Find the Length of the Diameter by First Finding the Measure of ST. The

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

3. SOLUTION:

11-7 Special Segments in a Circle

Find x. Assume that segments that appear to be tangent are tangent.

4. SOLUTION: 1. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

2. SOLUTION: SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

3. SOLUTION: Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. SOLUTION: eSolutions Manual - Powered by Cognero Page 1

4. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the 7. original pottery? Round to the nearest hundredth. SOLUTION:

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

8. SOLUTION: The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

9. SOLUTION: 6. SOLUTION:

10. SOLUTION:

7. SOLUTION:

11. SOLUTION:

Solve for x by using the Quadratic Formula. 8. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

9. 12. SOLUTION: SOLUTION:

10. SOLUTION:

13.

SOLUTION:

Solve for x using the Quadratic Formula.

11. SOLUTION:

Since lengths cannot be negative, the value of x is about 7.4. Solve for x by using the Quadratic Formula. 14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

Since lengths cannot be negative, the value of x is about 3.1. SOLUTION:

12. SOLUTION: Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake? 13. SOLUTION:

Solve for x using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is about 7.4. Since one of the perpendicular segments contains 14. BRIDGES What is the diameter of the circle the center of the cake, it will bisect the chord. To containing the arc of the Sydney Harbour Bridge? find the diameter of the cake, first use Theorem Round to the nearest tenth. 11.15 to find x. Refer to the photo on Page 772.

Add the two segments to find the diameter. SOLUTION: The diameter of the circle is 9 + 4 or 13 inches. CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the 16. diameter of the circle, first use Theorem 11.15 to find SOLUTION: the value of x.

Add the two segments to find the diameter. Solve for x using the Quadratic Formula. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

Since lengths cannot be negative, the value of x is 6.

17. SOLUTION: SOLUTION:

Solve for x using the Quadratic Formula.

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To find the diameter of the cake, first use Theorem

11.15 to find x. Since lengths cannot be negative, the value of x is about 7.1.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the 18. nearest tenth. Assume that segments that SOLUTION: appear to be tangent are tangent.

16. SOLUTION: First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

Solve for x using the Quadratic Formula.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of x is 6.

17. 19. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find a.

Solve for x using the Quadratic Formula.

Next, use the intersecting chords to find b.

Since lengths cannot be negative, the value of x is about 7.1.

18. 20. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find q.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent Since lengths cannot be negative, the value of q is 9. segments to the smaller circle. Use the intersecting secant segments to find r.

Solve for r using the Quadratic Formula. Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

19. SOLUTION: Use the intersecting tangent and secant segments to find a. 21. SOLUTION: Use the intersecting secant segments to find c.

Next, use the intersecting chords to find b.

Use the intersecting secant and tangent segments to find d.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is 20. standing next to the tree, as shown. The distance SOLUTION: between Gwendolyn and Chet is 27 feet. Draw a Use the intersecting tangent and secant segments to diagram of this situation, and then find the diameter find q. of the tree.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r. SOLUTION:

Solve for r using the Quadratic Formula.

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to

find d. Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8. So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Given: and intersect at B. 21. Prove: SOLUTION: Use the intersecting secant segments to find c.

Use the intersecting secant and tangent segments to SOLUTION: find d. Proof: Statements (Reasons) 1. and intersect at B. (Given) 2. , (Inscribed that Therefore, the values of the variables are c ≈ 22.8 intercept the same arc are and d ≈ 16.9. 3. (AA Similarity)

22. INDIRECT MEASUREMENT Gwendolyn is 4. (Definition of similar standing 16 feet from a giant sequoia tree and Chet is 5. (Cross products) standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter 24. paragraph proof of Theorem 11.16 of the tree. Given: Secants and Prove:

SOLUTION: SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that intercept the same arc are congruent. So, . By AA Similarity, By the definition of similar triangles, Let d be the diameter of the tree. Since the cross products of a proportion Use the intersecting tangent and secant segments to find d. are equal,

25. two-column proof of Theorem 11.17 Given: tangent secant Prove: So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Given: and intersect at B.

Prove: SOLUTION: Proof: Statements (Reasons) 1. tangent and secant (Given)

2. (The measure of an inscribed SOLUTION: Proof: ∠ equals half the measure of its intercept arc.) Statements (Reasons) 3. (The measure of an ∠ formed 1. and intersect at B. (Given) 2. , (Inscribed that by a secant and a tangent = half the measure of its intercept the same arc are intercepted arc.)

3. (AA Similarity) 4. (Substitution) 5. (Definition of 4. (Definition of similar 6. (Reflexive Property) 5. (Cross products) 7. (AA Similarity)

8. (Definition of 24. paragraph proof of Theorem 11.16 9. (Cross products) Given: Secants and Prove: 26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

SOLUTION: Proof:

Prove: SOLUTION: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant SOLUTION: and a tangent intersect at an exterior point, the product of the measures of the secant segment and Proof: its external segment is equal to the square of the Statements (Reasons) measure of the tangent segment, because for the

1. tangent and secant (Given) tangent the measures of the external segment and the whole segment are the same. 2. (The measure of an inscribed ∠ equals half the measure of its intercept arc.) 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the 3. (The measure of an ∠ formed steps that you used. by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property) 7. (AA Similarity) SOLUTION:

8. (Definition of Use the intersecting tangent and secant segments to find the value of a. 9. (Cross products)

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

29. REASONING When two chords intersect at the SOLUTION: center of a circle, are the measures of the Jun; the segments intersect outside of the circle, so intercepting arcs sometimes, always, or never equal the correct equation involves the products of the to each other? measures of a secant and its external secant SOLUTION: segment. Since the chords intersect at the center, the measure 27. WRITING IN MATH Compare and contrast the of each intercepted arc is equal to the measure of its methods for finding measures of segments when two related central angle. Since vertical angles are secants intersect in the exterior of a circle and when congruent, the opposite arcs will be congruent. All a secant and a tangent intersect in the exterior of a four arcs will only be congruent when all four central circle. angles are congruent. This will only happen when the chords are perpendicular and the angles are right SOLUTION: angles Sample answer: When two secants intersect in the Therefore, the measures of the intercepted arcs are exterior of a circle, the product of the measures of sometimes equal to each other. one secant segment and its external segment is equal to the product of the measures of the other secant 30. OPEN ENDED Investigate Theorem 11.17 by segment and its external segment. When a secant drawing and labeling a circle that has a secant and a and a tangent intersect at an exterior point, the tangent intersecting outside the circle. Measure and product of the measures of the secant segment and label the two parts of the secant segment to the its external segment is equal to the square of the nearest tenth of a centimeter. Use an equation to find measure of the tangent segment, because for the the measure of the tangent segment. Verify your tangent the measures of the external segment and answer by measuring the segment.

the whole segment are the same. SOLUTION: 28. CHALLENGE In the figure, a line tangent to circle Sample answer: M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION:

Use the intersecting tangent and secant segments to Therefore, the measure of the tangent segment is find the value of a. about 2.4 centimeters.

31. WRITING IN MATH Describe the relationship among segments in a circle when two secants intersect inside a circle. SOLUTION: Sample answer: The product of the parts on one intersecting chord equals the product of the parts of

the other chord.

29. REASONING When two chords intersect at the 32. is tangent to the circle, and R and S are points on center of a circle, are the measures of the the circle. What is the value of x to the nearest

intercepting arcs sometimes, always, or never equal tenth? to each other? SOLUTION: Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are A 7.6 congruent, the opposite arcs will be congruent. All B 6.4 four arcs will only be congruent when all four central C 5.7 angles are congruent. This will only happen when the D 4.8 chords are perpendicular and the angles are right angles SOLUTION: Therefore, the measures of the intercepted arcs are sometimes equal to each other.

30. OPEN ENDED Investigate Theorem 11.17 by drawing and labeling a circle that has a secant and a Solve for x using the Quadratic Formula. tangent intersecting outside the circle. Measure and label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: Sample answer: Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already Therefore, the measure of the tangent segment is discounted price. How much will you pay for a ring about 2.4 centimeters. with an original price of $200? F $80 31. WRITING IN MATH Describe the relationship G $96 among segments in a circle when two secants H $120 intersect inside a circle. J $140 SOLUTION: SOLUTION: Sample answer: The product of the parts on one intersecting chord equals the product of the parts of the other chord. Discount price = 200 – 80 = 120 32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest tenth? Discount price = 120 – 24 = 96 So, the correct choice is G.

A 7.6 34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, B 6.4 respectively. C 5.7 a. If m∠ABC = 70°, write two equations relating x D 4.8 and y. SOLUTION: b. Find x and y.

Solve for x using the Quadratic Formula.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 11.14, y – x = 140. b. Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

Therefore, x = 110° and y = 250°. 33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says 35. SAT/ACT During the first two weeks of summer you receive an additional 20% off the already vacation, Antonia earned $100 per week. During the discounted price. How much will you pay for a ring next six weeks, she earned $150 per week. What with an original price of $200? was her average weekly pay? F $80 A $50 G $96 B $112.50 H $120 C $125 J $140 D $135 SOLUTION: E $137.50 SOLUTION:

Discount price = 200 – 80 = 120 Simplify.

So, the correct choice is E.

Discount price = 120 – 24 36. WEAVING Once yarn is woven from wool fibers, it = 96 is often dyed and then threaded along a path of So, the correct choice is G. pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality 34. EXTENDED RESPONSE The degree measures of it does not. Use the information from the diagram to minor arc AC and major arc ADC are x and y, respectively. find a. If m∠ABC = 70°, write two equations relating x and y. b. Find x and y.

SOLUTION: First, use intersecting secants to find m∠GCD.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y = 360, By vertical angles, m∠BCH = m∠GCD. So, and by Theorem 11.14, y – x = 140. m∠BCH = 39. b. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141.

Therefore, x = 110° and y = 250°. Copy the figure shown and draw the common 35. SAT/ACT During the first two weeks of summer tangents. If no common tangent exists, state no vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What common tangent. was her average weekly pay? A $50 B $112.50 37. C $125 D $135 SOLUTION: E $137.50 Four common tangents can be drawn to these two circles. SOLUTION:

Simplify.

So, the correct choice is E. 36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to 38. find SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles. SOLUTION: First, use intersecting secants to find m∠GCD.

39. SOLUTION: By vertical angles, m∠BCH = m∠GCD. So, Three common tangents can be drawn to these two m∠BCH = 39. circles. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no common tangent. 40. SOLUTION: Two common tangents can be drawn to these two 37. circles. SOLUTION: Four common tangents can be drawn to these two circles.

Write an equation in slope-intercept form of the line having the given slope and y-intercept. 41. m: 3, y-intercept: –4 SOLUTION: The slope-intercept form of a line of slope m and y- 38. intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. SOLUTION: So, the equation of the line is y = 3x – 4. Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn 42. m: 2, (0, 8) to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at SOLUTION: exactly one point, no common tangent exists for To find the slope-intercept form of the equation of these two circles. this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8). 39. SOLUTION: Three common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

40. Here, and (x1, y1) = (0, –6). SOLUTION: Two common tangents can be drawn to these two circles.

42. m: 2, (0, 8) SOLUTION: To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8).

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

3. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent.

4. SOLUTION: 1. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

2. SOLUTION: SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

3. SOLUTION: Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. SOLUTION:

4. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter 7. of the circle. What was the circumference of the original pottery? Round to the nearest hundredth. SOLUTION:

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

SOLUTION: The diameter is equal to QS + ST or .

11-7 Special Segments in a Circle

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

9. SOLUTION: 6. SOLUTION:

10. SOLUTION:

7. SOLUTION:

11. SOLUTION:

Solve for x by using the Quadratic Formula. 8. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

eSolutions Manual - Powered by Cognero Page 2 9. 12. SOLUTION: SOLUTION:

10. SOLUTION:

13. SOLUTION:

Solve for x using the Quadratic Formula.

11. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1. SOLUTION:

12. SOLUTION: Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake? 13. SOLUTION:

Solve for x using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is

about 7.4. Since one of the perpendicular segments contains 14. BRIDGES What is the diameter of the circle the center of the cake, it will bisect the chord. To containing the arc of the Sydney Harbour Bridge? find the diameter of the cake, first use Theorem Round to the nearest tenth. 11.15 to find x. Refer to the photo on Page 772.

Add the two segments to find the diameter.

SOLUTION: The diameter of the circle is 9 + 4 or 13 inches. CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the 16. diameter of the circle, first use Theorem 11.15 to find SOLUTION: the value of x.

Solve for x using the Quadratic Formula. Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

Since lengths cannot be negative, the value of x is 6.

17. SOLUTION: SOLUTION:

Solve for x using the Quadratic Formula.

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To find the diameter of the cake, first use Theorem 11.15 to find x. Since lengths cannot be negative, the value of x is about 7.1.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the 18. nearest tenth. Assume that segments that SOLUTION: appear to be tangent are tangent.

16. SOLUTION: First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

Solve for x using the Quadratic Formula.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of x is 6.

17. 19. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find a.

Solve for x using the Quadratic Formula.

Next, use the intersecting chords to find b.

Since lengths cannot be negative, the value of x is about 7.1.

18. 20. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find q.

First, find the length, a, of the tangent segment to Since lengths cannot be negative, the value of q is 9. both circles using the intersecting secant and tangent Use the intersecting secant segments to find r. segments to the smaller circle.

Solve for r using the Quadratic Formula. Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

19. SOLUTION: Use the intersecting tangent and secant segments to find a. 21. SOLUTION: Use the intersecting secant segments to find c.

Next, use the intersecting chords to find b.

Use the intersecting secant and tangent segments to find d.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is 20. standing next to the tree, as shown. The distance SOLUTION: between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter Use the intersecting tangent and secant segments to of the tree. find q.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r. SOLUTION:

Solve for r using the Quadratic Formula.

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8. So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Given: and intersect at B. 21. Prove: SOLUTION: Use the intersecting secant segments to find c.

SOLUTION: Use the intersecting secant and tangent segments to Proof: find d. Statements (Reasons) 1. and intersect at B. (Given) 2. , (Inscribed that

Therefore, the values of the variables are c ≈ 22.8 intercept the same arc are and d ≈ 16.9. 3. (AA Similarity)

4. (Definition of similar 22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is 5. (Cross products) standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter 24. paragraph proof of Theorem 11.16 of the tree. Given: Secants and Prove:

Let d be the diameter of the tree. Since the cross products of a proportion Use the intersecting tangent and secant segments to find d. are equal,

25. two-column proof of Theorem 11.17 Given: tangent secant Prove: So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15

Given: and intersect at B. Prove: SOLUTION: Proof: Statements (Reasons) 1. tangent and secant (Given)

4. (Definition of similar 6. (Reflexive Property) 7. (AA Similarity)

5. (Cross products) 8. (Definition of 24. paragraph proof of Theorem 11.16 9. (Cross products) Given: Secants and Prove: 26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

SOLUTION: Proof:

and are secant segments. By the Reflexive Property, . Inscribed angles that SOLUTION: intercept the same arc are congruent. So, Jun; the segments intersect outside of the circle, so the correct equation involves the products of the . By AA Similarity, measures of a secant and its external secant By the definition of similar triangles, segment. Since the cross products of a proportion 27. WRITING IN MATH Compare and contrast the are equal, methods for finding measures of segments when two secants intersect in the exterior of a circle and when 25. two-column proof of Theorem 11.17 a secant and a tangent intersect in the exterior of a Given: tangent secant circle. Prove: SOLUTION: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant and a tangent intersect at an exterior point, the SOLUTION: product of the measures of the secant segment and Proof: its external segment is equal to the square of the Statements (Reasons) measure of the tangent segment, because for the 1. tangent and secant (Given) tangent the measures of the external segment and the whole segment are the same. 2. (The measure of an inscribed 28. CHALLENGE In the figure, a line tangent to circle ∠ equals half the measure of its intercept arc.) M and a secant line intersect at R. Find a. Show the 3. (The measure of an ∠ formed steps that you used. by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property) 7. (AA Similarity) SOLUTION: Use the intersecting tangent and secant segments to 8. (Definition of find the value of a. 9. (Cross products)

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

the correct equation involves the products of the to each other? measures of a secant and its external secant SOLUTION: segment. Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its WRITING IN MATH 27. Compare and contrast the related central angle. Since vertical angles are methods for finding measures of segments when two congruent, the opposite arcs will be congruent. All secants intersect in the exterior of a circle and when four arcs will only be congruent when all four central a secant and a tangent intersect in the exterior of a angles are congruent. This will only happen when the

circle. chords are perpendicular and the angles are right SOLUTION: angles Sample answer: When two secants intersect in the Therefore, the measures of the intercepted arcs are exterior of a circle, the product of the measures of sometimes equal to each other. one secant segment and its external segment is equal OPEN ENDED to the product of the measures of the other secant 30. Investigate Theorem 11.17 by segment and its external segment. When a secant drawing and labeling a circle that has a secant and a and a tangent intersect at an exterior point, the tangent intersecting outside the circle. Measure and product of the measures of the secant segment and label the two parts of the secant segment to the its external segment is equal to the square of the nearest tenth of a centimeter. Use an equation to find measure of the tangent segment, because for the the measure of the tangent segment. Verify your

tangent the measures of the external segment and answer by measuring the segment. the whole segment are the same. SOLUTION: Sample answer: 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION: Use the intersecting tangent and secant segments to Therefore, the measure of the tangent segment is find the value of a. about 2.4 centimeters.

32. is tangent to the circle, and R and S are points on 29. REASONING When two chords intersect at the the circle. What is the value of x to the nearest center of a circle, are the measures of the tenth? intercepting arcs sometimes, always, or never equal to each other? SOLUTION: Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its A related central angle. Since vertical angles are 7.6 congruent, the opposite arcs will be congruent. All B 6.4 four arcs will only be congruent when all four central C 5.7 angles are congruent. This will only happen when the D 4.8 chords are perpendicular and the angles are right SOLUTION: angles Therefore, the measures of the intercepted arcs are sometimes equal to each other.

Sample answer: Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

34. EXTENDED RESPONSE The degree measures of A 7.6 minor arc AC and major arc ADC are x and y, B 6.4 respectively. C 5.7 a. If m∠ABC = 70°, write two equations relating x D 4.8 and y. SOLUTION: b. Find x and y.

Solve for x using the Quadratic Formula.

Discount price = 200 – 80 = 120 Simplify.

So, the correct choice is E.

Discount price = 120 – 24 36. WEAVING Once yarn is woven from wool fibers, it = 96 is often dyed and then threaded along a path of So, the correct choice is G. pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality 34. EXTENDED RESPONSE The degree measures of it does not. Use the information from the diagram to minor arc AC and major arc ADC are x and y, find respectively. a. If m∠ABC = 70°, write two equations relating x and y. b. Find x and y.

SOLUTION: First, use intersecting secants to find m∠GCD.

Therefore, the measure of arc BH is 141.

Therefore, x = 110° and y = 250°. Copy the figure shown and draw the common 35. SAT/ACT During the first two weeks of summer tangents. If no common tangent exists, state no vacation, Antonia earned $100 per week. During the common tangent. next six weeks, she earned $150 per week. What was her average weekly pay? A $50 B $112.50 37. C $125 SOLUTION: D $135 Four common tangents can be drawn to these two E $137.50 circles. SOLUTION:

Simplify.

So, the correct choice is E. 36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality 38. it does not. Use the information from the diagram to find SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles. SOLUTION: First, use intersecting secants to find m∠GCD.

Therefore, the measure of arc BH is 141.

38. For this line, substitute m = 3 and b = –4. SOLUTION: So, the equation of the line is y = 3x – 4. Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn 42. m: 2, (0, 8) to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at SOLUTION: exactly one point, no common tangent exists for To find the slope-intercept form of the equation of these two circles. this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8). 39. SOLUTION: Three common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of

this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

40. Here, and (x1, y1) = (0, –6). SOLUTION: Two common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

Find x. Assume that segments that appear to be 3. tangent are tangent. SOLUTION:

1. SOLUTION:

4. SOLUTION:

SCIENCE 2. 5. A piece of broken pottery found at an archaeological site is shown. lies on a diameter SOLUTION: of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the 3. measure of ST. SOLUTION:

The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are 4. tangent. SOLUTION:

6. SOLUTION: 5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

SOLUTION: Let be the diameter of the circle. Then and 7. are intersecting chords. SOLUTION: Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that 8. segments that appear to be tangent are tangent. SOLUTION:

6. SOLUTION:

9. SOLUTION:

7. SOLUTION:

10. SOLUTION:

8. SOLUTION: 11. SOLUTION:

Solve for x by using the Quadratic Formula.

9. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

12. 10. SOLUTION: SOLUTION:

11. 13. SOLUTION: SOLUTION:

Solve for x by using the Quadratic Formula. Solve for x using the Quadratic Formula.

11-7Since Special lengths Segments cannot inbe anegative, Circle the value of x is Since lengths cannot be negative, the value of x is about 3.1. about 7.4.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772. 12. SOLUTION:

SOLUTION:

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

13. SOLUTION: Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the Solve for x using the Quadratic Formula. dimensions of the remaining cake are shown below, what was the original diameter of the cake?

Since lengths cannot be negative, the value of x is about 7.4.

14. BRIDGES What is the diameter of the circle SOLUTION: containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

SOLUTION: Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To find the diameter of the cake, first use Theorem 11.15 to find x. eSolutions Manual - Powered by Cognero Page 3

Since the height is perpendicular and bisects the Add the two segments to find the diameter. chord, it must be part of the diameter. To find the The diameter of the circle is 9 + 4 or 13 inches. diameter of the circle, first use Theorem 11.15 to find the value of x. CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters. 16. 15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, SOLUTION: what was the original diameter of the cake?

Solve for x using the Quadratic Formula.

SOLUTION:

Since lengths cannot be negative, the value of x is 6.

Since one of the perpendicular segments contains 17. the center of the cake, it will bisect the chord. To SOLUTION: find the diameter of the cake, first use Theorem 11.15 to find x.

Solve for x using the Quadratic Formula.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent. Since lengths cannot be negative, the value of x is about 7.1.

16. SOLUTION: 18. SOLUTION:

Solve for x using the Quadratic Formula.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent Since lengths cannot be negative, the value of x is 6. segments to the smaller circle.

Find x using the intersecting secant and tangent segments to the larger circle. 17. SOLUTION:

Solve for x using the Quadratic Formula.

19. SOLUTION: Use the intersecting tangent and secant segments to Since lengths cannot be negative, the value of x is find a. about 7.1.

Next, use the intersecting chords to find b. 18. SOLUTION:

First, find the length, a, of the tangent segment to 20. both circles using the intersecting secant and tangent segments to the smaller circle. SOLUTION: Use the intersecting tangent and secant segments to find q.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

Solve for r using the Quadratic Formula.

19. SOLUTION: Use the intersecting tangent and secant segments to find a. Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8. Next, use the intersecting chords to find b.

21. SOLUTION: Use the intersecting secant segments to find c.

20. Use the intersecting secant and tangent segments to find d. SOLUTION: Use the intersecting tangent and secant segments to find q.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is Since lengths cannot be negative, the value of q is 9. standing next to the tree, as shown. The distance Use the intersecting secant segments to find r. between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree.

Solve for r using the Quadratic Formula.

SOLUTION: Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to 21. find d. SOLUTION: Use the intersecting secant segments to find c.

So, the diameter is about 29.6 ft.

PROOF Prove each theorem. Use the intersecting secant and tangent segments to 23. two-column proof of Theorem 11.15 find d. Given: and intersect at B. Prove:

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is SOLUTION: standing next to the tree, as shown. The distance Proof: between Gwendolyn and Chet is 27 feet. Draw a Statements (Reasons) diagram of this situation, and then find the diameter 1. and intersect at B. (Given) of the tree. 2. , (Inscribed that intercept the same arc are 3. (AA Similarity)

4. (Definition of similar 5. (Cross products)

SOLUTION: 24. paragraph proof of Theorem 11.16 Given: Secants and Prove:

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d. SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that intercept the same arc are congruent. So, So, the diameter is about 29.6 ft. . By AA Similarity, By the definition of similar triangles, PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Since the cross products of a proportion Given: and intersect at B. are equal, Prove: 25. two-column proof of Theorem 11.17 Given: tangent secant Prove:

SOLUTION: Proof:

Statements (Reasons) 1. and intersect at B. (Given) SOLUTION: 2. , (Inscribed that Proof: intercept the same arc are Statements (Reasons) 3. (AA Similarity) 1. tangent and secant (Given) 4. (Definition of similar 2. (The measure of an inscribed 5. (Cross products) ∠ equals half the measure of its intercept arc.) 24. paragraph proof of Theorem 11.16 3. (The measure of an ∠ formed Given: Secants and by a secant and a tangent = half the measure of its Prove: intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property)

7. (AA Similarity)

SOLUTION: 8. (Definition of Proof: 9. (Cross products) and are secant segments. By the Reflexive Property, . Inscribed angles that 26. CRITIQUE Tiffany and Jun are finding the value of intercept the same arc are congruent. So, x in the figure. Tiffany wrote 3(5) = 2x, and Jun . By AA Similarity, wrote 3(8) = 2(2 + x). Is either of them correct? By the definition of similar triangles, Explain your reasoning. Since the cross products of a proportion are equal,

25. two-column proof of Theorem 11.17 Given: tangent secant SOLUTION: Prove: Jun; the segments intersect outside of the circle, so the correct equation involves the products of the measures of a secant and its external secant segment.

27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two SOLUTION: secants intersect in the exterior of a circle and when Proof: a secant and a tangent intersect in the exterior of a

Statements (Reasons) circle. 1. tangent and secant (Given) SOLUTION: Sample answer: When two secants intersect in the 2. (The measure of an inscribed exterior of a circle, the product of the measures of ∠ equals half the measure of its intercept arc.) one secant segment and its external segment is equal to the product of the measures of the other secant 3. (The measure of an ∠ formed segment and its external segment. When a secant and a tangent intersect at an exterior point, the by a secant and a tangent = half the measure of its product of the measures of the secant segment and intercepted arc.) its external segment is equal to the square of the 4. (Substitution) measure of the tangent segment, because for the 5. (Definition of tangent the measures of the external segment and 6. (Reflexive Property) the whole segment are the same. 7. (AA Similarity) 28. CHALLENGE In the figure, a line tangent to circle 8. (Definition of M and a secant line intersect at R. Find a. Show the steps that you used. 9. (Cross products)

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning. SOLUTION: Use the intersecting tangent and secant segments to find the value of a.

SOLUTION: Jun; the segments intersect outside of the circle, so the correct equation involves the products of the measures of a secant and its external secant segment.

27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two secants intersect in the exterior of a circle and when a secant and a tangent intersect in the exterior of a 29. REASONING When two chords intersect at the circle. center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal SOLUTION: to each other? Sample answer: When two secants intersect in the SOLUTION: exterior of a circle, the product of the measures of one secant segment and its external segment is equal Since the chords intersect at the center, the measure to the product of the measures of the other secant of each intercepted arc is equal to the measure of its segment and its external segment. When a secant related central angle. Since vertical angles are and a tangent intersect at an exterior point, the congruent, the opposite arcs will be congruent. All product of the measures of the secant segment and four arcs will only be congruent when all four central its external segment is equal to the square of the angles are congruent. This will only happen when the measure of the tangent segment, because for the chords are perpendicular and the angles are right tangent the measures of the external segment and angles the whole segment are the same. Therefore, the measures of the intercepted arcs are sometimes equal to each other. 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the 30. OPEN ENDED Investigate Theorem 11.17 by steps that you used. drawing and labeling a circle that has a secant and a tangent intersecting outside the circle. Measure and label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: SOLUTION: Sample answer: Use the intersecting tangent and secant segments to find the value of a.

Therefore, the measure of the tangent segment is

about 2.4 centimeters. 29. REASONING When two chords intersect at the 31. WRITING IN MATH Describe the relationship center of a circle, are the measures of the among segments in a circle when two secants intercepting arcs sometimes, always, or never equal intersect inside a circle. to each other? SOLUTION: SOLUTION: Sample answer: The product of the parts on one Since the chords intersect at the center, the measure intersecting chord equals the product of the parts of of each intercepted arc is equal to the measure of its the other chord. related central angle. Since vertical angles are congruent, the opposite arcs will be congruent. All four arcs will only be congruent when all four central 32. is tangent to the circle, and R and S are points on angles are congruent. This will only happen when the the circle. What is the value of x to the nearest chords are perpendicular and the angles are right tenth? angles Therefore, the measures of the intercepted arcs are sometimes equal to each other.

30. OPEN ENDED Investigate Theorem 11.17 by drawing and labeling a circle that has a secant and a A 7.6 tangent intersecting outside the circle. Measure and B 6.4 label the two parts of the secant segment to the C 5.7 nearest tenth of a centimeter. Use an equation to find D 4.8 the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: SOLUTION: Sample answer:

Solve for x using the Quadratic Formula.

Since length can not be negative, the value of x is Therefore, the measure of the tangent segment is 5.7. about 2.4 centimeters. Therefore, the correct answer is C. 31. WRITING IN MATH Describe the relationship among segments in a circle when two secants intersect inside a circle.

SOLUTION: 33. ALGEBRA A department store has all of its Sample answer: The product of the parts on one jewelry discounted 40%. It is having a sale that says intersecting chord equals the product of the parts of you receive an additional 20% off the already the other chord. discounted price. How much will you pay for a ring with an original price of $200? 32. is tangent to the circle, and R and S are points on F $80 the circle. What is the value of x to the nearest G $96 tenth? H $120 J $140 SOLUTION:

A 7.6 B 6.4 Discount price = 200 – 80 = 120 C 5.7 D 4.8 SOLUTION: Discount price = 120 – 24 = 96 So, the correct choice is G.

34. EXTENDED RESPONSE The degree measures of Solve for x using the Quadratic Formula. minor arc AC and major arc ADC are x and y, respectively. a. If m∠ABC = 70°, write two equations relating x and y. b. Find x and y.

Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

SOLUTION:

a. Since the sum of the arcs of a circle is 360, x + y = 360, 33. ALGEBRA A department store has all of its and by Theorem 11.14, y – x = 140. jewelry discounted 40%. It is having a sale that says b you receive an additional 20% off the already . discounted price. How much will you pay for a ring with an original price of $200? F $80 G $96 H $120 J $140 Therefore, x = 110° and y = 250°. SOLUTION: 35. SAT/ACT During the first two weeks of summer vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What Discount price = 200 – 80 = 120 was her average weekly pay? A $50 B $112.50 C $125 Discount price = 120 – 24 D $135 = 96 E $137.50 So, the correct choice is G. SOLUTION: 34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, respectively. Simplify. a. If m∠ABC = 70°, write two equations relating x and y. So, the correct choice is E. b. Find x and y. 36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to find

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 11.14, y – x = 140. b. SOLUTION: First, use intersecting secants to find m∠GCD.

Therefore, x = 110° and y = 250°. By vertical angles, m∠BCH = m∠GCD. So, m BCH = 39. 35. SAT/ACT During the first two weeks of summer ∠ vacation, Antonia earned $100 per week. During the Since major arc BAH has the same endpoints as next six weeks, she earned $150 per week. What minor arc BH, m(arc BAH) = 360 – m(arc BH). was her average weekly pay? A $50 B $112.50 C $125 Therefore, the measure of arc BH is 141. D $135 E $137.50 Copy the figure shown and draw the common SOLUTION: tangents. If no common tangent exists, state no common tangent.

Simplify.

So, the correct choice is E. 37. SOLUTION: 36. WEAVING Once yarn is woven from wool fibers, it Four common tangents can be drawn to these two is often dyed and then threaded along a path of circles. pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to find

SOLUTION: First, use intersecting secants to find m∠GCD. 38. SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at

any point. Since a tangent must intersect the circle at By vertical angles, m BCH = m GCD. So, ∠ ∠ exactly one point, no common tangent exists for m∠BCH = 39. these two circles. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

39. Therefore, the measure of arc BH is 141. SOLUTION: Three common tangents can be drawn to these two Copy the figure shown and draw the common circles. tangents. If no common tangent exists, state no common tangent.

37. SOLUTION: Four common tangents can be drawn to these two circles.

40. SOLUTION: Two common tangents can be drawn to these two circles.

38.

SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn Write an equation in slope-intercept form of the to the large circle will not intersect the small circle at line having the given slope and y-intercept. any point. Since a tangent must intersect the circle at 41. m: 3, y-intercept: –4 exactly one point, no common tangent exists for SOLUTION: these two circles. The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. So, the equation of the line is y = 3x – 4.

39. 42. m: 2, (0, 8) SOLUTION: SOLUTION: Three common tangents can be drawn to these two To find the slope-intercept form of the equation of circles. this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8).

40. 43. SOLUTION: SOLUTION: Two common tangents can be drawn to these two

circles. To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent.

1. 3. SOLUTION: SOLUTION:

4. 2. SOLUTION: SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

SOLUTION: SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or . 4. SOLUTION:

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent. 5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth. 6. SOLUTION:

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

SOLUTION: The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. 8. SOLUTION: SOLUTION:

7. 9. SOLUTION: SOLUTION:

10. SOLUTION:

8. SOLUTION:

11. SOLUTION:

9. SOLUTION: Solve for x by using the Quadratic Formula.

Since lengths cannot be negative, the value of x is 10. about 3.1. SOLUTION:

12. SOLUTION:

11. SOLUTION:

Solve for x by using the Quadratic Formula. 13. SOLUTION:

Since lengths cannot be negative, the value of x is Solve for x using the Quadratic Formula. about 3.1.

12. Since lengths cannot be negative, the value of x is SOLUTION: about 7.4.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

SOLUTION:

13. SOLUTION:

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Solve for x using the Quadratic Formula.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the Since lengths cannot be negative, the value of x is dimensions of the remaining cake are shown below, about 7.4. what was the original diameter of the cake? 14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

SOLUTION: SOLUTION:

Since the height is perpendicular and bisects the Since one of the perpendicular segments contains chord, it must be part of the diameter. To find the the center of the cake, it will bisect the chord. To diameter of the circle, first use Theorem 11.15 to find find the diameter of the cake, first use Theorem the value of x. 11.15 to find x.

Add the two segments to find the diameter. 11-7Therefore, Special Segments the diameter in aof Circle the circle is 528.01+ 60 or Add the two segments to find the diameter. about 588.1 meters. The diameter of the circle is 9 + 4 or 13 inches.

15. CAKES Sierra is serving cake at a party. If the CCSS STRUCTURE Find each variable to the dimensions of the remaining cake are shown below, nearest tenth. Assume that segments that what was the original diameter of the cake? appear to be tangent are tangent.

16. SOLUTION:

SOLUTION:

Solve for x using the Quadratic Formula.

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To Since lengths cannot be negative, the value of x is 6. find the diameter of the cake, first use Theorem 11.15 to find x.

Add the two segments to find the diameter. 17. The diameter of the circle is 9 + 4 or 13 inches. SOLUTION: CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent. Solve for x using the Quadratic Formula.

16. SOLUTION:

Since lengths cannot be negative, the value of x is about 7.1.

Solve for x using the Quadratic Formula.

18. SOLUTION: eSolutionsSinceManual lengths- Powered cannotby beCognero negative, the value of x is 6. Page 4

First, find the length, a, of the tangent segment to 17. both circles using the intersecting secant and tangent

SOLUTION: segments to the smaller circle.

Find x using the intersecting secant and tangent Solve for x using the Quadratic Formula. segments to the larger circle.

Since lengths cannot be negative, the value of x is about 7.1.

19. SOLUTION: 18. Use the intersecting tangent and secant segments to find a. SOLUTION:

Next, use the intersecting chords to find b.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

Find x using the intersecting secant and tangent segments to the larger circle. 20. SOLUTION: Use the intersecting tangent and secant segments to

find q.

19. Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r. SOLUTION: Use the intersecting tangent and secant segments to find a.

Solve for r using the Quadratic Formula.

Next, use the intersecting chords to find b.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

20. 21. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to Use the intersecting secant segments to find c. find q.

Use the intersecting secant and tangent segments to find d. Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9. Solve for r using the Quadratic Formula. 22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

SOLUTION: 21. SOLUTION: Use the intersecting secant segments to find c.

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to Use the intersecting secant and tangent segments to find d. find d.

Therefore, the values of the variables are c ≈ 22.8 So, the diameter is about 29.6 ft. and d ≈ 16.9. PROOF Prove each theorem. 22. INDIRECT MEASUREMENT Gwendolyn is 23. two-column proof of Theorem 11.15 standing 16 feet from a giant sequoia tree and Chet is Given: and intersect at B. standing next to the tree, as shown. The distance Prove: between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree.

SOLUTION: Proof: Statements (Reasons) 1. and intersect at B. (Given) 2. , (Inscribed that SOLUTION: intercept the same arc are 3. (AA Similarity)

4. (Definition of similar 5. (Cross products)

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to 24. paragraph proof of Theorem 11.16 find d. Given: Secants and Prove:

So, the diameter is about 29.6 ft.

PROOF Prove each theorem. SOLUTION: 23. two-column proof of Theorem 11.15 Proof: Given: and intersect at B. and are secant segments. By the Reflexive Prove: Property, . Inscribed angles that intercept the same arc are congruent. So, . By AA Similarity, By the definition of similar triangles, Since the cross products of a proportion SOLUTION: are equal, Proof: Statements (Reasons) 25. two-column proof of Theorem 11.17

1. and intersect at B. (Given) Given: tangent secant 2. , (Inscribed that Prove: intercept the same arc are 3. (AA Similarity)

4. (Definition of similar

5. (Cross products) SOLUTION: 24. paragraph proof of Theorem 11.16 Proof: Given: Secants and Statements (Reasons) Prove: 1. tangent and secant (Given) 2. (The measure of an inscribed ∠ equals half the measure of its intercept arc.) 3. (The measure of an ∠ formed SOLUTION: by a secant and a tangent = half the measure of its Proof: intercepted arc.) and are secant segments. By the Reflexive 4. (Substitution) Property, . Inscribed angles that 5. (Definition of intercept the same arc are congruent. So, 6. (Reflexive Property) . By AA Similarity, 7. (AA Similarity) By the definition of similar triangles, 8. (Definition of Since the cross products of a proportion 9. (Cross products) are equal, 26. CRITIQUE Tiffany and Jun are finding the value of 25. two-column proof of Theorem 11.17 x in the figure. Tiffany wrote 3(5) = 2x, and Jun Given: tangent secant wrote 3(8) = 2(2 + x). Is either of them correct? Prove: Explain your reasoning.

SOLUTION: SOLUTION: Jun; the segments intersect outside of the circle, so Proof: the correct equation involves the products of the Statements (Reasons) measures of a secant and its external secant

1. tangent and secant (Given) segment. 2. (The measure of an inscribed 27. WRITING IN MATH Compare and contrast the ∠ equals half the measure of its intercept arc.) methods for finding measures of segments when two secants intersect in the exterior of a circle and when 3. (The measure of an ∠ formed a secant and a tangent intersect in the exterior of a circle. by a secant and a tangent = half the measure of its intercepted arc.) SOLUTION: 4. (Substitution) Sample answer: When two secants intersect in the 5. (Definition of exterior of a circle, the product of the measures of 6. (Reflexive Property) one secant segment and its external segment is equal 7. (AA Similarity) to the product of the measures of the other secant segment and its external segment. When a secant 8. (Definition of and a tangent intersect at an exterior point, the product of the measures of the secant segment and 9. (Cross products) its external segment is equal to the square of the measure of the tangent segment, because for the 26. CRITIQUE Tiffany and Jun are finding the value of tangent the measures of the external segment and x in the figure. Tiffany wrote 3(5) = 2x, and Jun the whole segment are the same. wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning. 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION: Jun; the segments intersect outside of the circle, so the correct equation involves the products of the measures of a secant and its external secant SOLUTION: segment. Use the intersecting tangent and secant segments to find the value of a. 27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two secants intersect in the exterior of a circle and when a secant and a tangent intersect in the exterior of a circle. SOLUTION: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant 29. REASONING When two chords intersect at the and a tangent intersect at an exterior point, the center of a circle, are the measures of the product of the measures of the secant segment and intercepting arcs sometimes, always, or never equal its external segment is equal to the square of the to each other? measure of the tangent segment, because for the SOLUTION: tangent the measures of the external segment and the whole segment are the same. Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its 28. CHALLENGE In the figure, a line tangent to circle related central angle. Since vertical angles are M and a secant line intersect at R. Find a. Show the congruent, the opposite arcs will be congruent. All steps that you used. four arcs will only be congruent when all four central angles are congruent. This will only happen when the chords are perpendicular and the angles are right angles Therefore, the measures of the intercepted arcs are sometimes equal to each other.

30. OPEN ENDED Investigate Theorem 11.17 by SOLUTION: drawing and labeling a circle that has a secant and a Use the intersecting tangent and secant segments to tangent intersecting outside the circle. Measure and find the value of a. label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: Sample answer:

29. REASONING When two chords intersect at the center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal to each other? SOLUTION: Therefore, the measure of the tangent segment is Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its about 2.4 centimeters. related central angle. Since vertical angles are congruent, the opposite arcs will be congruent. All 31. WRITING IN MATH Describe the relationship four arcs will only be congruent when all four central among segments in a circle when two secants angles are congruent. This will only happen when the intersect inside a circle. chords are perpendicular and the angles are right SOLUTION:

angles Sample answer: The product of the parts on one Therefore, the measures of the intercepted arcs are intersecting chord equals the product of the parts of

sometimes equal to each other. the other chord. 30. OPEN ENDED Investigate Theorem 11.17 by drawing and labeling a circle that has a secant and a 32. is tangent to the circle, and R and S are points on tangent intersecting outside the circle. Measure and the circle. What is the value of x to the nearest

label the two parts of the secant segment to the tenth? nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: Sample answer: A 7.6 B 6.4 C 5.7 D 4.8 SOLUTION:

Solve for x using the Quadratic Formula. Therefore, the measure of the tangent segment is about 2.4 centimeters.

31. WRITING IN MATH Describe the relationship among segments in a circle when two secants intersect inside a circle. SOLUTION: Since length can not be negative, the value of x is Sample answer: The product of the parts on one 5.7. intersecting chord equals the product of the parts of the other chord. Therefore, the correct answer is C.

32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest tenth? 33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already discounted price. How much will you pay for a ring with an original price of $200? A 7.6 F $80 B 6.4 G $96 C 5.7 H $120 D 4.8 J $140 SOLUTION: SOLUTION:

Discount price = 200 – 80 = 120 Solve for x using the Quadratic Formula.

Discount price = 120 – 24 = 96 So, the correct choice is G.

34. EXTENDED RESPONSE The degree measures of Since length can not be negative, the value of x is minor arc AC and major arc ADC are x and y, 5.7. respectively. Therefore, the correct answer is C. a. If m∠ABC = 70°, write two equations relating x

and y. b. Find x and y.

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already discounted price. How much will you pay for a ring with an original price of $200? F $80 SOLUTION: G $96 a. Since the sum of the arcs of a circle is 360, x + y H $120 = 360, J $140 and by Theorem 11.14, y – x = 140. b. SOLUTION:

Discount price = 200 – 80 = 120

Therefore, x = 110° and y = 250°.

Discount price = 120 – 24 35. SAT/ACT During the first two weeks of summer = 96 vacation, Antonia earned $100 per week. During the So, the correct choice is G. next six weeks, she earned $150 per week. What was her average weekly pay? 34. EXTENDED RESPONSE The degree measures of A $50 minor arc AC and major arc ADC are x and y, B respectively. $112.50 C $125 a. If m∠ABC = 70°, write two equations relating x and y. D $135 b. Find x and y. E $137.50 SOLUTION:

Simplify.

So, the correct choice is E. SOLUTION: 36. WEAVING Once yarn is woven from wool fibers, it a. Since the sum of the arcs of a circle is 360, x + y is often dyed and then threaded along a path of = 360, pulleys to dry. One set of pulleys is shown. Note that and by Theorem 11.14, y – x = 140. the yarn appears to intersect itself at C, but in reality b. it does not. Use the information from the diagram to find

Therefore, x = 110° and y = 250°. SOLUTION: 35. SAT/ACT During the first two weeks of summer First, use intersecting secants to find m∠GCD. vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What was her average weekly pay? A $50 B $112.50 C $125 By vertical angles, m∠BCH = m∠GCD. So, D $135 m∠BCH = 39. E $137.50 Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH). SOLUTION:

Simplify. Therefore, the measure of arc BH is 141. So, the correct choice is E. Copy the figure shown and draw the common 36. WEAVING Once yarn is woven from wool fibers, it tangents. If no common tangent exists, state no is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that common tangent. the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to

find 37. SOLUTION: Four common tangents can be drawn to these two circles.

SOLUTION: First, use intersecting secants to find m∠GCD.

By vertical angles, m∠BCH = m∠GCD. So, m∠BCH = 39. 38. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH). SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at Therefore, the measure of arc BH is 141. exactly one point, no common tangent exists for these two circles. Copy the figure shown and draw the common tangents. If no common tangent exists, state no common tangent.

39.

37. SOLUTION: Three common tangents can be drawn to these two SOLUTION: circles. Four common tangents can be drawn to these two circles.

40. 38. SOLUTION: SOLUTION: Two common tangents can be drawn to these two

Every tangent drawn to the small circle will intersect circles. the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles.

Write an equation in slope-intercept form of the line having the given slope and y-intercept. 39. 41. m: 3, y-intercept: –4 SOLUTION: SOLUTION: Three common tangents can be drawn to these two The slope-intercept form of a line of slope m and y- circles. intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. So, the equation of the line is y = 3x – 4.

40. Here, m = 2 and (x1, y1) = (0, 8). SOLUTION: Two common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of Write an equation in slope-intercept form of the this line use the point-slope form of a line, y – y = m line having the given slope and y-intercept. 1 41. m: 3, y-intercept: –4 (x – x1) where m is the slope and (x1, y1) is a point on the line. SOLUTION:

The slope-intercept form of a line of slope m and y- Here, and (x1, y1) = (0, –6). intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. So, the equation of the line is y = 3x – 4.

42. m: 2, (0, 8)

SOLUTION: To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8).

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

3. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent.

4. SOLUTION: 1. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

2. SOLUTION: SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

3. SOLUTION:

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. SOLUTION:

4. SOLUTION:

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

8. SOLUTION: The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

9. SOLUTION: 6. SOLUTION:

10. SOLUTION:

7. SOLUTION:

11. SOLUTION:

8. Solve for x by using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

9. 12. SOLUTION: SOLUTION:

10. SOLUTION:

13. SOLUTION:

Solve for x using the Quadratic Formula.

11. SOLUTION:

Since lengths cannot be negative, the value of x is Solve for x by using the Quadratic Formula. about 7.4. 14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

Since lengths cannot be negative, the value of x is about 3.1. SOLUTION:

12.

SOLUTION: Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake? 13. SOLUTION:

Solve for x using the Quadratic Formula. SOLUTION:

Add the two segments to find the diameter. SOLUTION: The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the 16. diameter of the circle, first use Theorem 11.15 to find the value of x. SOLUTION:

Add the two segments to find the diameter. Solve for x using the Quadratic Formula. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

Since lengths cannot be negative, the value of x is 6.

17. SOLUTION: SOLUTION:

Solve for x using the Quadratic Formula.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the 18. nearest tenth. Assume that segments that appear to be tangent are tangent. SOLUTION:

16.

SOLUTION: First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

Solve for x using the Quadratic Formula.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of x is 6.

17. 19. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find a.

Solve for x using the Quadratic Formula.

Next, use the intersecting chords to find b.

11-7Since Special lengths Segments cannot inbe anegative, Circle the value of x is about 7.1.

18. 20. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find q.

Find x using the intersecting secant and tangent Solve for r using the Quadratic Formula. segments to the larger circle.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8. 19. SOLUTION: Use the intersecting tangent and secant segments to find a. 21. SOLUTION: Use the intersecting secant segments to find c.

Next, use the intersecting chords to find b.

Use the intersecting secant and tangent segments to find d.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9. eSolutions Manual - Powered by Cognero 22. INDIRECT MEASUREMENT Gwendolyn isPage 5 20. standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance SOLUTION: between Gwendolyn and Chet is 27 feet. Draw a Use the intersecting tangent and secant segments to diagram of this situation, and then find the diameter find q. of the tree.

Since lengths cannot be negative, the value of q is 9.

Use the intersecting secant segments to find r. SOLUTION:

Solve for r using the Quadratic Formula.

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d. Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 21. Given: and intersect at B. Prove: SOLUTION: Use the intersecting secant segments to find c.

22. INDIRECT MEASUREMENT Gwendolyn is 4. (Definition of similar standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance 5. (Cross products) between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter 24. paragraph proof of Theorem 11.16 of the tree. Given: Secants and Prove:

SOLUTION: SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that intercept the same arc are congruent. So, . By AA Similarity, By the definition of similar triangles, Let d be the diameter of the tree. Use the intersecting tangent and secant segments to Since the cross products of a proportion find d. are equal,

25. two-column proof of Theorem 11.17 Given: tangent secant

So, the diameter is about 29.6 ft. Prove:

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Given: and intersect at B. Prove: SOLUTION: Proof: Statements (Reasons)

1. tangent and secant (Given) SOLUTION: 2. (The measure of an inscribed Proof: ∠ equals half the measure of its intercept arc.) Statements (Reasons) 1. and intersect at B. (Given) 3. (The measure of an ∠ formed 2. , (Inscribed that by a secant and a tangent = half the measure of its intercept the same arc are intercepted arc.) 3. (AA Similarity) 4. (Substitution) 5. (Definition of 4. (Definition of similar 6. (Reflexive Property) 5. (Cross products) 7. (AA Similarity)

8. (Definition of 24. paragraph proof of Theorem 11.16 Given: Secants and 9. (Cross products) Prove: 26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that SOLUTION: intercept the same arc are congruent. So, Jun; the segments intersect outside of the circle, so . By AA Similarity, the correct equation involves the products of the By the definition of similar triangles, measures of a secant and its external secant segment. Since the cross products of a proportion are equal, 27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two 25. two-column proof of Theorem 11.17 secants intersect in the exterior of a circle and when a secant and a tangent intersect in the exterior of a Given: tangent secant circle. Prove: SOLUTION: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant SOLUTION: and a tangent intersect at an exterior point, the Proof: product of the measures of the secant segment and Statements (Reasons) its external segment is equal to the square of the 1. tangent and secant (Given) measure of the tangent segment, because for the tangent the measures of the external segment and 2. (The measure of an inscribed the whole segment are the same.

∠ equals half the measure of its intercept arc.) 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the 3. (The measure of an ∠ formed steps that you used. by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of

6. (Reflexive Property) 7. (AA Similarity) SOLUTION: 8. (Definition of Use the intersecting tangent and secant segments to find the value of a. 9. (Cross products)

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

29. REASONING When two chords intersect at the SOLUTION: center of a circle, are the measures of the Jun; the segments intersect outside of the circle, so intercepting arcs sometimes, always, or never equal the correct equation involves the products of the to each other? measures of a secant and its external secant segment. SOLUTION: Since the chords intersect at the center, the measure 27. WRITING IN MATH Compare and contrast the of each intercepted arc is equal to the measure of its methods for finding measures of segments when two related central angle. Since vertical angles are secants intersect in the exterior of a circle and when congruent, the opposite arcs will be congruent. All a secant and a tangent intersect in the exterior of a four arcs will only be congruent when all four central circle. angles are congruent. This will only happen when the chords are perpendicular and the angles are right SOLUTION: angles Sample answer: When two secants intersect in the Therefore, the measures of the intercepted arcs are exterior of a circle, the product of the measures of sometimes equal to each other. one secant segment and its external segment is equal to the product of the measures of the other secant 30. OPEN ENDED Investigate Theorem 11.17 by segment and its external segment. When a secant drawing and labeling a circle that has a secant and a and a tangent intersect at an exterior point, the tangent intersecting outside the circle. Measure and product of the measures of the secant segment and label the two parts of the secant segment to the its external segment is equal to the square of the nearest tenth of a centimeter. Use an equation to find measure of the tangent segment, because for the the measure of the tangent segment. Verify your tangent the measures of the external segment and answer by measuring the segment. the whole segment are the same. SOLUTION: 28. CHALLENGE In the figure, a line tangent to circle Sample answer: M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION: Use the intersecting tangent and secant segments to find the value of a. Therefore, the measure of the tangent segment is about 2.4 centimeters.

29. REASONING When two chords intersect at the 32. is tangent to the circle, and R and S are points on center of a circle, are the measures of the the circle. What is the value of x to the nearest intercepting arcs sometimes, always, or never equal tenth? to each other? SOLUTION: Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are A 7.6 congruent, the opposite arcs will be congruent. All B 6.4 four arcs will only be congruent when all four central C 5.7 angles are congruent. This will only happen when the chords are perpendicular and the angles are right D 4.8 angles SOLUTION: Therefore, the measures of the intercepted arcs are sometimes equal to each other.

30. OPEN ENDED Investigate Theorem 11.17 by drawing and labeling a circle that has a secant and a tangent intersecting outside the circle. Measure and Solve for x using the Quadratic Formula. label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: Sample answer: Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says

Therefore, the measure of the tangent segment is you receive an additional 20% off the already discounted price. How much will you pay for a ring about 2.4 centimeters. with an original price of $200? F $80 31. WRITING IN MATH Describe the relationship among segments in a circle when two secants G $96 intersect inside a circle. H $120 J $140 SOLUTION: Sample answer: The product of the parts on one SOLUTION: intersecting chord equals the product of the parts of the other chord.

32. is tangent to the circle, and R and S are points on Discount price = 200 – 80 = 120 the circle. What is the value of x to the nearest tenth?

Discount price = 120 – 24 = 96 So, the correct choice is G.

A 7.6 34. EXTENDED RESPONSE The degree measures of B 6.4 minor arc AC and major arc ADC are x and y, C 5.7 respectively. D 4.8 a. If m∠ABC = 70°, write two equations relating x and y. SOLUTION: b. Find x and y.

Solve for x using the Quadratic Formula.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 11.14, y – x = 140. Since length can not be negative, the value of x is b. 5.7. Therefore, the correct answer is C.

33. ALGEBRA A department store has all of its Therefore, x = 110° and y = 250°. jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already 35. SAT/ACT During the first two weeks of summer discounted price. How much will you pay for a ring vacation, Antonia earned $100 per week. During the with an original price of $200? next six weeks, she earned $150 per week. What

F $80 was her average weekly pay? A G $96 $50 B H $120 $112.50 J $140 C $125 D $135 SOLUTION: E $137.50

SOLUTION:

Discount price = 200 – 80 = 120 Simplify.

So, the correct choice is E. Discount price = 120 – 24 = 96 36. WEAVING Once yarn is woven from wool fibers, it So, the correct choice is G. is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that 34. EXTENDED RESPONSE The degree measures of the yarn appears to intersect itself at C, but in reality minor arc AC and major arc ADC are x and y, it does not. Use the information from the diagram to respectively. find a. If m∠ABC = 70°, write two equations relating x and y. b. Find x and y.

SOLUTION: First, use intersecting secants to find m∠GCD.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y

= 360, By vertical angles, m∠BCH = m∠GCD. So, and by Theorem 11.14, y – x = 140. b. m∠BCH = 39. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141. Therefore, x = 110° and y = 250°.

35. SAT/ACT During the first two weeks of summer Copy the figure shown and draw the common vacation, Antonia earned $100 per week. During the tangents. If no common tangent exists, state no next six weeks, she earned $150 per week. What common tangent. was her average weekly pay? A $50 B $112.50 C $125 37. D $135 SOLUTION: E $137.50 Four common tangents can be drawn to these two SOLUTION: circles.

Simplify.

So, the correct choice is E.

find SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for

SOLUTION: these two circles. First, use intersecting secants to find m∠GCD.

39.

SOLUTION: By vertical angles, m∠BCH = m∠GCD. So, Three common tangents can be drawn to these two m∠BCH = 39. circles. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no common tangent. 40. SOLUTION: 37. Two common tangents can be drawn to these two circles. SOLUTION: Four common tangents can be drawn to these two circles.

Write an equation in slope-intercept form of the line having the given slope and y-intercept. 41. m: 3, y-intercept: –4 SOLUTION: The slope-intercept form of a line of slope m and y- 38. intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. SOLUTION: So, the equation of the line is y = 3x – 4. Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at 42. m: 2, (0, 8) any point. Since a tangent must intersect the circle at SOLUTION: exactly one point, no common tangent exists for To find the slope-intercept form of the equation of these two circles. this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8). 39. SOLUTION: Three common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line. 40. Here, and (x1, y1) = (0, –6). SOLUTION: Two common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

3. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent. 4. SOLUTION:

1. SOLUTION: 5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. 2. Find the length of the diameter by first finding the SOLUTION: measure of ST.

The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that 3. segments that appear to be tangent are SOLUTION: tangent.

6. SOLUTION:

4. SOLUTION:

7. 5. SCIENCE A piece of broken pottery found at an SOLUTION: archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST. 8. SOLUTION:

The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are 9. tangent. SOLUTION:

6. SOLUTION:

10. SOLUTION:

7. SOLUTION:

11. SOLUTION:

Solve for x by using the Quadratic Formula.

8. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

12. SOLUTION:

9. SOLUTION:

10. SOLUTION: 13. SOLUTION:

Solve for x using the Quadratic Formula.

11.

SOLUTION: Since lengths cannot be negative, the value of x is about 7.4.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Solve for x by using the Quadratic Formula. Refer to the photo on Page 772.

SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the 12. diameter of the circle, first use Theorem 11.15 to find SOLUTION: the value of x.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

13. SOLUTION:

SOLUTION:

Solve for x using the Quadratic Formula.

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To Since lengths cannot be negative, the value of x is find the diameter of the cake, first use Theorem about 7.4. 11.15 to find x. 14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772. Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that SOLUTION: appear to be tangent are tangent.

16.

SOLUTION: Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x. Solve for x using the Quadratic Formula.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the Since lengths cannot be negative, the value of x is 6. dimensions of the remaining cake are shown below, what was the original diameter of the cake?

17. SOLUTION:

SOLUTION: Solve for x using the Quadratic Formula.

Since one of the perpendicular segments contains Since lengths cannot be negative, the value of x is the center of the cake, it will bisect the chord. To about 7.1. find the diameter of the cake, first use Theorem 11.15 to find x.

18. Add the two segments to find the diameter. SOLUTION: The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent 16. segments to the smaller circle. SOLUTION:

Find x using the intersecting secant and tangent Solve for x using the Quadratic Formula. segments to the larger circle.

Since lengths cannot be negative, the value of x is 6.

19. SOLUTION: Use the intersecting tangent and secant segments to 17. find a. SOLUTION:

Next, use the intersecting chords to find b. Solve for x using the Quadratic Formula.

Since lengths cannot be negative, the value of x is about 7.1.

20. SOLUTION: Use the intersecting tangent and secant segments to 18. find q. SOLUTION:

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle. Solve for r using the Quadratic Formula.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

19. 21. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to Use the intersecting secant segments to find c. find a.

Use the intersecting secant and tangent segments to

Next, use the intersecting chords to find b. find d.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter 20. of the tree. SOLUTION: Use the intersecting tangent and secant segments to find q.

SOLUTION:

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

Solve for r using the Quadratic Formula. Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d.

Since lengths cannot be negative, the value of r is So, the diameter is about 29.6 ft. about 1.8. Therefore, the values of the variables are q = 9 and r PROOF Prove each theorem. ≈ 1.8. 23. two-column proof of Theorem 11.15 Given: and intersect at B. Prove:

21.

SOLUTION: Use the intersecting secant segments to find c. SOLUTION: Proof: Statements (Reasons) 1. and intersect at B. (Given) Use the intersecting secant and tangent segments to 2. , (Inscribed that find d. intercept the same arc are 3. (AA Similarity)

4. (Definition of similar 11-7Therefore, Special Segments the values in of a theCircle variables are c ≈ 22.8 and d ≈ 16.9. 5. (Cross products)

22. INDIRECT MEASUREMENT Gwendolyn is 24. paragraph proof of Theorem 11.16 standing 16 feet from a giant sequoia tree and Chet is Given: Secants and standing next to the tree, as shown. The distance Prove: between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree.

SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that SOLUTION: intercept the same arc are congruent. So, . By AA Similarity, By the definition of similar triangles, Since the cross products of a proportion are equal,

Let d be the diameter of the tree. 25. two-column proof of Theorem 11.17 Use the intersecting tangent and secant segments to Given: tangent secant find d. Prove:

So, the diameter is about 29.6 ft.

PROOF Prove each theorem. SOLUTION: 23. two-column proof of Theorem 11.15 Proof: Given: and intersect at B. Statements (Reasons) Prove: 1. tangent and secant (Given) 2. (The measure of an inscribed ∠ equals half the measure of its intercept arc.)

3. (The measure of an ∠ formed SOLUTION: by a secant and a tangent = half the measure of its Proof: intercepted arc.) Statements (Reasons) 4. (Substitution)

1. and intersect at B. (Given) 5. (Definition of 2. , (Inscribed that 6. (Reflexive Property) intercept the same arc are 7. (AA Similarity) 3. (AA Similarity) 8. (Definition of 4. (Definition of similar 9. (Cross products) 5. (Cross products) 26. CRITIQUE Tiffany and Jun are finding the value of eSolutions24. paragraphManual proof- Powered of Theoremby Cognero 11.16 x in the figure. Tiffany wrote 3(5) = 2x, and Jun Page 6 Given: Secants and wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning. Prove:

SOLUTION: SOLUTION: Proof: Jun; the segments intersect outside of the circle, so the correct equation involves the products of the and are secant segments. By the Reflexive measures of a secant and its external secant Property, . Inscribed angles that segment. intercept the same arc are congruent. So, . By AA Similarity, 27. WRITING IN MATH Compare and contrast the By the definition of similar triangles, methods for finding measures of segments when two secants intersect in the exterior of a circle and when Since the cross products of a proportion a secant and a tangent intersect in the exterior of a circle. are equal, SOLUTION: 25. two-column proof of Theorem 11.17 Sample answer: When two secants intersect in the Given: tangent secant exterior of a circle, the product of the measures of Prove: one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant and a tangent intersect at an exterior point, the product of the measures of the secant segment and its external segment is equal to the square of the measure of the tangent segment, because for the SOLUTION: tangent the measures of the external segment and Proof: the whole segment are the same. Statements (Reasons) 28. CHALLENGE In the figure, a line tangent to circle 1. tangent and secant (Given) M and a secant line intersect at R. Find a. Show the 2. (The measure of an inscribed steps that you used. ∠ equals half the measure of its intercept arc.) 3. (The measure of an ∠ formed by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) SOLUTION: 5. (Definition of Use the intersecting tangent and secant segments to 6. (Reflexive Property) find the value of a. 7. (AA Similarity)

8. (Definition of

9. (Cross products)

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

29. REASONING When two chords intersect at the center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal to each other? SOLUTION: SOLUTION: Jun; the segments intersect outside of the circle, so Since the chords intersect at the center, the measure the correct equation involves the products of the of each intercepted arc is equal to the measure of its measures of a secant and its external secant related central angle. Since vertical angles are segment. congruent, the opposite arcs will be congruent. All four arcs will only be congruent when all four central 27. WRITING IN MATH Compare and contrast the angles are congruent. This will only happen when the methods for finding measures of segments when two chords are perpendicular and the angles are right secants intersect in the exterior of a circle and when angles a secant and a tangent intersect in the exterior of a Therefore, the measures of the intercepted arcs are circle. sometimes equal to each other.

SOLUTION: 30. OPEN ENDED Investigate Theorem 11.17 by Sample answer: When two secants intersect in the drawing and labeling a circle that has a secant and a exterior of a circle, the product of the measures of tangent intersecting outside the circle. Measure and one secant segment and its external segment is equal label the two parts of the secant segment to the to the product of the measures of the other secant nearest tenth of a centimeter. Use an equation to find segment and its external segment. When a secant the measure of the tangent segment. Verify your and a tangent intersect at an exterior point, the answer by measuring the segment. product of the measures of the secant segment and its external segment is equal to the square of the SOLUTION: measure of the tangent segment, because for the Sample answer: tangent the measures of the external segment and the whole segment are the same.

28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the steps that you used.

Therefore, the measure of the tangent segment is about 2.4 centimeters. SOLUTION: 31. WRITING IN MATH Describe the relationship Use the intersecting tangent and secant segments to among segments in a circle when two secants find the value of a. intersect inside a circle. SOLUTION: Sample answer: The product of the parts on one intersecting chord equals the product of the parts of the other chord.

32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest tenth?

29. REASONING When two chords intersect at the center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal to each other? A 7.6 SOLUTION: B 6.4 Since the chords intersect at the center, the measure C 5.7 of each intercepted arc is equal to the measure of its D 4.8 related central angle. Since vertical angles are congruent, the opposite arcs will be congruent. All SOLUTION: four arcs will only be congruent when all four central angles are congruent. This will only happen when the chords are perpendicular and the angles are right

angles Therefore, the measures of the intercepted arcs are Solve for x using the Quadratic Formula. sometimes equal to each other.

30. OPEN ENDED Investigate Theorem 11.17 by drawing and labeling a circle that has a secant and a tangent intersecting outside the circle. Measure and label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. Since length can not be negative, the value of x is 5.7. SOLUTION: Therefore, the correct answer is C. Sample answer:

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already discounted price. How much will you pay for a ring with an original price of $200? F $80 G $96 Therefore, the measure of the tangent segment is H $120 about 2.4 centimeters. J $140 31. WRITING IN MATH Describe the relationship SOLUTION: among segments in a circle when two secants intersect inside a circle. SOLUTION: Sample answer: The product of the parts on one Discount price = 200 – 80 = 120 intersecting chord equals the product of the parts of the other chord.

32. is tangent to the circle, and R and S are points on Discount price = 120 – 24 the circle. What is the value of x to the nearest = 96 tenth? So, the correct choice is G.

34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, respectively. a. If m∠ABC = 70°, write two equations relating x A 7.6 and y. B 6.4 b. Find x and y. C 5.7 D 4.8 SOLUTION:

SOLUTION: Solve for x using the Quadratic Formula. a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 11.14, y – x = 140. b.

Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C. Therefore, x = 110° and y = 250°.

35. SAT/ACT During the first two weeks of summer vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What 33. ALGEBRA A department store has all of its was her average weekly pay? jewelry discounted 40%. It is having a sale that says A $50 you receive an additional 20% off the already B discounted price. How much will you pay for a ring $112.50 with an original price of $200? C $125 F $80 D $135 G $96 E $137.50 H $120 SOLUTION: J $140 SOLUTION: Simplify.

So, the correct choice is E. Discount price = 200 – 80 = 120 36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality Discount price = 120 – 24 it does not. Use the information from the diagram to = 96

So, the correct choice is G. find

34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, respectively. a. If m∠ABC = 70°, write two equations relating x and y. b. Find x and y. SOLUTION: First, use intersecting secants to find m∠GCD.

By vertical angles, m∠BCH = m∠GCD. So, SOLUTION: m∠BCH = 39. Since major arc BAH has the same endpoints as a. Since the sum of the arcs of a circle is 360, x + y = 360, minor arc BH, m(arc BAH) = 360 – m(arc BH). and by Theorem 11.14, y – x = 140. b.

Therefore, the measure of arc BH is 141.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no

Therefore, x = 110° and y = 250°. common tangent.

35. SAT/ACT During the first two weeks of summer vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What 37. was her average weekly pay? SOLUTION: A $50 Four common tangents can be drawn to these two B $112.50 circles. C $125 D $135 E $137.50 SOLUTION:

Simplify.

So, the correct choice is E.

36. WEAVING Once yarn is woven from wool fibers, it 38. is often dyed and then threaded along a path of SOLUTION: pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality Every tangent drawn to the small circle will intersect it does not. Use the information from the diagram to the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at find any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles.

SOLUTION: First, use intersecting secants to find m∠GCD. 39. SOLUTION: Three common tangents can be drawn to these two circles.

By vertical angles, m∠BCH = m∠GCD. So, m∠BCH = 39. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141.

40. Copy the figure shown and draw the common SOLUTION: tangents. If no common tangent exists, state no common tangent. Two common tangents can be drawn to these two circles.

37. SOLUTION: Four common tangents can be drawn to these two

circles. Write an equation in slope-intercept form of the line having the given slope and y-intercept. 41. m: 3, y-intercept: –4 SOLUTION: The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. So, the equation of the line is y = 3x – 4.

38. 42. m: 2, (0, 8) SOLUTION: SOLUTION: Every tangent drawn to the small circle will intersect To find the slope-intercept form of the equation of the larger circle in two points. Every tangent drawn this line use the point-slope form of a line, y – y = m to the large circle will not intersect the small circle at 1 any point. Since a tangent must intersect the circle at (x – x1) where m is the slope and (x1, y1) is a point exactly one point, no common tangent exists for on the line. these two circles. Here, m = 2 and (x1, y1) = (0, 8).

39.

SOLUTION: Three common tangents can be drawn to these two 43. circles. SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

40. SOLUTION: Two common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent.

3. SOLUTION: 1. SOLUTION:

4. SOLUTION:

2. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

3. SOLUTION: SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

SOLUTION: Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the 6. original pottery? Round to the nearest hundredth. SOLUTION:

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST. 7. SOLUTION:

The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

8. SOLUTION: 6. SOLUTION:

9. SOLUTION: 7. SOLUTION:

10. SOLUTION:

8. SOLUTION:

11.

SOLUTION:

9. Solve for x by using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

10. SOLUTION:

12. SOLUTION:

11. SOLUTION:

13. SOLUTION: Solve for x by using the Quadratic Formula.

Solve for x using the Quadratic Formula.

Since lengths cannot be negative, the value of x is about 3.1.

Since lengths cannot be negative, the value of x is about 7.4. 12. 14. BRIDGES What is the diameter of the circle SOLUTION: containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

SOLUTION:

13. Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the SOLUTION: diameter of the circle, first use Theorem 11.15 to find the value of x.

Solve for x using the Quadratic Formula. Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

Since lengths cannot be negative, the value of x is about 7.4.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772. SOLUTION:

SOLUTION:

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To find the diameter of the cake, first use Theorem Since the height is perpendicular and bisects the 11.15 to find x. chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches. Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or CCSS STRUCTURE Find each variable to the about 588.1 meters. nearest tenth. Assume that segments that appear to be tangent are tangent. 15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

16. SOLUTION:

Solve for x using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is 6.

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To find the diameter of the cake, first use Theorem 11.15 to find x. 17. SOLUTION:

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the Solve for x using the Quadratic Formula. nearest tenth. Assume that segments that appear to be tangent are tangent.

16. Since lengths cannot be negative, the value of x is about 7.1. SOLUTION:

Solve for x using the Quadratic Formula. 18. SOLUTION:

Since lengths cannot be negative, the value of x is 6.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

17.

SOLUTION: Find x using the intersecting secant and tangent segments to the larger circle.

Solve for x using the Quadratic Formula.

Since lengths cannot be negative, the value of x is about 7.1. 19. SOLUTION: Use the intersecting tangent and secant segments to find a.

18. SOLUTION:

Next, use the intersecting chords to find b.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

20.

Find x using the intersecting secant and tangent SOLUTION:

segments to the larger circle. Use the intersecting tangent and secant segments to find q.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

19. SOLUTION: Solve for r using the Quadratic Formula. Use the intersecting tangent and secant segments to find a.

Next, use the intersecting chords to find b. Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

21. SOLUTION: 20. Use the intersecting secant segments to find c. SOLUTION: Use the intersecting tangent and secant segments to find q. Use the intersecting secant and tangent segments to find d.

Since lengths cannot be negative, the value of q is 9. Therefore, the values of the variables are c ≈ 22.8 Use the intersecting secant segments to find r. and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is Solve for r using the Quadratic Formula. standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8. SOLUTION:

21. SOLUTION: Use the intersecting secant segments to find c. Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d.

Use the intersecting secant and tangent segments to find d. So, the diameter is about 29.6 ft.

PROOF Prove each theorem. Therefore, the values of the variables are c ≈ 22.8 23. two-column proof of Theorem 11.15 and d ≈ 16.9. Given: and intersect at B. Prove: 22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree. SOLUTION: Proof: Statements (Reasons) 1. and intersect at B. (Given) 2. , (Inscribed that intercept the same arc are 3. (AA Similarity) SOLUTION: 4. (Definition of similar 5. (Cross products)

24. paragraph proof of Theorem 11.16

Given: Secants and Let d be the diameter of the tree. Prove: Use the intersecting tangent and secant segments to find d.

SOLUTION:

So, the diameter is about 29.6 ft. Proof: PROOF Prove each theorem. and are secant segments. By the Reflexive 23. two-column proof of Theorem 11.15 Property, . Inscribed angles that Given: and intersect at B. intercept the same arc are congruent. So, Prove: . By AA Similarity, By the definition of similar triangles, Since the cross products of a proportion are equal,

25. two-column proof of Theorem 11.17 SOLUTION:

Proof: Given: tangent secant Statements (Reasons) Prove: 1. and intersect at B. (Given) 2. , (Inscribed that intercept the same arc are 3. (AA Similarity)

4. (Definition of similar SOLUTION: 5. (Cross products) Proof: Statements (Reasons)

24. paragraph proof of Theorem 11.16 1. tangent and secant (Given) Given: Secants and 2. (The measure of an inscribed Prove: ∠ equals half the measure of its intercept arc.) 3. (The measure of an ∠ formed by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) SOLUTION: 5. (Definition of Proof: 6. (Reflexive Property) and are secant segments. By the Reflexive 7. (AA Similarity) Property, . Inscribed angles that intercept the same arc are congruent. So, 8. (Definition of . By AA Similarity, By the definition of similar triangles, 9. (Cross products) Since the cross products of a proportion 26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun are equal, wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning. 25. two-column proof of Theorem 11.17 Given: tangent secant Prove:

SOLUTION: Jun; the segments intersect outside of the circle, so the correct equation involves the products of the SOLUTION: measures of a secant and its external secant segment. Proof:

Statements (Reasons) 27. WRITING IN MATH Compare and contrast the 1. tangent and secant (Given) methods for finding measures of segments when two secants intersect in the exterior of a circle and when

2. (The measure of an inscribed a secant and a tangent intersect in the exterior of a ∠ equals half the measure of its intercept arc.) circle. SOLUTION: 3. (The measure of an ∠ formed Sample answer: When two secants intersect in the by a secant and a tangent = half the measure of its exterior of a circle, the product of the measures of intercepted arc.) one secant segment and its external segment is equal 4. (Substitution) to the product of the measures of the other secant 5. (Definition of segment and its external segment. When a secant 6. (Reflexive Property) and a tangent intersect at an exterior point, the 7. (AA Similarity) product of the measures of the secant segment and its external segment is equal to the square of the 8. (Definition of measure of the tangent segment, because for the 11-7 Special Segments in a Circle tangent the measures of the external segment and 9. (Cross products) the whole segment are the same.

26. CRITIQUE Tiffany and Jun are finding the value of 28. CHALLENGE In the figure, a line tangent to circle x in the figure. Tiffany wrote 3(5) = 2x, and Jun M and a secant line intersect at R. Find a. Show the wrote 3(8) = 2(2 + x). Is either of them correct? steps that you used. Explain your reasoning.

SOLUTION: SOLUTION: Jun; the segments intersect outside of the circle, so Use the intersecting tangent and secant segments to the correct equation involves the products of the find the value of a. measures of a secant and its external secant segment.

SOLUTION: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of 29. REASONING When two chords intersect at the one secant segment and its external segment is equal center of a circle, are the measures of the to the product of the measures of the other secant intercepting arcs sometimes, always, or never equal segment and its external segment. When a secant to each other? and a tangent intersect at an exterior point, the SOLUTION: product of the measures of the secant segment and Since the chords intersect at the center, the measure its external segment is equal to the square of the of each intercepted arc is equal to the measure of its measure of the tangent segment, because for the related central angle. Since vertical angles are tangent the measures of the external segment and congruent, the opposite arcs will be congruent. All the whole segment are the same. four arcs will only be congruent when all four central 28. CHALLENGE In the figure, a line tangent to circle angles are congruent. This will only happen when the M and a secant line intersect at R. Find a. Show the chords are perpendicular and the angles are right

steps that you used. angles Therefore, the measures of the intercepted arcs are sometimes equal to each other.

30. OPEN ENDED Investigate Theorem 11.17 by drawing and labeling a circle that has a secant and a tangent intersecting outside the circle. Measure and label the two parts of the secant segment to the SOLUTION: nearest tenth of a centimeter. Use an equation to find Use the intersecting tangent and secant segments to the measure of the tangent segment. Verify your find the value of a. answer by measuring the segment. SOLUTION: Sample answer:

eSolutions Manual - Powered by Cognero Page 7

29. REASONING When two chords intersect at the center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal Therefore, the measure of the tangent segment is to each other? about 2.4 centimeters. SOLUTION: WRITING IN MATH Since the chords intersect at the center, the measure 31. Describe the relationship of each intercepted arc is equal to the measure of its among segments in a circle when two secants

related central angle. Since vertical angles are intersect inside a circle. congruent, the opposite arcs will be congruent. All SOLUTION: four arcs will only be congruent when all four central Sample answer: The product of the parts on one angles are congruent. This will only happen when the intersecting chord equals the product of the parts of chords are perpendicular and the angles are right the other chord. angles Therefore, the measures of the intercepted arcs are 32. is tangent to the circle, and R and S are points on sometimes equal to each other. the circle. What is the value of x to the nearest tenth? 30. OPEN ENDED Investigate Theorem 11.17 by drawing and labeling a circle that has a secant and a tangent intersecting outside the circle. Measure and label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your A 7.6 answer by measuring the segment. B 6.4 SOLUTION: C 5.7 Sample answer: D 4.8 SOLUTION:

Solve for x using the Quadratic Formula.

Therefore, the measure of the tangent segment is about 2.4 centimeters.

31. WRITING IN MATH Describe the relationship among segments in a circle when two secants Since length can not be negative, the value of x is intersect inside a circle. 5.7. SOLUTION: Therefore, the correct answer is C. Sample answer: The product of the parts on one intersecting chord equals the product of the parts of the other chord.

32. is tangent to the circle, and R and S are points on 33. ALGEBRA A department store has all of its the circle. What is the value of x to the nearest jewelry discounted 40%. It is having a sale that says tenth? you receive an additional 20% off the already discounted price. How much will you pay for a ring with an original price of $200? F $80 G $96 H $120 A 7.6 J $140 B 6.4 SOLUTION: C 5.7 D 4.8 SOLUTION: Discount price = 200 – 80 = 120

Solve for x using the Quadratic Formula. Discount price = 120 – 24 = 96 So, the correct choice is G.

34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, respectively. a. If m∠ABC = 70°, write two equations relating x Since length can not be negative, the value of x is and y. 5.7. b. Find x and y. Therefore, the correct answer is C.

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already SOLUTION: discounted price. How much will you pay for a ring a. Since the sum of the arcs of a circle is 360, x + y with an original price of $200? = 360, F $80 and by Theorem 11.14, y – x = 140. G $96 b. H $120 J $140 SOLUTION:

Therefore, x = 110° and y = 250°. Discount price = 200 – 80 = 120 35. SAT/ACT During the first two weeks of summer vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What Discount price = 120 – 24 was her average weekly pay? = 96 A $50 So, the correct choice is G. B $112.50 34. EXTENDED RESPONSE The degree measures of C $125 minor arc AC and major arc ADC are x and y, D $135 respectively. E $137.50 a. If m∠ABC = 70°, write two equations relating x SOLUTION: and y.

b. Find x and y. Simplify.

So, the correct choice is E.

36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that SOLUTION: the yarn appears to intersect itself at C, but in reality a. Since the sum of the arcs of a circle is 360, x + y it does not. Use the information from the diagram to = 360, find and by Theorem 11.14, y – x = 140. b.

SOLUTION: First, use intersecting secants to find m∠GCD. Therefore, x = 110° and y = 250°.

35. SAT/ACT During the first two weeks of summer vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What was her average weekly pay? By vertical angles, m∠BCH = m∠GCD. So, A $50 m∠BCH = 39. B $112.50 Since major arc BAH has the same endpoints as C $125 minor arc BH, m(arc BAH) = 360 – m(arc BH). D $135 E $137.50 SOLUTION: Therefore, the measure of arc BH is 141.

Simplify. Copy the figure shown and draw the common tangents. If no common tangent exists, state no So, the correct choice is E. common tangent. 36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that 37. the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to SOLUTION: find Four common tangents can be drawn to these two circles.

SOLUTION:

First, use intersecting secants to find m∠GCD.

38. By vertical angles, m∠BCH = m∠GCD. So, SOLUTION: m∠BCH = 39. Every tangent drawn to the small circle will intersect Since major arc BAH has the same endpoints as the larger circle in two points. Every tangent drawn minor arc BH, m(arc BAH) = 360 – m(arc BH). to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles.

Therefore, the measure of arc BH is 141.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no 39. common tangent. SOLUTION: Three common tangents can be drawn to these two circles. 37. SOLUTION: Four common tangents can be drawn to these two circles.

40. SOLUTION: Two common tangents can be drawn to these two circles. 38. SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles. Write an equation in slope-intercept form of the line having the given slope and y-intercept. 41. m: 3, y-intercept: –4 SOLUTION: 39. The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. SOLUTION: For this line, substitute m = 3 and b = –4. Three common tangents can be drawn to these two So, the equation of the line is y = 3x – 4. circles. 42. m: 2, (0, 8) SOLUTION: To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m (x – x ) where m is the slope and (x , y ) is a point 1 1 1 on the line. Here, m = 2 and (x1, y1) = (0, 8).

40. SOLUTION:

Two common tangents can be drawn to these two circles. 43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point Write an equation in slope-intercept form of the on the line. line having the given slope and y-intercept. 41. m: 3, y-intercept: –4 Here, and (x1, y1) = (0, –6). SOLUTION: The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. So, the equation of the line is y = 3x – 4.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent.

3. 1. SOLUTION: SOLUTION:

4. 2. SOLUTION: SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

3. SOLUTION: SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or . 4. SOLUTION:

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

7. SOLUTION: The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. 8. SOLUTION: SOLUTION:

7. 9. SOLUTION: SOLUTION:

10. SOLUTION:

8. SOLUTION:

11. SOLUTION:

9. SOLUTION: Solve for x by using the Quadratic Formula.

Since lengths cannot be negative, the value of x is 10. about 3.1. SOLUTION:

12. SOLUTION:

11. SOLUTION:

Solve for x by using the Quadratic Formula. 13. SOLUTION:

Since lengths cannot be negative, the value of x is Solve for x using the Quadratic Formula. about 3.1.

12. Since lengths cannot be negative, the value of x is SOLUTION: about 7.4.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

SOLUTION:

13. SOLUTION: Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Solve for x using the Quadratic Formula.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

SOLUTION: SOLUTION:

Add the two segments to find the diameter. Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or

about 588.1 meters. The diameter of the circle is 9 + 4 or 13 inches. CCSS STRUCTURE Find each variable to the 15. CAKES Sierra is serving cake at a party. If the nearest tenth. Assume that segments that dimensions of the remaining cake are shown below, appear to be tangent are tangent. what was the original diameter of the cake?

16. SOLUTION:

SOLUTION: Solve for x using the Quadratic Formula.

17. Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches. SOLUTION:

CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent. Solve for x using the Quadratic Formula.

16. SOLUTION: Since lengths cannot be negative, the value of x is about 7.1.

Solve for x using the Quadratic Formula.

18. SOLUTION:

Since lengths cannot be negative, the value of x is 6.

First, find the length, a, of the tangent segment to 17. both circles using the intersecting secant and tangent segments to the smaller circle. SOLUTION:

Find x using the intersecting secant and tangent Solve for x using the Quadratic Formula. segments to the larger circle.

Since lengths cannot be negative, the value of x is about 7.1.

19. SOLUTION: Use the intersecting tangent and secant segments to 18. find a. SOLUTION:

Next, use the intersecting chords to find b.

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

Find x using the intersecting secant and tangent segments to the larger circle. 20. SOLUTION: Use the intersecting tangent and secant segments to find q.

19. Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r. SOLUTION: Use the intersecting tangent and secant segments to find a. Solve for r using the Quadratic Formula.

Next, use the intersecting chords to find b.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

20. 21. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to Use the intersecting secant segments to find c. find q.

Use the intersecting secant and tangent segments to find d.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

Solve for r using the Quadratic Formula. 22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

SOLUTION:

21. SOLUTION: Use the intersecting secant segments to find c.

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to Use the intersecting secant and tangent segments to find d. find d.

Therefore, the values of the variables are c ≈ 22.8 So, the diameter is about 29.6 ft. and d ≈ 16.9. PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is Given: and intersect at B. standing next to the tree, as shown. The distance Prove: between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree.

SOLUTION: Proof: Statements (Reasons) 1. and intersect at B. (Given)

2. , (Inscribed that SOLUTION: intercept the same arc are 3. (AA Similarity)

4. (Definition of similar 5. (Cross products)

Let d be the diameter of the tree. 24. paragraph proof of Theorem 11.16 Use the intersecting tangent and secant segments to Given: Secants and find d. Prove:

So, the diameter is about 29.6 ft. PROOF Prove each theorem. SOLUTION: 23. two-column proof of Theorem 11.15 Proof:

Given: and intersect at B. and are secant segments. By the Reflexive Prove: Property, . Inscribed angles that intercept the same arc are congruent. So, . By AA Similarity, By the definition of similar triangles, Since the cross products of a proportion SOLUTION: are equal, Proof: Statements (Reasons) 25. two-column proof of Theorem 11.17 1. and intersect at B. (Given) Given: tangent secant 2. , (Inscribed that Prove: intercept the same arc are 3. (AA Similarity)

4. (Definition of similar 5. (Cross products) SOLUTION: 24. paragraph proof of Theorem 11.16 Proof: Given: Secants and Statements (Reasons) Prove: 1. tangent and secant (Given) 2. (The measure of an inscribed ∠ equals half the measure of its intercept arc.) 3. (The measure of an ∠ formed SOLUTION: by a secant and a tangent = half the measure of its Proof: intercepted arc.) and are secant segments. By the Reflexive 4. (Substitution) Property, . Inscribed angles that 5. (Definition of intercept the same arc are congruent. So, 6. (Reflexive Property) . By AA Similarity, 7. (AA Similarity)

By the definition of similar triangles, 8. (Definition of Since the cross products of a proportion 9. (Cross products) are equal, 26. CRITIQUE Tiffany and Jun are finding the value of 25. two-column proof of Theorem 11.17 x in the figure. Tiffany wrote 3(5) = 2x, and Jun Given: tangent secant wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning. Prove:

SOLUTION: SOLUTION: Jun; the segments intersect outside of the circle, so Proof: the correct equation involves the products of the Statements (Reasons) measures of a secant and its external secant 1. tangent and secant (Given) segment. 2. (The measure of an inscribed 27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two ∠ equals half the measure of its intercept arc.) secants intersect in the exterior of a circle and when 3. (The measure of an ∠ formed a secant and a tangent intersect in the exterior of a circle. by a secant and a tangent = half the measure of its intercepted arc.) SOLUTION: 4. (Substitution) Sample answer: When two secants intersect in the 5. (Definition of exterior of a circle, the product of the measures of one secant segment and its external segment is equal 6. (Reflexive Property) to the product of the measures of the other secant 7. (AA Similarity) segment and its external segment. When a secant and a tangent intersect at an exterior point, the 8. (Definition of product of the measures of the secant segment and 9. (Cross products) its external segment is equal to the square of the measure of the tangent segment, because for the 26. CRITIQUE Tiffany and Jun are finding the value of tangent the measures of the external segment and x in the figure. Tiffany wrote 3(5) = 2x, and Jun the whole segment are the same. wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning. 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION: Jun; the segments intersect outside of the circle, so the correct equation involves the products of the SOLUTION: measures of a secant and its external secant Use the intersecting tangent and secant segments to segment. find the value of a. 27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two secants intersect in the exterior of a circle and when a secant and a tangent intersect in the exterior of a circle. SOLUTION:

Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant 29. REASONING When two chords intersect at the segment and its external segment. When a secant center of a circle, are the measures of the and a tangent intersect at an exterior point, the intercepting arcs sometimes, always, or never equal product of the measures of the secant segment and to each other? its external segment is equal to the square of the measure of the tangent segment, because for the SOLUTION: tangent the measures of the external segment and Since the chords intersect at the center, the measure the whole segment are the same. of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are 28. CHALLENGE In the figure, a line tangent to circle congruent, the opposite arcs will be congruent. All M and a secant line intersect at R. Find a. Show the four arcs will only be congruent when all four central steps that you used. angles are congruent. This will only happen when the chords are perpendicular and the angles are right angles Therefore, the measures of the intercepted arcs are sometimes equal to each other.

29. REASONING When two chords intersect at the center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal to each other?

SOLUTION: Therefore, the measure of the tangent segment is Since the chords intersect at the center, the measure about 2.4 centimeters. of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are 31. WRITING IN MATH Describe the relationship congruent, the opposite arcs will be congruent. All among segments in a circle when two secants four arcs will only be congruent when all four central intersect inside a circle. angles are congruent. This will only happen when the chords are perpendicular and the angles are right SOLUTION: angles Sample answer: The product of the parts on one 11-7Therefore, Special Segments the measures in a ofCircle the intercepted arcs are intersecting chord equals the product of the parts of sometimes equal to each other. the other chord.

30. OPEN ENDED Investigate Theorem 11.17 by 32. is tangent to the circle, and R and S are points on drawing and labeling a circle that has a secant and a the circle. What is the value of x to the nearest tangent intersecting outside the circle. Measure and tenth? label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: A 7.6 Sample answer: B 6.4 C 5.7 D 4.8 SOLUTION:

Solve for x using the Quadratic Formula.

Therefore, the measure of the tangent segment is about 2.4 centimeters.

31. WRITING IN MATH Describe the relationship among segments in a circle when two secants intersect inside a circle.

SOLUTION: Since length can not be negative, the value of x is Sample answer: The product of the parts on one 5.7. intersecting chord equals the product of the parts of Therefore, the correct answer is C. the other chord.

32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest

tenth? 33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already discounted price. How much will you pay for a ring with an original price of $200? F $80 A 7.6 G $96 B 6.4 H $120 C 5.7 J $140 D 4.8 SOLUTION: SOLUTION:

Discount price = 200 – 80 = 120

Solve for x using the Quadratic Formula. eSolutions Manual - Powered by Cognero Discount price = 120 – 24 Page 8 = 96 So, the correct choice is G.

34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, Since length can not be negative, the value of x is respectively. 5.7. a. If m∠ABC = 70°, write two equations relating x Therefore, the correct answer is C. and y. b. Find x and y.

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already discounted price. How much will you pay for a ring with an original price of $200? F $80 SOLUTION: G $96 a. Since the sum of the arcs of a circle is 360, x + y H $120 = 360, J $140 and by Theorem 11.14, y – x = 140. b. SOLUTION:

Discount price = 200 – 80 = 120

Therefore, x = 110° and y = 250°.

Discount price = 120 – 24 35. SAT/ACT During the first two weeks of summer = 96 vacation, Antonia earned $100 per week. During the So, the correct choice is G. next six weeks, she earned $150 per week. What was her average weekly pay? 34. EXTENDED RESPONSE The degree measures of A $50 minor arc AC and major arc ADC are x and y, B $112.50 respectively. C $125 a. If m∠ABC = 70°, write two equations relating x D $135

and y. E $137.50 b. Find x and y. SOLUTION:

Simplify.

So, the correct choice is E.

SOLUTION: 36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of a . Since the sum of the arcs of a circle is 360, x + y pulleys to dry. One set of pulleys is shown. Note that

= 360, the yarn appears to intersect itself at C, but in reality

and by Theorem 11.14, y – x = 140. it does not. Use the information from the diagram to b. find

B $112.50 By vertical angles, m∠BCH = m∠GCD. So, C $125 m BCH = 39. D $135 ∠ Since major arc BAH has the same endpoints as E $137.50 minor arc BH, m(arc BAH) = 360 – m(arc BH). SOLUTION:

Simplify. Therefore, the measure of arc BH is 141.

So, the correct choice is E. Copy the figure shown and draw the common 36. WEAVING Once yarn is woven from wool fibers, it tangents. If no common tangent exists, state no is often dyed and then threaded along a path of common tangent. pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to find 37. SOLUTION: Four common tangents can be drawn to these two circles.

SOLUTION: First, use intersecting secants to find m∠GCD.

By vertical angles, m∠BCH = m∠GCD. So, m∠BCH = 39. 38. Since major arc BAH has the same endpoints as SOLUTION: minor arc BH, m(arc BAH) = 360 – m(arc BH). Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at Therefore, the measure of arc BH is 141. exactly one point, no common tangent exists for these two circles.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no common tangent. 39. SOLUTION: 37. Three common tangents can be drawn to these two SOLUTION: circles. Four common tangents can be drawn to these two circles.

40.

38. SOLUTION: Two common tangents can be drawn to these two SOLUTION: circles. Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles.

42. m: 2, (0, 8) SOLUTION: To find the slope-intercept form of the equation of

this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8). 40. SOLUTION: Two common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of Write an equation in slope-intercept form of the this line use the point-slope form of a line, y – y1 = m line having the given slope and y-intercept. (x – x ) where m is the slope and (x , y ) is a point 41. m: 3, y-intercept: –4 1 1 1 on the line. SOLUTION: The slope-intercept form of a line of slope m and y- Here, and (x1, y1) = (0, –6). intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. So, the equation of the line is y = 3x – 4.

42. m: 2, (0, 8)

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

3. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent.

4. SOLUTION:

1. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

SOLUTION: 2. Let be the diameter of the circle. Then and SOLUTION: are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

SOLUTION: Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. SOLUTION:

SOLUTION:

5. SCIENCE A piece of broken pottery found at an

archaeological site is shown. lies on a diameter 7. of the circle. What was the circumference of the SOLUTION: original pottery? Round to the nearest hundredth.

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

8. SOLUTION:

The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

9. SOLUTION:

6. SOLUTION:

10. SOLUTION:

SOLUTION:

11. SOLUTION:

Solve for x by using the Quadratic Formula.

8. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

12. 9. SOLUTION: SOLUTION:

10. SOLUTION:

13. SOLUTION:

Solve for x using the Quadratic Formula.

11. SOLUTION:

Since lengths cannot be negative, the value of x is about 7.4.

Solve for x by using the Quadratic Formula. 14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

Since lengths cannot be negative, the value of x is SOLUTION: about 3.1.

12. SOLUTION: Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

13. SOLUTION:

Solve for x using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is Since one of the perpendicular segments contains about 7.4. the center of the cake, it will bisect the chord. To 14. BRIDGES What is the diameter of the circle find the diameter of the cake, first use Theorem containing the arc of the Sydney Harbour Bridge? 11.15 to find x. Round to the nearest tenth. Refer to the photo on Page 772.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches. SOLUTION: CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

Since the height is perpendicular and bisects the 16. chord, it must be part of the diameter. To find the SOLUTION: diameter of the circle, first use Theorem 11.15 to find the value of x.

Solve for x using the Quadratic Formula.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake? Since lengths cannot be negative, the value of x is 6.

17.

SOLUTION: SOLUTION:

Solve for x using the Quadratic Formula.

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To

find the diameter of the cake, first use Theorem Since lengths cannot be negative, the value of x is 11.15 to find x. about 7.1.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches. 18. CCSS STRUCTURE Find each variable to the SOLUTION: nearest tenth. Assume that segments that appear to be tangent are tangent.

16. SOLUTION: First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

Solve for x using the Quadratic Formula.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of x is 6.

19. 17. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find a.

Solve for x using the Quadratic Formula.

Next, use the intersecting chords to find b.

Since lengths cannot be negative, the value of x is about 7.1.

20. 18. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find q.

Since lengths cannot be negative, the value of q is 9. First, find the length, a, of the tangent segment to Use the intersecting secant segments to find r. both circles using the intersecting secant and tangent segments to the smaller circle.

Solve for r using the Quadratic Formula.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

19. SOLUTION: Use the intersecting tangent and secant segments to 21. find a. SOLUTION: Use the intersecting secant segments to find c.

Next, use the intersecting chords to find b. Use the intersecting secant and tangent segments to find d.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance 20. between Gwendolyn and Chet is 27 feet. Draw a SOLUTION: diagram of this situation, and then find the diameter

Use the intersecting tangent and secant segments to of the tree. find q.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r. SOLUTION:

Solve for r using the Quadratic Formula.

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8. So, the diameter is about 29.6 ft. PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Given: and intersect at B. Prove: 21. SOLUTION: Use the intersecting secant segments to find c.

SOLUTION:

Use the intersecting secant and tangent segments to Proof: Statements (Reasons) find d. 1. and intersect at B. (Given) 2. , (Inscribed that intercept the same arc are Therefore, the values of the variables are c ≈ 22.8 3. (AA Similarity) and d ≈ 16.9. 4. (Definition of similar 22. INDIRECT MEASUREMENT Gwendolyn is 5. (Cross products) standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a 24. paragraph proof of Theorem 11.16 diagram of this situation, and then find the diameter Given: Secants and of the tree. Prove:

SOLUTION: Proof: SOLUTION: and are secant segments. By the Reflexive Property, . Inscribed angles that intercept the same arc are congruent. So, . By AA Similarity, By the definition of similar triangles, Since the cross products of a proportion Let d be the diameter of the tree. Use the intersecting tangent and secant segments to are equal, find d. 25. two-column proof of Theorem 11.17 Given: tangent secant Prove:

So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15

Given: and intersect at B. Prove: SOLUTION: Proof: Statements (Reasons) 1. tangent and secant (Given)

2. (The measure of an inscribed SOLUTION: ∠ equals half the measure of its intercept arc.) Proof: 3. (The measure of an ∠ formed Statements (Reasons) 1. and intersect at B. (Given) by a secant and a tangent = half the measure of its 2. , (Inscribed that intercepted arc.) intercept the same arc are 4. (Substitution) 5. (Definition of 3. (AA Similarity) 6. (Reflexive Property) 4. (Definition of similar 7. (AA Similarity)

5. (Cross products) 8. (Definition of

24. paragraph proof of Theorem 11.16 9. (Cross products)

Given: Secants and 26. CRITIQUE Tiffany and Jun are finding the value of Prove: x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

SOLUTION: Proof: and are secant segments. By the Reflexive SOLUTION: Property, . Inscribed angles that Jun; the segments intersect outside of the circle, so intercept the same arc are congruent. So, the correct equation involves the products of the . By AA Similarity, measures of a secant and its external secant segment. By the definition of similar triangles, Since the cross products of a proportion 27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two are equal, secants intersect in the exterior of a circle and when a secant and a tangent intersect in the exterior of a 25. two-column proof of Theorem 11.17 circle. Given: tangent secant SOLUTION:

Prove: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant and a tangent intersect at an exterior point, the product of the measures of the secant segment and SOLUTION: its external segment is equal to the square of the Proof: measure of the tangent segment, because for the Statements (Reasons) tangent the measures of the external segment and 1. tangent and secant (Given) the whole segment are the same.

2. (The measure of an inscribed 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the ∠ equals half the measure of its intercept arc.) steps that you used. 3. (The measure of an ∠ formed by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property) SOLUTION: 7. (AA Similarity) Use the intersecting tangent and secant segments to

8. (Definition of find the value of a.

9. (Cross products)

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

29. REASONING When two chords intersect at the center of a circle, are the measures of the SOLUTION: intercepting arcs sometimes, always, or never equal Jun; the segments intersect outside of the circle, so to each other? the correct equation involves the products of the SOLUTION: measures of a secant and its external secant Since the chords intersect at the center, the measure

segment. of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are 27. WRITING IN MATH Compare and contrast the congruent, the opposite arcs will be congruent. All methods for finding measures of segments when two four arcs will only be congruent when all four central secants intersect in the exterior of a circle and when angles are congruent. This will only happen when the a secant and a tangent intersect in the exterior of a chords are perpendicular and the angles are right circle. angles SOLUTION: Therefore, the measures of the intercepted arcs are Sample answer: When two secants intersect in the sometimes equal to each other. exterior of a circle, the product of the measures of one secant segment and its external segment is equal 30. OPEN ENDED Investigate Theorem 11.17 by to the product of the measures of the other secant drawing and labeling a circle that has a secant and a segment and its external segment. When a secant tangent intersecting outside the circle. Measure and and a tangent intersect at an exterior point, the label the two parts of the secant segment to the product of the measures of the secant segment and nearest tenth of a centimeter. Use an equation to find its external segment is equal to the square of the the measure of the tangent segment. Verify your measure of the tangent segment, because for the answer by measuring the segment. tangent the measures of the external segment and SOLUTION:

the whole segment are the same. Sample answer: 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION: Therefore, the measure of the tangent segment is Use the intersecting tangent and secant segments to about 2.4 centimeters. find the value of a. 31. WRITING IN MATH Describe the relationship among segments in a circle when two secants intersect inside a circle. SOLUTION: Sample answer: The product of the parts on one intersecting chord equals the product of the parts of the other chord.

32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest REASONING 29. When two chords intersect at the tenth? center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal to each other? SOLUTION: Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its A 7.6 related central angle. Since vertical angles are B 6.4 congruent, the opposite arcs will be congruent. All C 5.7 four arcs will only be congruent when all four central D 4.8 angles are congruent. This will only happen when the chords are perpendicular and the angles are right SOLUTION: angles Therefore, the measures of the intercepted arcs are sometimes equal to each other.

30. OPEN ENDED Investigate Theorem 11.17 by Solve for x using the Quadratic Formula. drawing and labeling a circle that has a secant and a tangent intersecting outside the circle. Measure and label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. SOLUTION: Since length can not be negative, the value of x is Sample answer: 5.7. Therefore, the correct answer is C.

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already discounted price. How much will you pay for a ring Therefore, the measure of the tangent segment is with an original price of $200? about 2.4 centimeters. F $80 G $96 31. WRITING IN MATH Describe the relationship H $120 among segments in a circle when two secants J $140 intersect inside a circle. SOLUTION: SOLUTION: Sample answer: The product of the parts on one intersecting chord equals the product of the parts of

the other chord. Discount price = 200 – 80 = 120 32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest tenth? Discount price = 120 – 24 = 96 So, the correct choice is G.

34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, A 7.6 respectively. B 6.4 a. If m∠ABC = 70°, write two equations relating x C 5.7 and y. D 4.8 b. Find x and y. SOLUTION:

Solve for x using the Quadratic Formula.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 11.14, y – x = 140. b.

Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

11-7 Special Segments in a Circle Therefore, x = 110° and y = 250°.

33. ALGEBRA A department store has all of its 35. SAT/ACT During the first two weeks of summer jewelry discounted 40%. It is having a sale that says vacation, Antonia earned $100 per week. During the you receive an additional 20% off the already next six weeks, she earned $150 per week. What discounted price. How much will you pay for a ring was her average weekly pay? with an original price of $200? A $50 F $80 B $112.50 G $96 C $125 H $120 D $135 J $140 E $137.50 SOLUTION: SOLUTION:

Simplify. Discount price = 200 – 80 = 120 So, the correct choice is E.

36. WEAVING Once yarn is woven from wool fibers, it Discount price = 120 – 24 is often dyed and then threaded along a path of = 96 pulleys to dry. One set of pulleys is shown. Note that So, the correct choice is G. the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to 34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, find respectively. a. If m∠ABC = 70°, write two equations relating x and y. b. Find x and y.

SOLUTION: First, use intersecting secants to find m∠GCD.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y By vertical angles, m∠BCH = m∠GCD. So, = 360, m∠BCH = 39. and by Theorem 11.14, y – x = 140. Since major arc BAH has the same endpoints as b. minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141.

Therefore, x = 110° and y = 250°. Copy the figure shown and draw the common tangents. If no common tangent exists, state no SAT/ACT 35. During the first two weeks of summer common tangent. vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What eSolutions Manual - Powered by Cognero Page 9 was her average weekly pay? A $50 37. B $112.50 C $125 SOLUTION: D $135 Four common tangents can be drawn to these two

E $137.50 circles. SOLUTION:

Simplify.

So, the correct choice is E.

36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that 38. the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to SOLUTION: find Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles.

SOLUTION: First, use intersecting secants to find m∠GCD.

39. SOLUTION: Three common tangents can be drawn to these two By vertical angles, m∠BCH = m∠GCD. So, circles. m∠BCH = 39. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no 40. common tangent. SOLUTION: Two common tangents can be drawn to these two circles. 37. SOLUTION: Four common tangents can be drawn to these two circles.

Write an equation in slope-intercept form of the line having the given slope and y-intercept. 41. m: 3, y-intercept: –4 SOLUTION: The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. For this line, substitute m = 3 and b = –4. 38. So, the equation of the line is y = 3x – 4. SOLUTION: Every tangent drawn to the small circle will intersect 42. m: 2, (0, 8) the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at SOLUTION: any point. Since a tangent must intersect the circle at To find the slope-intercept form of the equation of exactly one point, no common tangent exists for this line use the point-slope form of a line, y – y1 = m these two circles. (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8).

39.

SOLUTION: Three common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6). 40. SOLUTION: Two common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

3. SOLUTION:

Find x. Assume that segments that appear to be tangent are tangent.

4. 1. SOLUTION: SOLUTION:

5. SCIENCE A piece of broken pottery found at an

archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

2. SOLUTION: SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

3. SOLUTION:

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. SOLUTION: 4. SOLUTION:

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or . SOLUTION:

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

9. SOLUTION: 6. SOLUTION:

10. SOLUTION:

7. SOLUTION:

11. SOLUTION:

8. Solve for x by using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is about 3.1.

9. 12. SOLUTION: SOLUTION:

10. SOLUTION:

13. SOLUTION:

Solve for x using the Quadratic Formula. 11. SOLUTION:

Since lengths cannot be negative, the value of x is Solve for x by using the Quadratic Formula. about 7.4.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

Since lengths cannot be negative, the value of x is about 3.1. SOLUTION:

12. SOLUTION:

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake? 13. SOLUTION:

Solve for x using the Quadratic Formula. SOLUTION:

Since lengths cannot be negative, the value of x is about 7.4.

Since one of the perpendicular segments contains 14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? the center of the cake, it will bisect the chord. To Round to the nearest tenth. find the diameter of the cake, first use Theorem Refer to the photo on Page 772. 11.15 to find x.

Add the two segments to find the diameter. SOLUTION: The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find 16. the value of x. SOLUTION:

Add the two segments to find the diameter. Solve for x using the Quadratic Formula. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

Since lengths cannot be negative, the value of x is 6.

17. SOLUTION: SOLUTION:

Solve for x using the Quadratic Formula.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the

nearest tenth. Assume that segments that 18. appear to be tangent are tangent. SOLUTION:

16. SOLUTION:

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle. Solve for x using the Quadratic Formula.

Find x using the intersecting secant and tangent segments to the larger circle.

Since lengths cannot be negative, the value of x is 6.

17. 19. SOLUTION: SOLUTION: Use the intersecting tangent and secant segments to find a.

Solve for x using the Quadratic Formula.

Next, use the intersecting chords to find b.

Since lengths cannot be negative, the value of x is about 7.1.

18. SOLUTION: 20. SOLUTION: Use the intersecting tangent and secant segments to find q.

Find x using the intersecting secant and tangent Solve for r using the Quadratic Formula. segments to the larger circle.

Next, use the intersecting chords to find b.

Use the intersecting secant and tangent segments to find d.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

22. INDIRECT MEASUREMENT Gwendolyn is 20. standing 16 feet from a giant sequoia tree and Chet is SOLUTION: standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a Use the intersecting tangent and secant segments to diagram of this situation, and then find the diameter find q. of the tree.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

SOLUTION:

Solve for r using the Quadratic Formula.

So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 21. Given: and intersect at B. SOLUTION: Prove: Use the intersecting secant segments to find c.

Use the intersecting secant and tangent segments to SOLUTION: find d. Proof: Statements (Reasons) 1. and intersect at B. (Given)

Therefore, the values of the variables are c ≈ 22.8 2. , (Inscribed that and d ≈ 16.9. intercept the same arc are 3. (AA Similarity) 22. INDIRECT MEASUREMENT Gwendolyn is 4. (Definition of similar standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance 5. (Cross products) between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree. 24. paragraph proof of Theorem 11.16 Given: Secants and Prove:

SOLUTION: SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that intercept the same arc are congruent. So, . By AA Similarity,

Let d be the diameter of the tree. By the definition of similar triangles, Use the intersecting tangent and secant segments to Since the cross products of a proportion find d. are equal,

25. two-column proof of Theorem 11.17

Given: tangent secant So, the diameter is about 29.6 ft. Prove: PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Given: and intersect at B. Prove: SOLUTION: Proof: Statements (Reasons) 1. tangent and secant (Given) SOLUTION: 2. (The measure of an inscribed Proof: Statements (Reasons) ∠ equals half the measure of its intercept arc.) 1. and intersect at B. (Given) 3. (The measure of an ∠ formed 2. , (Inscribed that by a secant and a tangent = half the measure of its intercept the same arc are intercepted arc.) 3. (AA Similarity) 4. (Substitution)

4. (Definition of similar 5. (Definition of 6. (Reflexive Property)

5. (Cross products) 7. (AA Similarity)

24. paragraph proof of Theorem 11.16 8. (Definition of

Given: Secants and 9. (Cross products) Prove: 26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that SOLUTION: intercept the same arc are congruent. So, Jun; the segments intersect outside of the circle, so . By AA Similarity, the correct equation involves the products of the By the definition of similar triangles, measures of a secant and its external secant Since the cross products of a proportion segment. are equal, 27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two 25. two-column proof of Theorem 11.17 secants intersect in the exterior of a circle and when Given: tangent secant a secant and a tangent intersect in the exterior of a circle. Prove: SOLUTION: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant SOLUTION: and a tangent intersect at an exterior point, the Proof: product of the measures of the secant segment and Statements (Reasons) its external segment is equal to the square of the 1. tangent and secant (Given) measure of the tangent segment, because for the tangent the measures of the external segment and 2. (The measure of an inscribed the whole segment are the same. ∠ equals half the measure of its intercept arc.) 28. CHALLENGE In the figure, a line tangent to circle 3. (The measure of an ∠ formed M and a secant line intersect at R. Find a. Show the steps that you used. by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property) 7. (AA Similarity) SOLUTION: 8. (Definition of Use the intersecting tangent and secant segments to 9. (Cross products) find the value of a.

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

SOLUTION: 29. REASONING When two chords intersect at the Jun; the segments intersect outside of the circle, so center of a circle, are the measures of the the correct equation involves the products of the intercepting arcs sometimes, always, or never equal

measures of a secant and its external secant to each other? segment. SOLUTION: Since the chords intersect at the center, the measure WRITING IN MATH 27. Compare and contrast the of each intercepted arc is equal to the measure of its methods for finding measures of segments when two related central angle. Since vertical angles are secants intersect in the exterior of a circle and when congruent, the opposite arcs will be congruent. All a secant and a tangent intersect in the exterior of a four arcs will only be congruent when all four central

circle. angles are congruent. This will only happen when the SOLUTION: chords are perpendicular and the angles are right Sample answer: When two secants intersect in the angles exterior of a circle, the product of the measures of Therefore, the measures of the intercepted arcs are one secant segment and its external segment is equal sometimes equal to each other. to the product of the measures of the other secant OPEN ENDED segment and its external segment. When a secant 30. Investigate Theorem 11.17 by and a tangent intersect at an exterior point, the drawing and labeling a circle that has a secant and a product of the measures of the secant segment and tangent intersecting outside the circle. Measure and its external segment is equal to the square of the label the two parts of the secant segment to the measure of the tangent segment, because for the nearest tenth of a centimeter. Use an equation to find tangent the measures of the external segment and the measure of the tangent segment. Verify your

the whole segment are the same. answer by measuring the segment. SOLUTION: CHALLENGE 28. In the figure, a line tangent to circle Sample answer: M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION: Use the intersecting tangent and secant segments to find the value of a. Therefore, the measure of the tangent segment is about 2.4 centimeters.

29. REASONING When two chords intersect at the 32. is tangent to the circle, and R and S are points on center of a circle, are the measures of the the circle. What is the value of x to the nearest intercepting arcs sometimes, always, or never equal tenth? to each other? SOLUTION: Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are A congruent, the opposite arcs will be congruent. All 7.6 four arcs will only be congruent when all four central B 6.4 angles are congruent. This will only happen when the C 5.7 chords are perpendicular and the angles are right D 4.8 angles SOLUTION: Therefore, the measures of the intercepted arcs are sometimes equal to each other.

Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

33. ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says Therefore, the measure of the tangent segment is you receive an additional 20% off the already about 2.4 centimeters. discounted price. How much will you pay for a ring with an original price of $200? 31. WRITING IN MATH Describe the relationship F $80 among segments in a circle when two secants G $96 intersect inside a circle. H $120 J $140 SOLUTION: Sample answer: The product of the parts on one SOLUTION: intersecting chord equals the product of the parts of the other chord.

32. is tangent to the circle, and R and S are points on Discount price = 200 – 80 = 120 the circle. What is the value of x to the nearest tenth?

Discount price = 120 – 24 = 96 So, the correct choice is G.

A 7.6 34. EXTENDED RESPONSE The degree measures of B 6.4 minor arc AC and major arc ADC are x and y, C 5.7 respectively. D 4.8 a. If m∠ABC = 70°, write two equations relating x

SOLUTION: and y. b. Find x and y.

Solve for x using the Quadratic Formula.

SOLUTION:

Discount price = 200 – 80 = 120 Simplify.

SOLUTION: First, use intersecting secants to find m∠GCD.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 11.14, y – x = 140. By vertical angles, m∠BCH = m∠GCD. So, b. m∠BCH = 39. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141. Therefore, x = 110° and y = 250°.

Simplify.

So, the correct choice is E.

36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to 38. find SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for SOLUTION: these two circles. First, use intersecting secants to find m∠GCD.

39.

By vertical angles, m∠BCH = m∠GCD. So, SOLUTION: m∠BCH = 39. Three common tangents can be drawn to these two Since major arc BAH has the same endpoints as circles. minor arc BH, m(arc BAH) = 360 – m(arc BH).

11-7Therefore, Special Segments the measure in a of Circle arc BH is 141.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no common tangent. 40. SOLUTION: 37. Two common tangents can be drawn to these two SOLUTION: circles. Four common tangents can be drawn to these two circles.

Write an equation in slope-intercept form of the line having the given slope and y-intercept. 41. m: 3, y-intercept: –4 SOLUTION: The slope-intercept form of a line of slope m and y-

38. intercept b is given by y = mx + b. SOLUTION: For this line, substitute m = 3 and b = –4. Every tangent drawn to the small circle will intersect So, the equation of the line is y = 3x – 4. the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at 42. m: 2, (0, 8) any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for SOLUTION: these two circles. To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m (x – x1) where m is the slope and (x1, y1) is a point on the line. Here, m = 2 and (x1, y1) = (0, 8). 39. SOLUTION: Three common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line. 40. eSolutions Manual - Powered by Cognero Here, and (x1, y1) = (0, –6). Page 10 SOLUTION: Two common tangents can be drawn to these two circles.

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).

Find x. Assume that segments that appear to be tangent are tangent.

1. SOLUTION:

2. SOLUTION:

3. SOLUTION:

4. SOLUTION:

5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth.

SOLUTION: Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST.

The diameter is equal to QS + ST or .

Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent.

6. SOLUTION:

7. SOLUTION:

8. SOLUTION:

9. SOLUTION:

10. SOLUTION:

11. SOLUTION:

Solve for x by using the Quadratic Formula.

Since lengths cannot be negative, the value of x is about 3.1.

12. SOLUTION:

13. SOLUTION:

Solve for x using the Quadratic Formula.

Since lengths cannot be negative, the value of x is about 7.4.

14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page 772.

SOLUTION:

Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem 11.15 to find the value of x.

Add the two segments to find the diameter. Therefore, the diameter of the circle is 528.01+ 60 or about 588.1 meters.

15. CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake?

SOLUTION:

Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To find the diameter of the cake, first use Theorem 11.15 to find x.

Add the two segments to find the diameter. The diameter of the circle is 9 + 4 or 13 inches.

CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent.

16. SOLUTION:

Solve for x using the Quadratic Formula.

Since lengths cannot be negative, the value of x is 6.

17. SOLUTION:

Solve for x using the Quadratic Formula.

Since lengths cannot be negative, the value of x is about 7.1.

18. SOLUTION:

First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle.

Find x using the intersecting secant and tangent segments to the larger circle.

19. SOLUTION: Use the intersecting tangent and secant segments to find a.

Next, use the intersecting chords to find b.

20. SOLUTION: Use the intersecting tangent and secant segments to find q.

Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r.

Solve for r using the Quadratic Formula.

Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r ≈ 1.8.

21. SOLUTION: Use the intersecting secant segments to find c.

Use the intersecting secant and tangent segments to find d.

Therefore, the values of the variables are c ≈ 22.8 and d ≈ 16.9.

SOLUTION:

Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d.

So, the diameter is about 29.6 ft.

PROOF Prove each theorem. 23. two-column proof of Theorem 11.15 Given: and intersect at B. Prove:

SOLUTION: Proof: Statements (Reasons) 1. and intersect at B. (Given) 2. , (Inscribed that intercept the same arc are 3. (AA Similarity)

4. (Definition of similar 5. (Cross products)

24. paragraph proof of Theorem 11.16 Given: Secants and Prove:

SOLUTION: Proof: and are secant segments. By the Reflexive Property, . Inscribed angles that intercept the same arc are congruent. So, . By AA Similarity, By the definition of similar triangles, Since the cross products of a proportion are equal,

25. two-column proof of Theorem 11.17 Given: tangent secant Prove:

SOLUTION: Proof: Statements (Reasons) 1. tangent and secant (Given) 2. (The measure of an inscribed ∠ equals half the measure of its intercept arc.) 3. (The measure of an ∠ formed by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property) 7. (AA Similarity)

8. (Definition of

9. (Cross products)

26. CRITIQUE Tiffany and Jun are finding the value of x in the figure. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning.

SOLUTION: Jun; the segments intersect outside of the circle, so the correct equation involves the products of the measures of a secant and its external secant segment.

27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two secants intersect in the exterior of a circle and when a secant and a tangent intersect in the exterior of a circle. SOLUTION: Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant and a tangent intersect at an exterior point, the product of the measures of the secant segment and its external segment is equal to the square of the measure of the tangent segment, because for the tangent the measures of the external segment and the whole segment are the same.

28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the steps that you used.

SOLUTION: Use the intersecting tangent and secant segments to find the value of a.

29. REASONING When two chords intersect at the center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal to each other? SOLUTION: Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are congruent, the opposite arcs will be congruent. All four arcs will only be congruent when all four central angles are congruent. This will only happen when the chords are perpendicular and the angles are right angles Therefore, the measures of the intercepted arcs are sometimes equal to each other.

Therefore, the measure of the tangent segment is about 2.4 centimeters.

32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest tenth?

A 7.6 B 6.4 C 5.7 D 4.8 SOLUTION:

Solve for x using the Quadratic Formula.

Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C.

Discount price = 200 – 80 = 120

Discount price = 120 – 24 = 96 So, the correct choice is G.

34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, respectively. a. If m∠ABC = 70°, write two equations relating x and y. b. Find x and y.

SOLUTION: a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 11.14, y – x = 140. b.

Therefore, x = 110° and y = 250°.

35. SAT/ACT During the first two weeks of summer vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What was her average weekly pay? A $50 B $112.50 C $125 D $135 E $137.50 SOLUTION:

Simplify.

So, the correct choice is E.

SOLUTION: First, use intersecting secants to find m∠GCD.

By vertical angles, m∠BCH = m∠GCD. So, m∠BCH = 39. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 – m(arc BH).

Therefore, the measure of arc BH is 141.

Copy the figure shown and draw the common tangents. If no common tangent exists, state no common tangent.

37. SOLUTION: Four common tangents can be drawn to these two circles.

38. SOLUTION: Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles.

39. SOLUTION: Three common tangents can be drawn to these two circles.

40. SOLUTION: Two common tangents can be drawn to these two circles.

11-7 Special Segments in a Circle

43.

SOLUTION:

To find the slope-intercept form of the equation of this line use the point-slope form of a line, y – y1 = m

(x – x1) where m is the slope and (x1, y1) is a point on the line.

Here, and (x1, y1) = (0, –6).