The diameter of the circle is 12 feet. The circumference C of a circle is given by

Therefore, the circumference of the circle is 17 centimeters.

ANSWER: Therefore, the circumference of the circle is 12 feet. ANSWER:

29. 30. SOLUTION: SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the Each diagonal of the inscribed rectangle will pass diameter. through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is 12 feet. The circumference C of a circle is given by

The diameter of the circle is inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 12 feet.

ANSWER: Therefore, the circumference of the circle is inches.

ANSWER:

30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter 31.

of the circle. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is inches. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is inches.

ANSWER: Therefore, the circumference of the circle is 10 inches. ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass 32. through the origin, so it is a diameter of the circle. SOLUTION: Use the Pythagorean Theorem to find the diameter A diameter perpendicular to a side of the square and of the circle. the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The circumference C of a circle is given by Therefore, the circumference of the circle is 25 millimeters.

ANSWER: Therefore, the circumference of the circle is 10 inches.

ANSWER:

33. SOLUTION: 32. A diameter perpendicular to a side of the square and SOLUTION: the length of each side will measure the same. So, d A diameter perpendicular to a side of the square and = 14 yards. the length of each side will measure the same. So, d The circumference C of a circle is given by = 25 millimeters. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 yards. Therefore, the circumference of the circle is 25 millimeters. ANSWER:

ANSWER: 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: 33. The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 66.92 cm. Use the formula to find the A diameter perpendicular to a side of the square and diameter. the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

The diameter of the disc is about 21.3 centimeters Therefore, the circumference of the circle is 14 and the maximum weight allowed is 8.3 times the yards. diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or ANSWER: about 176.8 grams.

ANSWER: 34. DISC GOLF Disc golf is similar to regular golf, 176.8 g except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum 35. PATIOS Mr. Martinez is going to build the patio weight of a disc in grams is 8.3 times its diameter in shown. centimeters. What is the maximum allowable weight a. What is the patio’s approximate circumference? for a disc with circumference 66.92 centimeters? b. If Mr. Martinez changes the plans so that the Round to the nearest tenth. inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the SOLUTION: nearest foot? The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter.

SOLUTION: a. The diameter of the disc is about 21.3 centimeters The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by and the maximum weight allowed is 8.3 times the diameter in centimeters. Use the formula to find the circumference. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g Therefore, the circumference is about 31.42 feet. b. Here, C = 25 ft. Use the formula to find the 35. PATIOS Mr. Martinez is going to build the patio inner shown. radius of the inner circle. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to SOLUTION: the nearest hundredth. a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by 36.

Use the formula to find the circumference. SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the radius of the inner circle is about 4 feet. Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. ANSWER: ANSWER: a. 31.42 ft 4.25 in.; 26.70 in. b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to 37. the nearest hundredth. SOLUTION: 36. The diameter is twice the radius and the circumference C of a circle with diameter d is given SOLUTION: by The radius is half the diameter and the circumference C of a circle with diameter d is given by

So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: Therefore, the radius is 4.25 inches and the 22.80 ft; 71.63 ft circumference is about 26.70 inches. 38. ANSWER: 4.25 in.; 26.70 in. SOLUTION: The circumference C of a circle with diameter d is

given by 37. Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters

and the radius is about 5.57x centimeters. So, the diameter is 22.80 feet and the circumference is about 71.63 feet. ANSWER: ANSWER: 11.14x cm; 5.57x cm 22.80 ft; 71.63 ft 39. 38. SOLUTION: SOLUTION: The diameter is twice the radius and the The circumference C of a circle with diameter d is circumference C of a circle with diameter d is given given by by Here, C = 35x cm. Use the formula to find the diameter.Then find the radius.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: Therefore, the diameter is about 11.14x centimeters 0.25x; 0.79x and the radius is about 5.57x centimeters. Determine whether the circles in the figures ANSWER: below appear to be congruent, concentric, or 11.14x cm; 5.57x cm neither. 40. Refer to the photo on page 703.

39. SOLUTION: The circles have the same center and they are in the SOLUTION: same plane. So, they are concentric circles. The diameter is twice the radius and the circumference C of a circle with diameter d is given ANSWER: by concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither So, the diameter is 0.25x units and the circumference is about 0.79x units. 42. Refer to the photo on page 703. ANSWER: SOLUTION: 0.25x; 0.79x The circles appear to have the same radius. So, they appear to be congruent. Determine whether the circles in the figures below appear to be congruent, concentric, or ANSWER: neither. congruent 40. Refer to the photo on page 703. 43. HISTORY The Indian Shell Ring on Hilton Head SOLUTION: Island approximates a circle. If each unit on the The circles have the same center and they are in the coordinate grid represents 25 feet, how far would same plane. So, they are concentric circles. someone have to walk to go completely around the ring? Round to the nearest tenth. ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION: SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, The circles appear to have the same radius. So, they the diameter is 150 ft. The circumference C of a appear to be congruent. circle with diameter d is given by So, the circumference is ANSWER: Therefore, someone have to congruent walk about 471.2 feet to go completely around the 43. HISTORY The Indian Shell Ring on Hilton Head ring. Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would ANSWER: someone have to walk to go completely around the 471.2 ft ring? Round to the nearest tenth. 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way around. What is the approximate circumference of the path? Round to the nearest hundredth. SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a circle with diameter d is given by So, the circumference is So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 Therefore, someone have to = 14.82 ft. Use the radius to find the circumference walk about 471.2 feet to go completely around the of the brick path. ring.

ANSWER:

471.2 ft Therefore, the circumference of the path is about 44. CCSS MODELING A brick path is being installed 93.13 feet. around a circular pond. The pond has a ANSWER: circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way 93.13 ft around. What is the approximate circumference of MULTIPLE REPRESENTATIONS the path? Round to the nearest hundredth. 45. In this problem, you will explore changing dimensions in SOLUTION: circles. The circumference C of a circle with radius r is a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to given by Here, Cpond = 68 ft. Use the the next larger circle is 1 : 2. formula to find the radius of the pond. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio So, the radius of the pond is about 10.82 feet and the between the circumferences of two circles when the radius to the outer edge of the path will be 10.82 + 4 ratio between their radii is 2. = 14.82 ft. Use the radius to find the circumference e. ANALYTICAL The scale factor from to of the brick path. is . Write an equation relating the circumference (C ) of to the circumference A Therefore, the circumference of the path is about (CB) of . 93.13 feet. f. NUMERICAL If the scale factor from to

ANSWER: is , and the circumference of is 12 93.13 ft inches, what is the circumference of ? 45. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will explore changing dimensions in a. Sample answer: circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. c. VERBAL Explain why these three circles are b. Sample answer. geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to is . Write an equation relating the c. They all have the same shape–circular. circumference (CA) of to the circumference d. The ratio of their circumferences is also 2. (CB) of . e. The circumference is directly proportional to radii. f. NUMERICAL If the scale factor from to So, if the radii are in the ratio b : a, so will the

circumference. Therefore, . is , and the circumference of is 12 f. Use the formula from part e. Here, the inches, what is the circumference of ? circumference of is 12 inches and . SOLUTION: a. Sample answer:

Therefore, the circumference of is 4 inches.

ANSWER: a. Sample answer: b. Sample answer.

b. Sample answer. c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the

circumference. Therefore, .

f. Use the formula from part e. Here, the c. They all have the same shape—circular. circumference of is 12 inches and . d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a Therefore, the circumference of is 4 inches. needle (or toothpick) in centimeters. Next, draw a set ANSWER: of horizontal lines that are centimeters apart on a a. Sample answer: sheet of plain white paper.

b. Sample answer.

a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2. b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 e. drops.

f. 4 in. c. How are the values you found in part b related to 46. BUFFON’S NEEDLE Measure the length of a π? needle (or toothpick) in centimeters. Next, draw a set SOLUTION: of horizontal lines that are centimeters apart on a a. See students’ work. sheet of plain white paper. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work.

b. See students’ work. c. Sample answer: The values are approaching 3.14, a. Drop the needle onto the paper. When the needle which is approximately equal to π. lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops. 47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π? SOLUTION: a. See students’ work. b. See students’ work. a. How much greater is the circumference of the c. Sample answer: The values are approaching 3.14, outermost circle than the circumference of the center which is approximately equal to π. circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? ANSWER: a. See students’ work. SOLUTION: b. See students’ work. a. The radius of the outermost circle is 30 miles and c. Sample answer: The values are approaching 3.14, that of the center circle is 5 miles. Find the which is approximately equal to π. circumference of each circle.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

The difference of the circumferences of the outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle.

a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase?

SOLUTION: The difference of the two circumferences is (2xπ + a. The radius of the outermost circle is 30 miles and 10π) - (2xπ) = 10π or about 31.4 miles. that of the center circle is 5 miles. Find the Therefore, as the radius of each circle increases by 5 circumference of each circle. miles, the circumference increases by about 31.4 miles.

ANSWER: a. 157.1 mi b. by about 31.4 mi

WRITING IN MATH How can we describe the 48. relationships that exist between The difference of the circumferences of the circles and lines? outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. SOLUTION: b. Let the radius of one circle be x miles and the Sample answer: A line and a circle may intersect in radius of the next larger circle be x + 5 miles. Find one point, two points, or may not intersect at all. A

the circumference of each circle. line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a The difference of the two circumferences is (2xπ + 10π) - (2xπ) = 10π or about 31.4 miles. radius. Therefore, as the radius of each circle increases by 5 ANSWER: miles, the circumference increases by about 31.4 miles. Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that ANSWER: intersects a circle in exactly two points can be a. 157.1 mi described as a secant. A line segment with b. by about 31.4 mi endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, 48. WRITING IN MATH How can we describe the it can be described as a diameter. A line segment relationships that exist between with endpoints at the center and on the circle can be circles and lines? described as a radius. SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle 49. in exactly two points can be described as a secant. REASONING In the figure, a circle with radius r is A line segment with endpoints on a circle can be inscribed in a regular polygon and circumscribed described as a chord. If the chord passes through about another. the center of the circle, it can be described as a a. What are the perimeters of the circumscribed and diameter. A line segment with endpoints at the inscribed polygons in terms of r? Explain. center and on the circle can be described as a b. Is the circumference C of the circle greater or radius. less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality ANSWER: comparing C to these perimeters. Sample answer: A line that intersects a circle in one c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. point can be described as a tangent. A line that d. As the number of sides of both the circumscribed intersects a circle in exactly two points can be and inscribed polygons increase, what will happen to described as a secant. A line segment with the upper and lower limits of the inequality from part endpoints on a circle can be described as a chord. c, and what does this imply? If the chord passes through the center of the circle, it can be described as a diameter. A line segment SOLUTION: with endpoints at the center and on the circle can be a. The circumscribed polygon is a square with sides described as a radius. of length 2r.

The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle 49. will be equilateral.) REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality comparing C to these perimeters. c. b Rewrite the inequality from part in terms of the diameter d of the circle and interpret its meaning. So, the perimeters are 8r and 6r. d. As the number of sides of both the circumscribed b. and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part c, and what does this imply? The circumference is less than the perimeter of the SOLUTION: circumscribed polygon and greater than the a. The circumscribed polygon is a square with sides perimeter of the inscribed polygon. of length 2r. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the The inscribed polygon is a hexagon with sides of circumference of the circle is between 3 and 4 times length r. (Each triangle formed using a central angle its diameter. will be equilateral.) d. These limits will approach a value of πd, implying that C = πd.

ANSWER: a. 8r and 6r ; Twice the radius of the circle, 2r is the side length of the square, so the perimeter of the square is 4(2r) or 8r. The regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the So, the perimeters are 8r and 6r. hexagon is 6(r) or 6r. b. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. The circumference is less than the perimeter of the d. These limits will approach a value of πd, implying circumscribed polygon and greater than the that C = πd. perimeter of the inscribed polygon. 50. CHALLENGE The sum of the circumferences of 8r < C < 6r circles H, J, and K shown at the right is units. c. Since the diameter is twice the radius, 8r = 4(2r) Find KJ. or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: SOLUTION: a. 8r and 6r ; Twice the radius of the circle, 2r is The radius of is 4x units, the radius of is x the side length of the square, so the perimeter of the units, and the radius of is 2x units. Find the square is 4(2r) or 8r. The circumference of each circle. regular hexagon is made up of six equilateral triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r c. 3d < C < 4d; The circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying that C = πd.

50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. Find KJ. The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 units. SOLUTION: The measure of KJ is equal to the sum of the radii The radius of is 4x units, the radius of is x of and . units, and the radius of is 2x units. Find the Therefore, KJ = 16 + 8 or 24 units. circumference of each circle. ANSWER: 24 units

51. REASONING Is the distance from the center of a circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? Explain. SOLUTION: A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle. The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius So, the radius of is 4 units, the radius of is of the circle. 2(4) or 8 units, and the radius of is 4(4) or 16 units. 52. PROOF Use the locus definition of a circle and The measure of KJ is equal to the sum of the radii dilations to prove that all circles are similar. of and . SOLUTION: Therefore, KJ = 16 + 8 or 24 units. A circle is a locus of points in a plane equidistant ANSWER: from a given point. For any two circles and 24 units , there exists a translation that maps center A onto center B, moving so that it 51. REASONING Is the distance from the center of a is concentric with . There also exists a dilation circle to a point in the interior of a circle sometimes, with scale factor k such always, or never less than the radius of the circle? that each point that makes up is moved to be the Explain. same distance from center A as the points that make up are from center B. SOLUTION: For some value k, r2 = A radius is a segment drawn between the center of kr . the circle and a point on the circle. A segment drawn 1 from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the Therefore, is mapped onto . Since there circle will always have a length less than the radius exists a rigid motion followed by a scaling that maps of the circle. onto , the circles are similar. Thus, all circles are similar. 52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: ANSWER: A circle is a locus of points in a plane equidistant A circle is a locus of points in a plane equidistant from a given point. For any two circles and from a given point. For any two circles and , there exists a translation that , there exists a translation that maps center A maps center A onto center B, moving so that it onto center B, moving so that it is concentric is concentric with . There also exists a dilation with . There also exists a dilation with scale with scale factor k such factor k such that each point that makes up is that each point that makes up is moved to be the moved to be the same distance from center A as the same distance from center A as the points that make points that make up are from center B. up are from center B. Therefore, is mapped onto . Since there For some value k, r2 = exists a rigid motion followed by a scaling that maps kr . 1 onto , the circles are similar. Thus, all circles are similar.

53. CHALLENGE In the figure, is inscribed in

equilateral triangle LMN. What is the circumference

of ? Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar.

ANSWER: SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and Draw a perpendicular to the side Also join , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps Since is equilateral triangle, MN = 8 inches onto , the circles are similar. Thus, all and RN = 4 inches. Also, bisects circles are similar. Use the definition of tangent of an angle to find PR, which is the radius of 53. CHALLENGE In the figure, is inscribed in equilateral triangle LMN. What is the circumference

of ?

Therefore, the radius of is The circumference C of a circle with radius r is given by SOLUTION: Draw a perpendicular to the side Also join

The circumference of is inches.

ANSWER:

Since is equilateral triangle, MN = 8 inches in. and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry.

SOLUTION: See students’ work.

ANSWER: See students’ work. Therefore, the radius of is 55. GRIDDED RESPONSE What is the The circumference C of a circle with radius r is circumference of ? Round to the nearest tenth. given by

SOLUTION: The circumference of is inches. Use the Pythagorean Theorem to find the diameter

ANSWER: of the circle.

in. The circumference C of a circle with diameter d is given by Use the formula to find the circumference. 54. WRITING IN MATH Research and write about

the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

ANSWER: See students’ work. ANSWER: 55. GRIDDED RESPONSE What is the 40.8 circumference of ? Round to the nearest tenth. 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft SOLUTION: SOLUTION: Use the Pythagorean Theorem to find the diameter The circumference C of a circle with radius r is of the circle. given by Here, C = 10 ft. Use the formula to find the radius. The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: A

ANSWER: 57. ALGEBRA Bill is planning a circular vegetable 40.8 garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the 56. What is the radius of a table with a circumference of garden? 10 feet? F 10 A 1.6 ft G 9 B 2.5 ft H 8 C 3.2 ft J 7 D 5 ft SOLUTION: SOLUTION: The circumference C of a circle with radius r is The circumference C of a circle with radius r is given by given by Because 50 feet of fence can be used to border the Here, C = 10 ft. Use the formula to find the radius. garden, Cmax = 50. Use the circumference formula to find the maximum radius.

The radius of the circle is about 1.6 feet, so the correct choice is A. Therefore, the radius of the garden must be less than ANSWER: or equal to about 7.96 feet. Therefore, the only A correct choice is J. ANSWER: 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use J up to 50 feet of fence, what radius can he use for the garden? 58. SAT/ACT What is the radius of a circle with an

F 10 area of square units? G 9 H 8 A 0.4 units J 7 B 0.5 units C 2 units SOLUTION: D 4 units The circumference C of a circle with radius r is E 16 units given by Because 50 feet of fence can be used to border the SOLUTION: The area of a circle of radius r is given by the garden, Cmax = 50. Use the circumference formula to find the maximum radius. formula Here A = square units. Use the formula to find the radius.

Therefore, the radius of the garden must be less than

or equal to about 7.96 feet. Therefore, the only Therefore, the correct choice is B. correct choice is J. ANSWER: ANSWER: B J Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation 58. SAT/ACT What is the radius of a circle with an with center B and the scale factor r indicated.

area of square units? 59. A 0.4 units B 0.5 units C 2 units D 4 units E 16 units SOLUTION: SOLUTION: The area of a circle of radius r is given by the Draw a ray from point B through a vertex of the formula Here A = square units. Use the hexagon. Use the ruler to measure the distance from formula to find the radius. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the correct choice is B.

ANSWER: B Copy each figure and point B. Then use a ruler to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the 60. hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the SOLUTION: hexagon. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER: ANSWER: 10-1 Circles and Circumference

61. 60.

SOLUTION: SOLUTION: Draw a ray from point B through a vertex of the Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for B to the vertex. Mark a point on the ray that is of rays through the other three vertices. Connect the that distance from B. Repeat the process for rays four points on the rays to construct the dilated image through the other two vertices. Connect the three of the trapezoid. points on the rays to construct the dilated image of the triangle.

ANSWER: ANSWER:

61.

62. SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the SOLUTION: four points on the rays to construct the dilated image Draw a ray from point B through a vertex of the of the trapezoid. triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

eSolutions Manual - Powered by Cognero Page 19

62. ANSWER:

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER:

No

ANSWER: 64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry is 4.

ANSWER: Yes; 4, 90

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any 65. rotation. So, it does not have a rotational symmetry. SOLUTION: ANSWER: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. No ANSWER: 64. Refer to the image on page 705. No

SOLUTION: 66. Refer to the image on page 705. The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the SOLUTION: magnitude of symmetry is 90º and order of symmetry The given figure cannot be mapped onto itself by any is 4. rotation. So, it does not have a rotational symmetry.

ANSWER: ANSWER: Yes; 4, 90 No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true 65. and the conclusion is true, then the statement is true SOLUTION: for the conditions. The given figure cannot be mapped onto itself by any ANSWER:

rotation. So, it does not have a rotational symmetry. True; sample answer: Since the hypothesis is true ANSWER: and the conclusion is true, then the statement is true for the conditions. No

66. Refer to the image on page 705. 68. You are 21 years old and do not vote. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any False; sample answer: Since the hypothesis is true rotation. So, it does not have a rotational symmetry. and the conclusion is false, then the statement is false for the conditions. ANSWER: No ANSWER: False; sample answer: Since the hypothesis is true Determine the truth value of the following and the conclusion is false, then the statement is false statement for each set of conditions. Explain for the conditions. your reasoning. If you are over 18 years old, then you vote in all Find x. elections. 67. You are 19 years old and you vote. SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. 69.

ANSWER: SOLUTION: True; sample answer: Since the hypothesis is true The sum of the measures of angles around a point is and the conclusion is true, then the statement is true 360º. for the conditions.

68. You are 21 years old and do not vote. SOLUTION: ANSWER: False; sample answer: Since the hypothesis is true 90 and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. 70. Find x. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

69. ANSWER: SOLUTION: 60 The sum of the measures of angles around a point is 360º.

ANSWER:

90 71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

70. SOLUTION: ANSWER: The angles with the measures xº and 2xº form a 20 straight line. So, the sum is 180.

ANSWER: 60 72. SOLUTION: The sum of the three angles is 180.

71. ANSWER: SOLUTION: 45 The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: Here, is a radius and the diameter is twice the radius.

For Exercises 1–4, refer to . Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure. 1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

5. FG 2. Identify each. a. a chord SOLUTION: b. a diameter Since the diameter of B is 18 inches and A is 8 c. a radius inches, AG = 18 and AF = (8) or 4. SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes Therefore, FG = 14 in. through the center. Here, is a diameter. ANSWER: c. A radius is a segment with endpoints at the center 14 in. and on the circle. Here, is radius. 6. FB ANSWER: SOLUTION: a. b. Since the diameter of B is 18 inches and A is 4

c. inches, AB = (18) or 9 and AF = (8) or 4.

3. If CN = 8 centimeters, find DN. SOLUTION:

Here, are radii of the same circle. So, Therefore, FB = 5 inches. they are equal in length. Therefore, DN = 8 cm. ANSWER: ANSWER: 5 in. 8 cm 7. RIDES The circular ride described at the beginning 4. If EN = 13 feet, what is the diameter of the circle? of the lesson has a diameter of 44 feet. What are the SOLUTION: radius and circumference of the ride? Round to the nearest hundredth, if necessary. Here, is a radius and the diameter is twice the radius. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. Therefore, the diameter is 26 ft.

ANSWER: 26 ft

The diameters of , , and are 8 Therefore, the circumference of the ride is about inches, 18 inches, and 11 inches, respectively. 138.23 feet. Find each measure. ANSWER: 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth.

5. FG SOLUTION: Since the diameter of B is 18 inches and A is 8 inches, AG = 18 and AF = (8) or 4.

SOLUTION: Therefore, FG = 14 in.

ANSWER: 14 in.

6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4 inches, AB = (18) or 9 and AF = (8) or 4.

The diameter of the pool is about 17.98 feet and the radius of the pool is about 8.99 feet.

Therefore, FB = 5 inches. ANSWER: 17.98 ft; 8.99 ft ANSWER: 5 in. 9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of 7. RIDES The circular ride described at the beginning . of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. SOLUTION: The radius is half the diameter. So, the radius of the circular ride is 22 feet. SOLUTION: The diameter of the circle is the hypotenuse of the right triangle.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: The diameter of the circle is 4 centimeters. 22 ft; 138.23 ft

8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. What are the diameter and radius of the pool? Round to the nearest hundredth. The circumference of the circle is 4π centimeters.

ANSWER:

For Exercises 10–13, refer to .

SOLUTION:

10. Name the center of the circle.

SOLUTION:

R ANSWER: R

The diameter of the pool is about 17.98 feet and the 11. Identify a chord that is also a diameter. radius of the pool is about 8.99 feet. SOLUTION: ANSWER: Two chords are shown: and . goes 17.98 ft; 8.99 ft through the center, R, so is a diameter.

9. SHORT RESPONSE The right triangle shown is inscribed in . Find the exact circumference of ANSWER: .

12. Is a radius? Explain. SOLUTION: A radius is a segment with endpoints at the center and on the circle. But has both the end points on SOLUTION: the circle, so it is a chord. The diameter of the circle is the hypotenuse of the ANSWER: right triangle. No; it is a chord. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: Here, is a diameter and is a radius. The diameter of the circle is 4 centimeters.

The circumference of the circle is 4π Therefore, RT = 8.1 cm. centimeters. ANSWER: ANSWER: 8.1 cm For Exercises 14–17, refer to . For Exercises 10–13, refer to .

14. Identify a chord that is not a diameter.

10. Name the center of the circle. SOLUTION: SOLUTION: The chords do not pass through the center. So, they are not diameters. R ANSWER: ANSWER: R

11. Identify a chord that is also a diameter. 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Two chords are shown: and . goes Here, is a radius and the diameter is twice the radius. . through the center, R, so is a diameter.

ANSWER: Therefore, the diameter of the circle is 28 inches. ANSWER:

12. Is a radius? Explain. 28 in. SOLUTION:

A radius is a segment with endpoints at the center 16. Is ? Explain. and on the circle. But has both the end points on SOLUTION: the circle, so it is a chord. All radii of a circle are congruent. Since

ANSWER: are both radii of , . No; it is a chord. ANSWER: 13. If SU = 16.2 centimeters, what is RT? Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? Here, is a diameter and is a radius. SOLUTION: Here, is a diameter and is a radius.

Therefore, RT = 8.1 cm.

ANSWER: Therefore, EF = 3.7 cm. 8.1 cm ANSWER: For Exercises 14–17, refer to . 3.7 cm Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each measure.

14. Identify a chord that is not a diameter. SOLUTION:

The chords do not pass through the center. So, they are not diameters. 18. CK ANSWER: SOLUTION: Since the radius of K is 8 units, BK = 8.

15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: Therefore, CK = 2.6 units.

Here, is a radius and the diameter is twice the ANSWER: radius. . 2.6

19. AB

Therefore, the diameter of the circle is 28 inches. SOLUTION: Since is a diameter of circle J and the radius is ANSWER: 10, AC = 2(10) or 20 units. 28 in.

16. Is ? Explain. SOLUTION: Therefore, AB = 14.6 units. All radii of a circle are congruent. Since ANSWER:

are both radii of , . 14.6 ANSWER: 20. JK Yes; they are both radii of . SOLUTION: 17. If DA = 7.4 centimeters, what is EF? First find CK. Since circle K has a radius of 8 units, BK = 8. SOLUTION: Here, is a diameter and is a radius.

Since circle J has a radius of 10 units, JC = 10.

Therefore, EF = 3.7 cm. Therefore, JK = 12.6 units.

ANSWER: ANSWER: 3.7 cm 12.6

Circle J has a radius of 10 units, has a 21. AD radius of 8 units, and BC = 5.4 units. Find each SOLUTION: measure. We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

18. CK SOLUTION: Since the radius of K is 8 units, BK = 8. Therefore, AD = 30.6 units. ANSWER: 30.6 Therefore, CK = 2.6 units. 22. PIZZA Find the radius and circumference of the ANSWER: pizza shown. Round to the nearest hundredth, if

2.6 necessary.

19. AB SOLUTION: Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

SOLUTION: Therefore, AB = 14.6 units. The diameter of the pizza is 16 inches. ANSWER: 14.6 20. JK SOLUTION: . So, the radius of the pizza is 8 inches. First find CK. Since circle K has a radius of 8 units, The circumference C of a circle of diameter d is BK = 8. given by

Since circle J has a radius of 10 units, JC = 10. Therefore, the circumference of the pizza is about 50.27 inches.

Therefore, JK = 12.6 units. ANSWER: 8 in.; 50.27 in. ANSWER: 12.6 23. BICYCLES A bicycle has tires with a diameter of 26 inches. Find the radius and circumference of a 21. AD tire. Round to the nearest hundredth, if necessary. SOLUTION: SOLUTION: We can find AD using AD = AC + CK + KD. The The diameter of a tire is 26 inches. The radius is half radius of circle K is 8 units and the radius of circle J the diameter. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by Therefore, AD = 30.6 units.

ANSWER: 30.6 Therefore, the circumference of the bicycle tire is 22. PIZZA Find the radius and circumference of the about 81.68 inches. pizza shown. Round to the nearest hundredth, if necessary. ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. 24. C = 18 in. SOLUTION: The circumference C of a circle with diameter d is given by SOLUTION: Here, C = 18 in. Use the formula to find the The diameter of the pizza is 16 inches. diameter. Then find the radius.

. So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the diameter is about 5.73 inches and the Therefore, the circumference of the pizza is about radius is about 2.86 inches. 50.27 inches. ANSWER: ANSWER: 5.73 in.; 2.86 in. 8 in.; 50.27 in.

23. BICYCLES A bicycle has tires with a diameter of 25. C = 124 ft 26 inches. Find the radius and circumference of a SOLUTION: tire. Round to the nearest hundredth, if necessary. The circumference C of a circle with diameter d is SOLUTION: given by The diameter of a tire is 26 inches. The radius is half Here, C = 124 ft. Use the formula to find the the diameter. diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the diameter is about 39.47 feet and the Therefore, the circumference of the bicycle tire is radius is about 19.74 feet. about 81.68 inches. ANSWER: ANSWER: 39.47 ft; 19.74 ft 13 in.; 81.68 in. 26. C = 375.3 cm Find the diameter and radius of a circle by with the given circumference. Round to the nearest SOLUTION: hundredth. The circumference C of a circle with diameter d is 24. C = 18 in. given by Here, C = 375.3 cm. Use the formula to find the SOLUTION: diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. ANSWER: Therefore, the diameter is about 5.73 inches and the 119.46 cm; 59.73 cm radius is about 2.86 inches. 27. C = 2608.25 m ANSWER: 5.73 in.; 2.86 in. SOLUTION: The circumference C of a circle with diameter d is given by

25. C = 124 ft Here, C = 2608.25 meters. Use the formula to find SOLUTION: the diameter. Then find the radius. The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters.

ANSWER: 830.23 m; 415.12 m Therefore, the diameter is about 39.47 feet and the radius is about 19.74 feet. CCSS SENSE-MAKING Find the exact circumference of each circle by using the given ANSWER: inscribed or 39.47 ft; 19.74 ft circumscribed polygon.

26. C = 375.3 cm SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the 28. diameter. Then find the radius. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters.

ANSWER: Therefore, the circumference of the circle is 17 119.46 cm; 59.73 cm centimeters.

27. C = 2608.25 m ANSWER: SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 830.23 meters and the radius is about 415.12 meters. The diameter of the circle is 12 feet. The ANSWER: circumference C of a circle is given by 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact circumference of each circle by using the given inscribed or Therefore, the circumference of the circle is 12 circumscribed polygon. feet.

ANSWER:

28. SOLUTION: The hypotenuse of the right triangle is a diameter of 30. the circle. Use the Pythagorean Theorem to find the diameter. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

The diameter of the circle is 17 cm. The circumference C of a circle is given by

The diameter of the circle is inches. The Therefore, the circumference of the circle is 17 circumference C of a circle is given by centimeters.

ANSWER:

Therefore, the circumference of the circle is inches.

ANSWER:

29. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter. 31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter The diameter of the circle is 12 feet. The of the circle. circumference C of a circle is given by

Therefore, the circumference of the circle is 12 The diameter of the circle is 10 inches. The feet. circumference C of a circle is given by ANSWER:

Therefore, the circumference of the circle is 10 inches.

ANSWER: 30. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. 32. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The diameter of the circle is inches. The The circumference C of a circle is given by circumference C of a circle is given by

Therefore, the circumference of the circle is 25 millimeters. Therefore, the circumference of the circle is inches. ANSWER:

ANSWER:

31. 33. SOLUTION: SOLUTION: Each diagonal of the inscribed rectangle will pass A diameter perpendicular to a side of the square and through the origin, so it is a diameter of the circle. the length of each side will measure the same. So, d Use the Pythagorean Theorem to find the diameter = 14 yards. of the circle. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 The diameter of the circle is 10 inches. The yards. circumference C of a circle is given by ANSWER:

34. DISC GOLF Disc golf is similar to regular golf, Therefore, the circumference of the circle is 10 except that a flying disc is used instead of a ball and inches. clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in ANSWER: centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the 32. diameter. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by The diameter of the disc is about 21.3 centimeters and the maximum weight allowed is 8.3 times the diameter in centimeters. Therefore, the maximum weight is 8.3 × 21.3 or Therefore, the circumference of the circle is 25 about 176.8 grams. millimeters. ANSWER: ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? 33. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The circumference C of a circle with radius r is given by Therefore, the circumference of the circle is 14 yards. Use the formula to find the circumference.

ANSWER:

Therefore, the circumference is about 31.42 feet. 34. DISC GOLF Disc golf is similar to regular golf, b. Here, C = 25 ft. Use the formula to find the except that a flying disc is used instead of a ball and inner clubs. For professional competitions, the maximum radius of the inner circle. weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. SOLUTION: So, the radius of the inner circle is about 4 feet. The circumference C of a circle with diameter d is given by ANSWER: Here, C = 66.92 cm. Use the formula to find the a. 31.42 ft diameter. b. 4 ft

The radius, diameter, or circumference of a circle is given. Find each missing measure to the nearest hundredth.

The diameter of the disc is about 21.3 centimeters 36. and the maximum weight allowed is 8.3 times the diameter in centimeters. SOLUTION: Therefore, the maximum weight is 8.3 × 21.3 or The radius is half the diameter and the about 176.8 grams. circumference C of a circle with diameter d is given by ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot? Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in.

37. SOLUTION: SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The The diameter is twice the radius and the circumference C of a circle with radius r is given by circumference C of a circle with diameter d is given

by Use the formula to find the circumference.

Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle. So, the diameter is 22.80 feet and the circumference is about 71.63 feet.

ANSWER: 22.80 ft; 71.63 ft So, the radius of the inner circle is about 4 feet. 38. ANSWER: a. 31.42 ft SOLUTION: b. 4 ft The circumference C of a circle with diameter d is given by The radius, diameter, or circumference of a Here, C = 35x cm. Use the formula to find the circle is given. Find each missing measure to diameter.Then find the radius. the nearest hundredth.

36.

SOLUTION:

The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. ANSWER: 11.14x cm; 5.57x cm

Therefore, the radius is 4.25 inches and the 39. circumference is about 26.70 inches.

ANSWER: SOLUTION: 4.25 in.; 26.70 in. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

37.

SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference

is about 0.79x units. ANSWER: 0.25x; 0.79x

So, the diameter is 22.80 feet and the circumference Determine whether the circles in the figures is about 71.63 feet. below appear to be congruent, concentric, or neither. ANSWER: 40. Refer to the photo on page 703. 22.80 ft; 71.63 ft SOLUTION: 38. The circles have the same center and they are in the same plane. So, they are concentric circles. SOLUTION: ANSWER: The circumference C of a circle with diameter d is concentric given by Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. 41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they Therefore, the diameter is about 11.14x centimeters appear to be congruent. and the radius is about 5.57x centimeters. ANSWER: ANSWER: congruent 11.14x cm; 5.57x cm 43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the 39. coordinate grid represents 25 feet, how far would someone have to walk to go completely around the SOLUTION: ring? Round to the nearest tenth. The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. SOLUTION: The radius of the circle is 3 units, that is 75 feet. So, ANSWER: the diameter is 150 ft. The circumference C of a 0.25x; 0.79x circle with diameter d is given by So, the circumference is Determine whether the circles in the figures below appear to be congruent, concentric, or Therefore, someone have to neither. walk about 471.2 feet to go completely around the 40. Refer to the photo on page 703. ring. SOLUTION: ANSWER: The circles have the same center and they are in the 471.2 ft same plane. So, they are concentric circles. 44. CCSS MODELING A brick path is being installed ANSWER: around a circular pond. The pond has a concentric circumference of 68 feet. The outer edge of the path is going to be 4 feet from the pond all the way around. What is the approximate circumference of 41. Refer to the photo on page 703. the path? Round to the nearest hundredth. SOLUTION: SOLUTION: The circles neither have the same radius nor are they The circumference C of a circle with radius r is concentric. given by Here, Cpond = 68 ft. Use the ANSWER: formula to find the radius of the pond. neither

42. Refer to the photo on page 703. SOLUTION: The circles appear to have the same radius. So, they appear to be congruent. So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 ANSWER: = 14.82 ft. Use the radius to find the circumference congruent of the brick path.

43. HISTORY The Indian Shell Ring on Hilton Head Island approximates a circle. If each unit on the coordinate grid represents 25 feet, how far would Therefore, the circumference of the path is about someone have to walk to go completely around the ring? Round to the nearest tenth. 93.13 feet. ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth) of each circle. Record your results in a table. SOLUTION: c. VERBAL Explain why these three circles are The radius of the circle is 3 units, that is 75 feet. So, geometrically similar. the diameter is 150 ft. The circumference C of a d. VERBAL Make a conjecture about the ratio circle with diameter d is given by between the circumferences of two circles when the So, the circumference is ratio between their radii is 2. Therefore, someone have to e. ANALYTICAL The scale factor from to walk about 471.2 feet to go completely around the is . Write an equation relating the ring. circumference (C ) of to the circumference ANSWER: A (C ) of . 471.2 ft B f. NUMERICAL If the scale factor from to 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a is , and the circumference of is 12 circumference of 68 feet. The outer edge of the path inches, what is the circumference of ? is going to be 4 feet from the pond all the way around. What is the approximate circumference of SOLUTION: the path? Round to the nearest hundredth. a. Sample answer: SOLUTION: The circumference C of a circle with radius r is given by Here, Cpond = 68 ft. Use the formula to find the radius of the pond.

b. Sample answer.

So, the radius of the pond is about 10.82 feet and the radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path. c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. Therefore, the circumference of the path is about So, if the radii are in the ratio b : a, so will the 93.13 feet. circumference. Therefore, . ANSWER: f. Use the formula from part e. Here, the 93.13 ft circumference of is 12 inches and . 45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three circles in which the scale factor from each circle to Therefore, the circumference of is 4 inches. the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest ANSWER: tenth) and circumference (to the nearest hundredth) a. Sample answer: of each circle. Record your results in a table. c. VERBAL Explain why these three circles are geometrically similar. d. VERBAL Make a conjecture about the ratio between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to b. Sample answer. is . Write an equation relating the

circumference (CA) of to the circumference (CB) of . f. NUMERICAL If the scale factor from to is , and the circumference of is 12 c. They all have the same shape—circular. inches, what is the circumference of ? d. The ratio of their circumferences is also 2.

SOLUTION: e. a. Sample answer: f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

b. Sample answer.

c. They all have the same shape–circular. a. Drop the needle onto the paper. When the needle d. The ratio of their circumferences is also 2. lands, record whether it touches one of the lines as a e. The circumference is directly proportional to radii. hit. Record the number of hits after 25, 50, and 100 So, if the radii are in the ratio b : a, so will the drops.

circumference. Therefore, . f. Use the formula from part e. Here, the b. Calculate the ratio of two times the total number circumference of is 12 inches and . of drops to the number of hits after 25, 50, and 100 drops.

c. How are the values you found in part b related to π?

Therefore, the circumference of is 4 inches. SOLUTION: a. See students’ work. ANSWER: b. See students’ work. a. Sample answer: c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. b. Sample answer. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

47. M APS The concentric circles on the map below show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix.

c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper. a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase? SOLUTION: a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the circumference of each circle. a. Drop the needle onto the paper. When the needle lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 drops.

b. Calculate the ratio of two times the total number The difference of the circumferences of the of drops to the number of hits after 25, 50, and 100 outermost circle and the center circle is 60π - 10π = drops. 50π or about 157.1 miles.

b. Let the radius of one circle be x miles and the c. How are the values you found in part b related to radius of the next larger circle be x + 5 miles. Find π? the circumference of each circle. SOLUTION: a. See students’ work. b. See students’ work.

c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

ANSWER: The difference of the two circumferences is (2xπ + a. See students’ work. 10π) - (2xπ) = 10π or about 31.4 miles. b. See students’ work. Therefore, as the radius of each circle increases by 5 c. Sample answer: The values are approaching 3.14, miles, the circumference increases by about 31.4 which is approximately equal to π. miles.

ANSWER: 47. M APS The concentric circles on the map below a. 157.1 mi show the areas that are 5, 10, 15, 20, 25, and 30 b. by about 31.4 mi miles from downtown Phoenix. 48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. a. How much greater is the circumference of the A line segment with endpoints on a circle can be outermost circle than the circumference of the center described as a chord. If the chord passes through

circle? the center of the circle, it can be described as a b. As the radii of the circles increases by 5 miles, by diameter. A line segment with endpoints at the how much does the circumference increase? center and on the circle can be described as a SOLUTION: radius. a. The radius of the outermost circle is 30 miles and that of the center circle is 5 miles. Find the ANSWER: circumference of each circle. Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with

endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be The difference of the circumferences of the described as a radius. outermost circle and the center circle is 60π - 10π = 50π or about 157.1 miles. b. Let the radius of one circle be x miles and the radius of the next larger circle be x + 5 miles. Find the circumference of each circle. 49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? The difference of the two circumferences is (2xπ + the inscribed polygon? Write a compound inequality 10π) - (2xπ) = 10π or about 31.4 miles. comparing C to these perimeters. Therefore, as the radius of each circle increases by 5 c. Rewrite the inequality from part b in terms of the miles, the circumference increases by about 31.4 diameter d of the circle and interpret its meaning. miles. d. As the number of sides of both the circumscribed ANSWER: and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part a. 157.1 mi c, and what does this imply? b. by about 31.4 mi SOLUTION: 48. WRITING IN MATH How can we describe the a. The circumscribed polygon is a square with sides relationships that exist between of length 2r. circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in one point, two points, or may not intersect at all. A The inscribed polygon is a hexagon with sides of line that intersects a circle in one point can be length r. (Each triangle formed using a central angle described as a tangent. A line that intersects a circle will be equilateral.) in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. If the chord passes through the center of the circle, it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER:

Sample answer: A line that intersects a circle in one So, the perimeters are 8r and 6r. point can be described as a tangent. A line that b. intersects a circle in exactly two points can be described as a secant. A line segment with

endpoints on a circle can be described as a chord. The circumference is less than the perimeter of the If the chord passes through the center of the circle, circumscribed polygon and greater than the it can be described as a diameter. A line segment perimeter of the inscribed polygon. with endpoints at the center and on the circle can be 8r < C < 6r described as a radius. c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. 49. d. These limits will approach a value of πd, implying REASONING In the figure, a circle with radius r is that C = πd. inscribed in a regular polygon and circumscribed about another. a. What are the perimeters of the circumscribed and ANSWER: inscribed polygons in terms of r? Explain. a. 8r and 6r ; Twice the radius of the circle, 2r is b. Is the circumference C of the circle greater or the side length of the square, so the perimeter of the less than the perimeter of the circumscribed polygon? square is 4(2r) or 8r. The the inscribed polygon? Write a compound inequality regular hexagon is made up of six equilateral comparing C to these perimeters. triangles with side length r, so the perimeter of the c. Rewrite the inequality from part b in terms of the diameter d of the circle and interpret its meaning. hexagon is 6(r) or 6r. d. As the number of sides of both the circumscribed b. less; greater; 6r < C < 8r and inscribed polygons increase, what will happen to c. 3d < C < 4d; The circumference of the circle is the upper and lower limits of the inequality from part between 3 and 4 times its diameter. c, and what does this imply? d. These limits will approach a value of πd, implying that C = πd. SOLUTION: a. The circumscribed polygon is a square with sides 50. CHALLENGE The sum of the circumferences of of length 2r. circles H, J, and K shown at the right is units. Find KJ.

The inscribed polygon is a hexagon with sides of length r. (Each triangle formed using a central angle will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the circumference of each circle.

So, the perimeters are 8r and 6r. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) The sum of the circumferences is 56π units. or 4d and 6r = 3(2r) or 3d. Substitute the three circumferences and solve for x. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying So, the radius of is 4 units, the radius of is that C = πd. 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii ANSWER: of and . a. 8r and 6r ; Twice the radius of the circle, 2r is Therefore, KJ = 16 + 8 or 24 units. the side length of the square, so the perimeter of the square is 4(2r) or 8r. The ANSWER: regular hexagon is made up of six equilateral 24 units triangles with side length r, so the perimeter of the hexagon is 6(r) or 6r. 51. REASONING Is the distance from the center of a b. less; greater; 6r < C < 8r circle to a point in the interior of a circle sometimes, always, or never less than the radius of the circle? c. 3d < C < 4d; The circumference of the circle is Explain. between 3 and 4 times its diameter. d. These limits will approach a value of πd, implying SOLUTION: that C = πd. A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn 50. CHALLENGE The sum of the circumferences of from the center to a point inside the circle will always circles H, J, and K shown at the right is units. have a length less than the radius of the circle. Find KJ. ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the A circle is a locus of points in a plane equidistant circumference of each circle. from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

The sum of the circumferences is 56π units. Substitute the three circumferences and solve for x.

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are So, the radius of is 4 units, the radius of is similar. Thus, all circles are similar. 2(4) or 8 units, and the radius of is 4(4) or 16 units. The measure of KJ is equal to the sum of the radii ANSWER: of and . Therefore, KJ = 16 + 8 or 24 units. A circle is a locus of points in a plane equidistant from a given point. For any two circles and ANSWER: , there exists a translation that maps center A 24 units onto center B, moving so that it is concentric with . There also exists a dilation with scale REASONING 51. Is the distance from the center of a factor k such that each point that makes up is circle to a point in the interior of a circle sometimes, moved to be the same distance from center A as the always, or never less than the radius of the circle? Explain. points that make up are from center B. Therefore, is mapped onto . Since there SOLUTION: exists a rigid motion followed by a scaling that maps A radius is a segment drawn between the center of onto , the circles are similar. Thus, all the circle and a point on the circle. A segment drawn circles are similar. from the center to a point inside the circle will always have a length less than the radius of the circle. 53. CHALLENGE In the figure, is inscribed in ANSWER: equilateral triangle LMN. What is the circumference

Always; a radius is a segment drawn between the of ? center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: SOLUTION: A circle is a locus of points in a plane equidistant Draw a perpendicular to the side Also join from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = Since is equilateral triangle, MN = 8 inches kr1. and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of

Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps onto , the circles are similar. Thus, all circles are similar. Therefore, the radius of is The circumference C of a circle with radius r is given by ANSWER: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale

factor k such that each point that makes up is The circumference of is inches. moved to be the same distance from center A as the points that make up are from center B. ANSWER: Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps in. onto , the circles are similar. Thus, all circles are similar. 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of 53. CHALLENGE In the figure, is inscribed in geometry. equilateral triangle LMN. What is the circumference of ? SOLUTION: See students’ work.

ANSWER: See students’ work.

55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth. SOLUTION: Draw a perpendicular to the side Also join

SOLUTION: Use the Pythagorean Theorem to find the diameter of the circle.

Since is equilateral triangle, MN = 8 inches The circumference C of a circle with diameter d is

and RN = 4 inches. Also, bisects given by Use the formula to find the circumference. Use the definition of

tangent of an angle to find PR, which is the radius of

ANSWER: Therefore, the radius of is 40.8 The circumference C of a circle with radius r is 56. What is the radius of a table with a circumference of given by 10 feet? A 1.6 ft B 2.5 ft C 3.2 ft D 5 ft SOLUTION:

The circumference of is inches. The circumference C of a circle with radius r is given by ANSWER: Here, C = 10 ft. Use the formula to find the radius.

in.

54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. The radius of the circle is about 1.6 feet, so the correct choice is A. SOLUTION: See students’ work. ANSWER: A ANSWER: See students’ work. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use 55. GRIDDED RESPONSE What is the up to 50 feet of fence, what radius can he use for the circumference of ? Round to the nearest tenth. garden? F 10 G 9 H 8 J 7

SOLUTION: The circumference C of a circle with radius r is SOLUTION: given by Use the Pythagorean Theorem to find the diameter Because 50 feet of fence can be used to border the of the circle. garden, C = 50. Use the circumference formula to max find the maximum radius. The circumference C of a circle with diameter d is given by Use the formula to find the circumference.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: ANSWER: J 40.8 58. SAT/ACT What is the radius of a circle with an 56. What is the radius of a table with a circumference of 10 feet? area of square units? A 1.6 ft A 0.4 units B 2.5 ft B 0.5 units C 3.2 ft C 2 units D 5 ft D 4 units SOLUTION: E 16 units The circumference C of a circle with radius r is SOLUTION: given by The area of a circle of radius r is given by the Here, C = 10 ft. Use the formula to find the radius. formula Here A = square units. Use the formula to find the radius.

The radius of the circle is about 1.6 feet, so the correct choice is A. Therefore, the correct choice is B. ANSWER: ANSWER: A B ALGEBRA 57. Bill is planning a circular vegetable Copy each figure and point B. Then use a ruler garden with a fence around the border. If he can use to draw the image of the figure under a dilation up to 50 feet of fence, what radius can he use for the with center B and the scale factor r indicated. garden?

F 10 59. G 9 H 8 J 7 SOLUTION: The circumference C of a circle with radius r is given by SOLUTION: Because 50 feet of fence can be used to border the Draw a ray from point B through a vertex of the garden, Cmax = 50. Use the circumference formula to hexagon. Use the ruler to measure the distance from

find the maximum radius. B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon.

Therefore, the radius of the garden must be less than or equal to about 7.96 feet. Therefore, the only correct choice is J.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units?

A 0.4 units B 0.5 units ANSWER: C 2 units D 4 units E 16 units SOLUTION: 60. The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from Therefore, the correct choice is B. B to the vertex. Mark a point on the ray that is of ANSWER: that distance from B. Repeat the process for rays B through the other two vertices. Connect the three points on the rays to construct the dilated image of Copy each figure and point B. Then use a ruler the triangle. to draw the image of the figure under a dilation with center B and the scale factor r indicated.

59.

ANSWER:

SOLUTION: 61. Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays SOLUTION: through the other five vertices. Connect the six points Draw a ray from point B through a vertex of the on the rays to construct the dilated image of the trapezoid. Use the ruler to measure the distance from hexagon. B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER: ANSWER:

60.

62.

SOLUTION: Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from SOLUTION: B to the vertex. Mark a point on the ray that is of Draw a ray from point B through a vertex of the that distance from B. Repeat the process for rays triangle. Use the ruler to measure the distance from through the other two vertices. Connect the three B to the vertex. Mark a point on the ray that is 3 points on the rays to construct the dilated image of times that distance from B. Repeat the process for the triangle. rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

ANSWER:

61.

ANSWER: SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. ANSWER: SOLUTION: The figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: 10-1 Circles and Circumference No

62. 64. Refer to the image on page 705. SOLUTION: The figure can be mapped onto itself by a rotation of 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry SOLUTION: is 4. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from ANSWER: B to the vertex. Mark a point on the ray that is 3 Yes; 4, 90 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle.

65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: ANSWER: No 66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. State whether each figure has rotational If you are over 18 years old, then you vote in all symmetry. Write yes or no. If so, copy the elections. figure, locate the center of symmetry, and state 67. You are 19 years old and you vote. the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true The figure cannot be mapped onto itself by any for the conditions. rotation. So, it does not have a rotational symmetry. ANSWER: ANSWER: True; sample answer: Since the hypothesis is true No and the conclusion is true, then the statement is true for the conditions. Refer to the image on page 705. eSolutions64. Manual - Powered by Cognero Page 20 68. You are 21 years old and do not vote. SOLUTION: The figure can be mapped onto itself by a rotation of SOLUTION: 90º. So, it has a rotational symmetry. So, the False; sample answer: Since the hypothesis is true magnitude of symmetry is 90º and order of symmetry and the conclusion is false, then the statement is false is 4. for the conditions.

ANSWER: ANSWER: Yes; 4, 90 False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. 69. SOLUTION: SOLUTION: The given figure cannot be mapped onto itself by any The sum of the measures of angles around a point is rotation. So, it does not have a rotational symmetry. 360º. ANSWER: No

66. Refer to the image on page 705. ANSWER: SOLUTION: 90 The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No Determine the truth value of the following statement for each set of conditions. Explain 70. your reasoning. SOLUTION: If you are over 18 years old, then you vote in all elections. The angles with the measures xº and 2xº form a

67. You are 19 years old and you vote. straight line. So, the sum is 180.

SOLUTION: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions. ANSWER: ANSWER: 60 True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true for the conditions.

68. You are 21 years old and do not vote. SOLUTION: 71. False; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is false, then the statement is false for the conditions. The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180. ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. ANSWER: Find x. 20

69.

SOLUTION: 72. The sum of the measures of angles around a point is SOLUTION: 360º. The sum of the three angles is 180.

ANSWER: ANSWER: 90 45

70. SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45 For Exercises 1–4, refer to .

1. Name the circle. SOLUTION: The center of the circle is N. So, the circle is

ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius SOLUTION: a. A chord is a segment with endpoints on the circle. So, here are chords. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius.

ANSWER: a. b. c.

3. If CN = 8 centimeters, find DN. SOLUTION: Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER: 8 cm 4. If EN = 13 feet, what is the diameter of the circle? SOLUTION: For Exercises 1–4, refer to . Here, is a radius and the diameter is twice the radius.

Therefore, the diameter is 26 ft.

ANSWER: 26 ft 1. Name the circle. SOLUTION: The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. The center of the circle is N. So, the circle is Find each measure. ANSWER:

2. Identify each. a. a chord b. a diameter c. a radius 5. FG SOLUTION: SOLUTION: a. A chord is a segment with endpoints on the circle. Since the diameter of B is 18 inches and A is 8 So, here are chords. inches, AG = 18 and AF = (8) or 4. b. A diameter of a circle is a chord that passes through the center. Here, is a diameter. c. A radius is a segment with endpoints at the center and on the circle. Here, is radius. Therefore, FG = 14 in.

ANSWER: ANSWER: a. 14 in. b. c. 6. FB SOLUTION: 3. If CN = 8 centimeters, find DN. Since the diameter of B is 18 inches and A is 4 SOLUTION: inches, AB = (18) or 9 and AF = (8) or 4. Here, are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

ANSWER:

8 cm Therefore, FB = 5 inches.

4. If EN = 13 feet, what is the diameter of the circle? ANSWER: SOLUTION: 5 in.

Here, is a radius and the diameter is twice the 7. RIDES The circular ride described at the beginning

radius. of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the nearest hundredth, if necessary. Therefore, the diameter is 26 ft. SOLUTION: ANSWER: The radius is half the diameter. So, the radius of the 26 ft circular ride is 22 feet. The diameters of , , and are 8 inches, 18 inches, and 11 inches, respectively. Find each measure.

Therefore, the circumference of the ride is about 138.23 feet.

ANSWER: 22 ft; 138.23 ft

5. FG 8. CCSS MODELING The circumference of the circular swimming pool shown is about 56.5 feet. SOLUTION: What are the diameter and radius of the pool? Round Since the diameter of B is 18 inches and A is 8 to the nearest hundredth. inches, AG = 18 and AF = (8) or 4.

Therefore, FG = 14 in.

ANSWER: 14 in. SOLUTION:

6. FB SOLUTION: Since the diameter of B is 18 inches and A is 4

inches, AB = (18) or 9 and AF = (8) or 4.

Therefore, FB = 5 inches. The diameter of the pool is about 17.98 feet and the ANSWER: radius of the pool is about 8.99 feet. 5 in. ANSWER: 7. RIDES The circular ride described at the beginning 17.98 ft; 8.99 ft of the lesson has a diameter of 44 feet. What are the radius and circumference of the ride? Round to the 9. SHORT RESPONSE The right triangle shown is nearest hundredth, if necessary. inscribed in . Find the exact circumference of SOLUTION: . The radius is half the diameter. So, the radius of the circular ride is 22 feet.

SOLUTION: about Therefore, the circumference of the ride is The diameter of the circle is the hypotenuse of the 138.23 feet. right triangle. ANSWER: 22 ft; 138.23 ft

CCSS MODELING 8. The circumference of the circular swimming pool shown is about 56.5 feet. The diameter of the circle is 4 centimeters. What are the diameter and radius of the pool? Round to the nearest hundredth.

The circumference of the circle is 4π centimeters.

ANSWER:

SOLUTION: For Exercises 10–13, refer to .

10. Name the center of the circle. SOLUTION: R The diameter of the pool is about 17.98 feet and the ANSWER: radius of the pool is about 8.99 feet. R ANSWER: 17.98 ft; 8.99 ft 11. Identify a chord that is also a diameter.

9. SHORT RESPONSE The right triangle shown is SOLUTION: inscribed in . Find the exact circumference of Two chords are shown: and . goes . through the center, R, so is a diameter.

ANSWER:

12. Is a radius? Explain. SOLUTION: SOLUTION: The diameter of the circle is the hypotenuse of the A radius is a segment with endpoints at the center right triangle. and on the circle. But has both the end points on the circle, so it is a chord.

ANSWER: No; it is a chord. The diameter of the circle is 4 centimeters. 13. If SU = 16.2 centimeters, what is RT? SOLUTION: Here, is a diameter and is a radius.

The circumference of the circle is 4π centimeters.

ANSWER: Therefore, RT = 8.1 cm. ANSWER: For Exercises 10–13, refer to . 8.1 cm

For Exercises 14–17, refer to .

10. Name the center of the circle. SOLUTION: R 14. Identify a chord that is not a diameter. SOLUTION: ANSWER: R The chords do not pass through the center. So, they are not diameters.

11. Identify a chord that is also a diameter. ANSWER: SOLUTION: Two chords are shown: and . goes 15. If CF = 14 inches, what is the diameter of the circle? through the center, R, so is a diameter. SOLUTION: ANSWER: Here, is a radius and the diameter is twice the radius. .

12. Is a radius? Explain.

SOLUTION: Therefore, the diameter of the circle is 28 inches. A radius is a segment with endpoints at the center ANSWER: and on the circle. But has both the end points on 28 in. the circle, so it is a chord.

ANSWER: 16. Is ? Explain. No; it is a chord. SOLUTION: 13. If SU = 16.2 centimeters, what is RT? All radii of a circle are congruent. Since are both radii of , . SOLUTION: Here, is a diameter and is a radius. ANSWER: Yes; they are both radii of .

17. If DA = 7.4 centimeters, what is EF? SOLUTION:

Therefore, RT = 8.1 cm. Here, is a diameter and is a radius.

ANSWER: 8.1 cm

For Exercises 14–17, refer to . Therefore, EF = 3.7 cm.

ANSWER: 3.7 cm

Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each 14. Identify a chord that is not a diameter. measure. SOLUTION: The chords do not pass through the center. So, they are not diameters.

ANSWER:

18. CK 15. If CF = 14 inches, what is the diameter of the circle? SOLUTION: SOLUTION: Since the radius of K is 8 units, BK = 8. Here, is a radius and the diameter is twice the radius. . Therefore, CK = 2.6 units.

ANSWER: Therefore, the diameter of the circle is 28 inches. 2.6 ANSWER: 28 in. 19. AB SOLUTION:

16. Is ? Explain. Since is a diameter of circle J and the radius is SOLUTION: 10, AC = 2(10) or 20 units. All radii of a circle are congruent. Since are both radii of , .

ANSWER: Therefore, AB = 14.6 units. Yes; they are both radii of . ANSWER: 14.6 17. If DA = 7.4 centimeters, what is EF? SOLUTION: 20. JK Here, is a diameter and is a radius. SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8.

Therefore, EF = 3.7 cm. Since circle J has a radius of 10 units, JC = 10.

ANSWER: 3.7 cm Therefore, JK = 12.6 units. Circle J has a radius of 10 units, has a radius of 8 units, and BC = 5.4 units. Find each ANSWER: measure. 12.6 21. AD SOLUTION: We can find AD using AD = AC + CK + KD. The radius of circle K is 8 units and the radius of circle J is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. 18. CK SOLUTION: Since the radius of K is 8 units, BK = 8.

Therefore, CK = 2.6 units. Therefore, AD = 30.6 units.

ANSWER: ANSWER: 2.6 30.6 22. PIZZA Find the radius and circumference of the 19. AB pizza shown. Round to the nearest hundredth, if SOLUTION: necessary. Since is a diameter of circle J and the radius is 10, AC = 2(10) or 20 units.

Therefore, AB = 14.6 units.

ANSWER: 14.6 SOLUTION: 20. JK The diameter of the pizza is 16 inches. SOLUTION: First find CK. Since circle K has a radius of 8 units, BK = 8. . So, the radius of the pizza is 8 inches.

Since circle J has a radius of 10 units, JC = 10. The circumference C of a circle of diameter d is given by

Therefore, JK = 12.6 units. Therefore, the circumference of the pizza is about ANSWER: 50.27 inches. 12.6 ANSWER: 21. AD 8 in.; 50.27 in. SOLUTION: 23. BICYCLES A bicycle has tires with a diameter of We can find AD using AD = AC + CK + KD. The 26 inches. Find the radius and circumference of a radius of circle K is 8 units and the radius of circle J tire. Round to the nearest hundredth, if necessary. is 10 units, so KD = 8 and AC = 2(10) or 20. Before finding AD, we need to find CK. SOLUTION: The diameter of a tire is 26 inches. The radius is half the diameter.

Therefore, AD = 30.6 units. So, the radius of the tires is 13 inches. ANSWER: The circumference C of a circle with diameter d is 30.6 given by 22. PIZZA Find the radius and circumference of the pizza shown. Round to the nearest hundredth, if necessary. Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: 13 in.; 81.68 in. Find the diameter and radius of a circle by with the given circumference. Round to the nearest hundredth. SOLUTION: 24. C = 18 in. The diameter of the pizza is 16 inches. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 18 in. Use the formula to find the diameter. Then find the radius. . So, the radius of the pizza is 8 inches. The circumference C of a circle of diameter d is given by

Therefore, the circumference of the pizza is about 50.27 inches.

ANSWER: Therefore, the diameter is about 5.73 inches and the 8 in.; 50.27 in. radius is about 2.86 inches.

23. BICYCLES A bicycle has tires with a diameter of ANSWER: 26 inches. Find the radius and circumference of a 5.73 in.; 2.86 in. tire. Round to the nearest hundredth, if necessary.

SOLUTION: 25. C = 124 ft The diameter of a tire is 26 inches. The radius is half the diameter. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 124 ft. Use the formula to find the diameter. Then find the radius.

So, the radius of the tires is 13 inches. The circumference C of a circle with diameter d is given by

Therefore, the circumference of the bicycle tire is about 81.68 inches.

ANSWER: Therefore, the diameter is about 39.47 feet and the 13 in.; 81.68 in. radius is about 19.74 feet.

Find the diameter and radius of a circle by with ANSWER: the given circumference. Round to the nearest 39.47 ft; 19.74 ft hundredth. 24. C = 18 in. 26. C = 375.3 cm SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is given by given by Here, C = 18 in. Use the formula to find the Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius. diameter. Then find the radius.

Therefore, the diameter is about 5.73 inches and the Therefore, the diameter is about 119.46 centimeters radius is about 2.86 inches. and the radius is about 59.73 centimeters.

ANSWER: ANSWER: 5.73 in.; 2.86 in. 119.46 cm; 59.73 cm

25. C = 124 ft 27. C = 2608.25 m SOLUTION: SOLUTION: The circumference C of a circle with diameter d is The circumference C of a circle with diameter d is given by given by Here, C = 124 ft. Use the formula to find the Here, C = 2608.25 meters. Use the formula to find diameter. Then find the radius. the diameter. Then find the radius.

Therefore, the diameter is about 830.23 meters and Therefore, the diameter is about 39.47 feet and the the radius is about 415.12 meters. radius is about 19.74 feet. ANSWER: ANSWER: 830.23 m; 415.12 m 39.47 ft; 19.74 ft CCSS SENSE-MAKING Find the exact 26. C = 375.3 cm circumference of each circle by using the given inscribed or SOLUTION: circumscribed polygon. The circumference C of a circle with diameter d is given by Here, C = 375.3 cm. Use the formula to find the diameter. Then find the radius.

28. SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

Therefore, the diameter is about 119.46 centimeters and the radius is about 59.73 centimeters. The diameter of the circle is 17 cm. The ANSWER: circumference C of a circle is given by 119.46 cm; 59.73 cm

27. C = 2608.25 m SOLUTION: Therefore, the circumference of the circle is 17 The circumference C of a circle with diameter d is centimeters. given by ANSWER: Here, C = 2608.25 meters. Use the formula to find the diameter. Then find the radius.

29. SOLUTION: The hypotenuse of the right triangle is a diameter of Therefore, the diameter is about 830.23 meters and the circle. Use the Pythagorean Theorem to find the the radius is about 415.12 meters. diameter. ANSWER: 830.23 m; 415.12 m CCSS SENSE-MAKING Find the exact The diameter of the circle is 12 feet. The circumference of each circle by using the given inscribed or circumference C of a circle is given by circumscribed polygon.

Therefore, the circumference of the circle is 12 feet.

ANSWER: 28.

SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

30. SOLUTION:

The diameter of the circle is 17 cm. The Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. circumference C of a circle is given by Use the Pythagorean Theorem to find the diameter of the circle.

Therefore, the circumference of the circle is 17 centimeters.

ANSWER: The diameter of the circle is inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is inches.

29. ANSWER: SOLUTION: The hypotenuse of the right triangle is a diameter of the circle. Use the Pythagorean Theorem to find the diameter.

31. The diameter of the circle is 12 feet. The SOLUTION: circumference C of a circle is given by Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle. Therefore, the circumference of the circle is 12 feet.

ANSWER:

The diameter of the circle is 10 inches. The circumference C of a circle is given by

30. Therefore, the circumference of the circle is 10 inches. SOLUTION: Each diagonal of the inscribed rectangle will pass ANSWER: through the origin, so it is a diameter of the circle. Use the Pythagorean Theorem to find the diameter of the circle.

32. The diameter of the circle is inches. The SOLUTION: circumference C of a circle is given by A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 25 millimeters. The circumference C of a circle is given by

Therefore, the circumference of the circle is inches. Therefore, the circumference of the circle is 25 ANSWER: millimeters.

ANSWER:

31. SOLUTION: Each diagonal of the inscribed rectangle will pass through the origin, so it is a diameter of the circle. 33. Use the Pythagorean Theorem to find the diameter of the circle. SOLUTION: A diameter perpendicular to a side of the square and the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

The diameter of the circle is 10 inches. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 yards.

Therefore, the circumference of the circle is 10 ANSWER: inches. ANSWER: 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in centimeters. What is the maximum allowable weight for a disc with circumference 66.92 centimeters? Round to the nearest tenth. 32. SOLUTION: SOLUTION: The circumference C of a circle with diameter d is

A diameter perpendicular to a side of the square and given by the length of each side will measure the same. So, d Here, C = 66.92 cm. Use the formula to find the

= 25 millimeters. diameter. The circumference C of a circle is given by

Therefore, the circumference of the circle is 25 The diameter of the disc is about 21.3 centimeters millimeters. and the maximum weight allowed is 8.3 times the diameter in centimeters. ANSWER: Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams.

ANSWER: 176.8 g

35. PATIOS Mr. Martinez is going to build the patio shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the 33. inner circle has a circumference of approximately 25 SOLUTION: feet, what should the radius of the circle be to the A diameter perpendicular to a side of the square and nearest foot? the length of each side will measure the same. So, d = 14 yards. The circumference C of a circle is given by

Therefore, the circumference of the circle is 14 SOLUTION: yards. a. The radius of the patio is 3 + 2 = 5 ft. The ANSWER: circumference C of a circle with radius r is given by

Use the formula to find the circumference. 34. DISC GOLF Disc golf is similar to regular golf, except that a flying disc is used instead of a ball and clubs. For professional competitions, the maximum weight of a disc in grams is 8.3 times its diameter in Therefore, the circumference is about 31.42 feet. centimeters. What is the maximum allowable weight b. Here, C = 25 ft. Use the formula to find the for a disc with circumference 66.92 centimeters? inner Round to the nearest tenth. radius of the inner circle. SOLUTION: The circumference C of a circle with diameter d is given by Here, C = 66.92 cm. Use the formula to find the diameter. So, the radius of the inner circle is about 4 feet.

ANSWER: a. 31.42 ft b. 4 ft

The diameter of the disc is about 21.3 centimeters The radius, diameter, or circumference of a and the maximum weight allowed is 8.3 times the circle is given. Find each missing measure to diameter in centimeters. the nearest hundredth. Therefore, the maximum weight is 8.3 × 21.3 or about 176.8 grams. 36.

ANSWER: SOLUTION: 176.8 g The radius is half the diameter and the circumference C of a circle with diameter d is given 35. PATIOS Mr. Martinez is going to build the patio by shown. a. What is the patio’s approximate circumference? b. If Mr. Martinez changes the plans so that the inner circle has a circumference of approximately 25 feet, what should the radius of the circle be to the nearest foot?

Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches.

ANSWER: 4.25 in.; 26.70 in. SOLUTION: a. The radius of the patio is 3 + 2 = 5 ft. The

circumference C of a circle with radius r is given by 37.

Use the formula to find the circumference. SOLUTION: The diameter is twice the radius and the circumference C of a circle with diameter d is given by Therefore, the circumference is about 31.42 feet.

b. Here, Cinner = 25 ft. Use the formula to find the radius of the inner circle.

So, the diameter is 22.80 feet and the circumference So, the radius of the inner circle is about 4 feet. is about 71.63 feet.

ANSWER: ANSWER: a. 31.42 ft 22.80 ft; 71.63 ft b. 4 ft 38. The radius, diameter, or circumference of a circle is given. Find each missing measure to SOLUTION: the nearest hundredth. The circumference C of a circle with diameter d is given by 36. Here, C = 35x cm. Use the formula to find the diameter.Then find the radius. SOLUTION: The radius is half the diameter and the circumference C of a circle with diameter d is given by

Therefore, the diameter is about 11.14x centimeters and the radius is about 5.57x centimeters. Therefore, the radius is 4.25 inches and the circumference is about 26.70 inches. ANSWER: 11.14x cm; 5.57x cm ANSWER: 4.25 in.; 26.70 in. 39.

SOLUTION: 37. The diameter is twice the radius and the circumference C of a circle with diameter d is given SOLUTION: by The diameter is twice the radius and the circumference C of a circle with diameter d is given by

So, the diameter is 0.25x units and the circumference is about 0.79x units. So, the diameter is 22.80 feet and the circumference is about 71.63 feet. ANSWER: 0.25x; 0.79x ANSWER: 22.80 ft; 71.63 ft Determine whether the circles in the figures below appear to be congruent, concentric, or 38. neither. 40. Refer to the photo on page 703. SOLUTION: The circumference C of a circle with diameter d is SOLUTION: given by The circles have the same center and they are in the Here, C = 35x cm. Use the formula to find the same plane. So, they are concentric circles. diameter.Then find the radius. ANSWER: concentric

41. Refer to the photo on page 703. SOLUTION: The circles neither have the same radius nor are they concentric.

ANSWER: neither Therefore, the diameter is about 11.14x centimeters Refer to the photo on page 703. and the radius is about 5.57x centimeters. 42. SOLUTION: ANSWER: The circles appear to have the same radius. So, they 11.14x cm; 5.57x cm appear to be congruent.

ANSWER: 39. congruent SOLUTION: 43. HISTORY The Indian Shell Ring on Hilton Head The diameter is twice the radius and the Island approximates a circle. If each unit on the circumference C of a circle with diameter d is given coordinate grid represents 25 feet, how far would by someone have to walk to go completely around the ring? Round to the nearest tenth.

So, the diameter is 0.25x units and the circumference is about 0.79x units.

ANSWER: 0.25x; 0.79x SOLUTION: Determine whether the circles in the figures below appear to be congruent, concentric, or The radius of the circle is 3 units, that is 75 feet. So, the diameter is 150 ft. The circumference C of a neither. circle with diameter d is given by 40. Refer to the photo on page 703. So, the circumference is SOLUTION: Therefore, someone have to The circles have the same center and they are in the walk about 471.2 feet to go completely around the same plane. So, they are concentric circles. ring.

ANSWER: ANSWER: concentric 471.2 ft

41. Refer to the photo on page 703. 44. CCSS MODELING A brick path is being installed around a circular pond. The pond has a SOLUTION: circumference of 68 feet. The outer edge of the path The circles neither have the same radius nor are they is going to be 4 feet from the pond all the way concentric. around. What is the approximate circumference of the path? Round to the nearest hundredth. ANSWER: SOLUTION: neither The circumference C of a circle with radius r is 42. Refer to the photo on page 703. given by Here, Cpond = 68 ft. Use the

SOLUTION: formula to find the radius of the pond. The circles appear to have the same radius. So, they appear to be congruent.

ANSWER: congruent So, the radius of the pond is about 10.82 feet and the 43. HISTORY The Indian Shell Ring on Hilton Head radius to the outer edge of the path will be 10.82 + 4 Island approximates a circle. If each unit on the = 14.82 ft. Use the radius to find the circumference coordinate grid represents 25 feet, how far would of the brick path. someone have to walk to go completely around the ring? Round to the nearest tenth.

Therefore, the circumference of the path is about 93.13 feet.

ANSWER: 93.13 ft

45. MULTIPLE REPRESENTATIONS In this problem, you will explore changing dimensions in circles. a. GEOMETRIC Use a compass to draw three

circles in which the scale factor from each circle to SOLUTION: the next larger circle is 1 : 2. The radius of the circle is 3 units, that is 75 feet. So, b. TABULAR Calculate the radius (to the nearest the diameter is 150 ft. The circumference C of a tenth) and circumference (to the nearest hundredth) circle with diameter d is given by of each circle. Record your results in a table. So, the circumference is c. VERBAL Explain why these three circles are Therefore, someone have to geometrically similar. d. VERBAL walk about 471.2 feet to go completely around the Make a conjecture about the ratio ring. between the circumferences of two circles when the ratio between their radii is 2. ANSWER: e. ANALYTICAL The scale factor from to 471.2 ft is . Write an equation relating the 44. CCSS MODELING A brick path is being installed circumference (C ) of to the circumference around a circular pond. The pond has a A circumference of 68 feet. The outer edge of the path (CB) of . is going to be 4 feet from the pond all the way f. NUMERICAL If the scale factor from to around. What is the approximate circumference of the path? Round to the nearest hundredth. is , and the circumference of is 12 SOLUTION: inches, what is the circumference of ? The circumference C of a circle with radius r is SOLUTION: given by Here, C = 68 ft. Use the pond a. Sample answer: formula to find the radius of the pond.

So, the radius of the pond is about 10.82 feet and the b. Sample answer. radius to the outer edge of the path will be 10.82 + 4 = 14.82 ft. Use the radius to find the circumference of the brick path.

Therefore, the circumference of the path is about 93.13 feet. c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. ANSWER: e. The circumference is directly proportional to radii. 93.13 ft So, if the radii are in the ratio b : a, so will the

45. MULTIPLE REPRESENTATIONS In this circumference. Therefore, . problem, you will explore changing dimensions in f. e. circles. Use the formula from part Here, the

a. GEOMETRIC Use a compass to draw three circumference of is 12 inches and . circles in which the scale factor from each circle to the next larger circle is 1 : 2. b. TABULAR Calculate the radius (to the nearest tenth) and circumference (to the nearest hundredth)

of each circle. Record your results in a table. Therefore, the circumference of is 4 inches. c. VERBAL Explain why these three circles are geometrically similar. ANSWER: d. VERBAL Make a conjecture about the ratio a. Sample answer: between the circumferences of two circles when the ratio between their radii is 2. e. ANALYTICAL The scale factor from to is . Write an equation relating the circumference (C ) of to the circumference A b. (CB) of . Sample answer. f. NUMERICAL If the scale factor from to is , and the circumference of is 12 inches, what is the circumference of ? SOLUTION:

a. Sample answer: c. They all have the same shape—circular. d. The ratio of their circumferences is also 2.

e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a b. Sample answer. needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

c. They all have the same shape–circular. d. The ratio of their circumferences is also 2. e. The circumference is directly proportional to radii. So, if the radii are in the ratio b : a, so will the a. Drop the needle onto the paper. When the needle circumference. Therefore, . lands, record whether it touches one of the lines as a hit. Record the number of hits after 25, 50, and 100 f. Use the formula from part e. Here, the drops. circumference of is 12 inches and .

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 Therefore, the circumference of is 4 inches. drops.

ANSWER: c. How are the values you found in part b related to a. Sample answer: π? SOLUTION: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

b. Sample answer. ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π.

c. They all have the same shape—circular. 47. M APS The concentric circles on the map below d. The ratio of their circumferences is also 2. show the areas that are 5, 10, 15, 20, 25, and 30 miles from downtown Phoenix. e. f. 4 in.

46. BUFFON’S NEEDLE Measure the length of a needle (or toothpick) in centimeters. Next, draw a set of horizontal lines that are centimeters apart on a sheet of plain white paper.

a. How much greater is the circumference of the outermost circle than the circumference of the center circle? b. As the radii of the circles increases by 5 miles, by how much does the circumference increase?

a. Drop the needle onto the paper. When the needle SOLUTION: lands, record whether it touches one of the lines as a a. The radius of the outermost circle is 30 miles and hit. Record the number of hits after 25, 50, and 100 that of the center circle is 5 miles. Find the drops. circumference of each circle.

b. Calculate the ratio of two times the total number of drops to the number of hits after 25, 50, and 100 drops.

c. b How are the values you found in part related to The difference of the circumferences of the π? outermost circle and the center circle is 60π - 10π = SOLUTION: 50π or about 157.1 miles. a. See students’ work. b. Let the radius of one circle be x miles and the b. See students’ work. radius of the next larger circle be x + 5 miles. Find c. Sample answer: The values are approaching 3.14, the circumference of each circle. which is approximately equal to π.

ANSWER: a. See students’ work. b. See students’ work. c. Sample answer: The values are approaching 3.14, which is approximately equal to π. The difference of the two circumferences is (2xπ + 10π) - (2xπ) = 10π or about 31.4 miles. Therefore, as the radius of each circle increases by 5 47. M APS The concentric circles on the map below miles, the circumference increases by about 31.4 show the areas that are 5, 10, 15, 20, 25, and 30 miles. miles from downtown Phoenix. ANSWER: a. 157.1 mi b. by about 31.4 mi

48. WRITING IN MATH How can we describe the relationships that exist between circles and lines? SOLUTION: Sample answer: A line and a circle may intersect in a. How much greater is the circumference of the one point, two points, or may not intersect at all. A outermost circle than the circumference of the center line that intersects a circle in one point can be circle? described as a tangent. A line that intersects a circle b. As the radii of the circles increases by 5 miles, by in exactly two points can be described as a secant. how much does the circumference increase? A line segment with endpoints on a circle can be SOLUTION: described as a chord. If the chord passes through a. The radius of the outermost circle is 30 miles and the center of the circle, it can be described as a that of the center circle is 5 miles. Find the diameter. A line segment with endpoints at the circumference of each circle. center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with The difference of the circumferences of the endpoints on a circle can be described as a chord. outermost circle and the center circle is 60π - 10π = If the chord passes through the center of the circle, 50π or about 157.1 miles. it can be described as a diameter. A line segment b. Let the radius of one circle be x miles and the with endpoints at the center and on the circle can be radius of the next larger circle be x + 5 miles. Find described as a radius. the circumference of each circle.

49. REASONING In the figure, a circle with radius r is inscribed in a regular polygon and circumscribed

The difference of the two circumferences is (2xπ + about another. 10π) - (2xπ) = 10π or about 31.4 miles. a. What are the perimeters of the circumscribed and inscribed polygons in terms of r? Explain. Therefore, as the radius of each circle increases by 5 b. miles, the circumference increases by about 31.4 Is the circumference C of the circle greater or miles. less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality ANSWER: comparing C to these perimeters. c. b a. 157.1 mi Rewrite the inequality from part in terms of the diameter d of the circle and interpret its meaning. b. by about 31.4 mi d. As the number of sides of both the circumscribed 48. WRITING IN MATH How can we describe the and inscribed polygons increase, what will happen to relationships that exist between the upper and lower limits of the inequality from part c, and what does this imply? circles and lines? SOLUTION: SOLUTION: a. The circumscribed polygon is a square with sides Sample answer: A line and a circle may intersect in of length 2r. one point, two points, or may not intersect at all. A line that intersects a circle in one point can be described as a tangent. A line that intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be The inscribed polygon is a hexagon with sides of described as a chord. If the chord passes through length r. (Each triangle formed using a central angle the center of the circle, it can be described as a will be equilateral.) diameter. A line segment with endpoints at the center and on the circle can be described as a radius.

ANSWER: Sample answer: A line that intersects a circle in one point can be described as a tangent. A line that

intersects a circle in exactly two points can be described as a secant. A line segment with endpoints on a circle can be described as a chord. So, the perimeters are 8r and 6r. If the chord passes through the center of the circle, b. it can be described as a diameter. A line segment with endpoints at the center and on the circle can be described as a radius. The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) 49. or 4d and 6r = 3(2r) or 3d. REASONING In the figure, a circle with radius r is The inequality is then 4d < C < 3d. Therefore, the inscribed in a regular polygon and circumscribed circumference of the circle is between 3 and 4 times about another. its diameter. a. What are the perimeters of the circumscribed and d. These limits will approach a value of πd, implying inscribed polygons in terms of r? Explain. that C = πd. b. Is the circumference C of the circle greater or less than the perimeter of the circumscribed polygon? the inscribed polygon? Write a compound inequality ANSWER: comparing C to these perimeters. a. 8r and 6r ; Twice the radius of the circle, 2r is c. Rewrite the inequality from part b in terms of the the side length of the square, so the perimeter of the diameter d of the circle and interpret its meaning. square is 4(2r) or 8r. The d. As the number of sides of both the circumscribed regular hexagon is made up of six equilateral and inscribed polygons increase, what will happen to the upper and lower limits of the inequality from part triangles with side length r, so the perimeter of the c, and what does this imply? hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r SOLUTION: c. 3d < C < 4d; The circumference of the circle is a. The circumscribed polygon is a square with sides between 3 and 4 times its diameter. of length 2r. d. These limits will approach a value of πd, implying that C = πd.

CHALLENGE 50. The sum of the circumferences of The inscribed polygon is a hexagon with sides of circles H, J, and K shown at the right is units. length r. (Each triangle formed using a central angle Find KJ. will be equilateral.)

SOLUTION: The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the So, the perimeters are 8r and 6r. circumference of each circle. b.

The circumference is less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. 8r < C < 6r c. Since the diameter is twice the radius, 8r = 4(2r) or 4d and 6r = 3(2r) or 3d. The inequality is then 4d < C < 3d. Therefore, the circumference of the circle is between 3 and 4 times The sum of the circumferences is 56π units. its diameter. Substitute the three circumferences and solve for x. d. These limits will approach a value of πd, implying that C = πd.

ANSWER: So, the radius of is 4 units, the radius of is a. 8r and 6r ; Twice the radius of the circle, 2r is 2(4) or 8 units, and the radius of is 4(4) or 16 the side length of the square, so the perimeter of the units. square is 4(2r) or 8r. The The measure of KJ is equal to the sum of the radii regular hexagon is made up of six equilateral of and . triangles with side length r, so the perimeter of the Therefore, KJ = 16 + 8 or 24 units. hexagon is 6(r) or 6r. b. less; greater; 6r < C < 8r ANSWER: c. 3d < C < 4d; The circumference of the circle is 24 units between 3 and 4 times its diameter. 51. REASONING Is the distance from the center of a d. These limits will approach a value of πd, implying circle to a point in the interior of a circle sometimes, that C = πd. always, or never less than the radius of the circle? Explain. 50. CHALLENGE The sum of the circumferences of circles H, J, and K shown at the right is units. SOLUTION: Find KJ. A radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius of the circle.

ANSWER: Always; a radius is a segment drawn between the center of the circle and a point on the circle. A segment drawn from the center to a point inside the circle will always have a length less than the radius SOLUTION: of the circle. The radius of is 4x units, the radius of is x units, and the radius of is 2x units. Find the 52. PROOF Use the locus definition of a circle and circumference of each circle. dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = The sum of the circumferences is 56π units. kr1. Substitute the three circumferences and solve for x.

So, the radius of is 4 units, the radius of is 2(4) or 8 units, and the radius of is 4(4) or 16 Therefore, is mapped onto . Since there units. exists a rigid motion followed by a scaling that maps The measure of KJ is equal to the sum of the radii onto , the circles are of and . similar. Thus, all circles are similar. Therefore, KJ = 16 + 8 or 24 units.

ANSWER: 24 units ANSWER: A circle is a locus of points in a plane equidistant 51. REASONING Is the distance from the center of a from a given point. For any two circles and circle to a point in the interior of a circle sometimes, , there exists a translation that maps center A always, or never less than the radius of the circle? onto center B, moving so that it is concentric

Explain. with . There also exists a dilation with scale SOLUTION: factor k such that each point that makes up is A radius is a segment drawn between the center of moved to be the same distance from center A as the the circle and a point on the circle. A segment drawn points that make up are from center B. from the center to a point inside the circle will always Therefore, is mapped onto . Since there have a length less than the radius of the circle. exists a rigid motion followed by a scaling that maps ANSWER: onto , the circles are similar. Thus, all Always; a radius is a segment drawn between the circles are similar. center of the circle and a point on the circle. A CHALLENGE segment drawn from the center to a point inside the 53. In the figure, is inscribed in circle will always have a length less than the radius equilateral triangle LMN. What is the circumference of the circle. of ?

52. PROOF Use the locus definition of a circle and dilations to prove that all circles are similar. SOLUTION: A circle is a locus of points in a plane equidistant from a given point. For any two circles and , there exists a translation that maps center A onto center B, moving so that it SOLUTION: is concentric with . There also exists a dilation Draw a perpendicular to the side Also join with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. For some value k, r2 = kr1.

Since is equilateral triangle, MN = 8 inches and RN = 4 inches. Also, bisects Use the definition of tangent of an angle to find PR, which is the radius of Therefore, is mapped onto . Since there exists a rigid motion followed by a scaling that maps

onto , the circles are similar. Thus, all circles are similar.

ANSWER: A circle is a locus of points in a plane equidistant Therefore, the radius of is from a given point. For any two circles and The circumference C of a circle with radius r is , there exists a translation that maps center A given by onto center B, moving so that it is concentric with . There also exists a dilation with scale factor k such that each point that makes up is moved to be the same distance from center A as the points that make up are from center B. Therefore, is mapped onto . Since there

exists a rigid motion followed by a scaling that maps The circumference of is inches. onto , the circles are similar. Thus, all circles are similar. ANSWER:

53. CHALLENGE In the figure, is inscribed in in. equilateral triangle LMN. What is the circumference of ? 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work.

ANSWER: See students’ work. SOLUTION: Draw a perpendicular to the side Also join 55. GRIDDED RESPONSE What is the circumference of ? Round to the nearest tenth.

SOLUTION: Since is equilateral triangle, MN = 8 inches Use the Pythagorean Theorem to find the diameter and RN = 4 inches. Also, bisects of the circle. Use the definition of tangent of an angle to find PR, which is the radius of The circumference C of a circle with diameter d is

given by Use the formula to find the circumference.

Therefore, the radius of is The circumference C of a circle with radius r is given by ANSWER: 40.8 56. What is the radius of a table with a circumference of 10 feet? A 1.6 ft

B 2.5 ft The circumference of is inches. C 3.2 ft D 5 ft ANSWER: SOLUTION: in. The circumference C of a circle with radius r is given by Here, C = 10 ft. Use the formula to find the radius. 54. WRITING IN MATH Research and write about the history of pi and its importance to the study of geometry. SOLUTION: See students’ work. The radius of the circle is about 1.6 feet, so the ANSWER: correct choice is A. See students’ work. ANSWER: 55. GRIDDED RESPONSE What is the A circumference of ? Round to the nearest tenth. 57. ALGEBRA Bill is planning a circular vegetable garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the garden? F 10 G 9 SOLUTION: H 8 Use the Pythagorean Theorem to find the diameter J 7 of the circle. SOLUTION: The circumference C of a circle with radius r is The circumference C of a circle with diameter d is given by given by Because 50 feet of fence can be used to border the

Use the formula to find the circumference. garden, Cmax = 50. Use the circumference formula to find the maximum radius.

Therefore, the radius of the garden must be less than ANSWER: or equal to about 7.96 feet. Therefore, the only 40.8 correct choice is J. 56. What is the radius of a table with a circumference of ANSWER: 10 feet? J A 1.6 ft B 2.5 ft 58. SAT/ACT What is the radius of a circle with an C 3.2 ft area of square units? D 5 ft A 0.4 units SOLUTION: B 0.5 units The circumference C of a circle with radius r is C 2 units given by D 4 units Here, C = 10 ft. Use the formula to find the radius. E 16 units SOLUTION: The area of a circle of radius r is given by the formula Here A = square units. Use the formula to find the radius. The radius of the circle is about 1.6 feet, so the correct choice is A.

ANSWER: A

57. ALGEBRA Bill is planning a circular vegetable Therefore, the correct choice is B. garden with a fence around the border. If he can use up to 50 feet of fence, what radius can he use for the ANSWER: garden? B F 10 Copy each figure and point B. Then use a ruler G 9 to draw the image of the figure under a dilation H 8 with center B and the scale factor r indicated. J 7 59. SOLUTION: The circumference C of a circle with radius r is given by Because 50 feet of fence can be used to border the garden, Cmax = 50. Use the circumference formula to find the maximum radius. SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays Therefore, the radius of the garden must be less than through the other five vertices. Connect the six points or equal to about 7.96 feet. Therefore, the only on the rays to construct the dilated image of the correct choice is J. hexagon.

ANSWER: J

58. SAT/ACT What is the radius of a circle with an

area of square units? A 0.4 units B 0.5 units C 2 units D 4 units E 16 units ANSWER: SOLUTION: The area of a circle of radius r is given by the

formula Here A = square units. Use the

formula to find the radius. 60.

Therefore, the correct choice is B. SOLUTION: ANSWER: Draw a ray from point B through a vertex of the B triangle. Use the ruler to measure the distance from Copy each figure and point B. Then use a ruler B to the vertex. Mark a point on the ray that is of to draw the image of the figure under a dilation that distance from B. Repeat the process for rays with center B and the scale factor r indicated. through the other two vertices. Connect the three points on the rays to construct the dilated image of 59. the triangle.

ANSWER: SOLUTION: Draw a ray from point B through a vertex of the hexagon. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of 61. that distance from B. Repeat the process for rays through the other five vertices. Connect the six points on the rays to construct the dilated image of the hexagon. SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

ANSWER:

ANSWER: 60.

SOLUTION: Draw a ray from point B through a vertex of the 62. triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is of that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of SOLUTION: the triangle. Draw a ray from point B through a vertex of the triangle. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 3 times that distance from B. Repeat the process for rays through the other two vertices. Connect the three points on the rays to construct the dilated image of the triangle. ANSWER:

61.

SOLUTION: Draw a ray from point B through a vertex of the trapezoid. Use the ruler to measure the distance from B to the vertex. Mark a point on the ray that is 2 ANSWER: times that distance from B. Repeat the process for rays through the other three vertices. Connect the four points on the rays to construct the dilated image of the trapezoid.

State whether each figure has rotational ANSWER: symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 63. Refer to the image on page 705. SOLUTION: The figure cannot be mapped onto itself by any

rotation. So, it does not have a rotational symmetry. 62. ANSWER: No

64. Refer to the image on page 705. SOLUTION: SOLUTION: Draw a ray from point B through a vertex of the The figure can be mapped onto itself by a rotation of triangle. Use the ruler to measure the distance from 90º. So, it has a rotational symmetry. So, the B to the vertex. Mark a point on the ray that is 3 magnitude of symmetry is 90º and order of symmetry times that distance from B. Repeat the process for is 4. rays through the other two vertices. Connect the three points on the rays to construct the dilated image ANSWER: of the triangle. Yes; 4, 90

65.

SOLUTION: ANSWER: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: No

66. Refer to the image on page 705. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry.

ANSWER: State whether each figure has rotational No symmetry. Write yes or no. If so, copy the Determine the truth value of the following figure, locate the center of symmetry, and state statement for each set of conditions. Explain the order and magnitude of symmetry. your reasoning. 63. Refer to the image on page 705. If you are over 18 years old, then you vote in all SOLUTION: elections. The figure cannot be mapped onto itself by any 67. You are 19 years old and you vote. rotation. So, it does not have a rotational symmetry. SOLUTION: ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true No for the conditions.

64. Refer to the image on page 705. ANSWER: True; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is true, then the statement is true The figure can be mapped onto itself by a rotation of for the conditions. 90º. So, it has a rotational symmetry. So, the magnitude of symmetry is 90º and order of symmetry 68. You are 21 years old and do not vote. is 4. SOLUTION: ANSWER: False; sample answer: Since the hypothesis is true Yes; 4, 90 and the conclusion is false, then the statement is false for the conditions.

ANSWER: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions.

Find x.

65. SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. 69. ANSWER: SOLUTION: No The sum of the measures of angles around a point is 66. Refer to the image on page 705. 360º.

SOLUTION: The given figure cannot be mapped onto itself by any rotation. So, it does not have a rotational symmetry. ANSWER: 10-1ANSWER: Circles and Circumference 90 No Determine the truth value of the following statement for each set of conditions. Explain your reasoning. If you are over 18 years old, then you vote in all elections. 67. You are 19 years old and you vote. 70. SOLUTION: SOLUTION: True; sample answer: Since the hypothesis is true The angles with the measures xº and 2xº form a and the conclusion is true, then the statement is true straight line. So, the sum is 180. for the conditions. ANSWER: True; sample answer: Since the hypothesis is true and the conclusion is true, then the statement is true ANSWER: for the conditions. 60 68. You are 21 years old and do not vote. SOLUTION: False; sample answer: Since the hypothesis is true and the conclusion is false, then the statement is false for the conditions. 71. ANSWER: False; sample answer: Since the hypothesis is true SOLUTION: and the conclusion is false, then the statement is false The angles with the measures 6xº and 3xº form a for the conditions. straight line. So, the sum is 180.

Find x.

ANSWER: 20

69. SOLUTION: The sum of the measures of angles around a point is 360º.

72. SOLUTION: The sum of the three angles is 180. ANSWER: 90

ANSWER: 45 eSolutions70. Manual - Powered by Cognero Page 21 SOLUTION: The angles with the measures xº and 2xº form a straight line. So, the sum is 180.

ANSWER: 60

71. SOLUTION: The angles with the measures 6xº and 3xº form a straight line. So, the sum is 180.

ANSWER: 20

72. SOLUTION: The sum of the three angles is 180.

ANSWER: 45