<<

8-2 The Pythagorean and Its Converse

Find x.

1. 2. SOLUTION: In a right , the sum of the of the SOLUTION: of the legs is equal to the of the In a , the sum of the squares of the of the . The length of the hypotenuse is 13 lengths of the legs is equal to the square of the length and the lengths of the legs are 5 and x. of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 12.

eSolutions Manual - Powered by Cognero Page 1 8-2 The and Its Converse

4. Use a to find x. Explain your reasoning.

3.

SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length SOLUTION: of the hypotenuse. The length of the hypotenuse is 16 35 is the hypotenuse, so it is the greatest value in the and the lengths of the legs are 4 and x. Pythagorean Triple.

Find the common factors of 35 and 21.

The GCF of 35 and 21 is 7. Divide this out.

Check to see if 5 and 3 are part of a Pythagorean triple with 5 as the largest value.

ANSWER: We have one Pythagorean triple 3-4-5. The multiples of this triple also will be Pythagorean triple. So, x = 7(4) = 28. Use the Pythagorean Theorem to check it.

ANSWER: 28; Since and and 3-4-5 is a Pythagorean triple, or 28.

eSolutions Manual - Powered by Cognero Page 2 8-2 The Pythagorean Theorem and Its Converse

5. MULTIPLE CHOICE The mainsail of a boat is Determine whether each of numbers can be shown. What is the length, in feet, of ? the measures of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. Justify your answer. 6. 15, 36, 39

SOLUTION: By the triangle theorem, the sum of the lengths of any two sides should be greater than the length of the third side. A 52.5 B 65 C 72.5 D 75 Therefore, the set of numbers can be measures of a triangle. SOLUTION: The mainsail is in the form of a right triangle. The Classify the triangle by comparing the square of the length of the hypotenuse is x and the lengths of the longest side to the sum of the squares of the other legs are 45 and 60. two sides.

Therefore, by the converse of Pythagorean Theorem, a triangle with the given measures will be a right triangle.

eSolutions Manual - Powered by Cognero Page 3 8-2 The Pythagorean Theorem and Its Converse

7. 16, 18, 26 8. 15, 20, 24

SOLUTION: SOLUTION: By the theorem, the sum of the By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the lengths of any two sides should be greater than the length of the third side. length of the third side.

Therefore, the set of numbers can be measures of a Therefore, the set of numbers can be measures of a triangle. triangle.

Now, classify the triangle by comparing the square of Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other the longest side to the sum of the squares of the other two sides. two sides.

Therefore, by Pythagorean Inequality Theorem, a Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an obtuse triangle with the given measures will be an acute triangle. triangle.

eSolutions Manual - Powered by Cognero Page 4 8-2 The Pythagorean Theorem and Its Converse

Find x.

10.

9. SOLUTION: In a right triangle, the sum of the squares of the SOLUTION: lengths of the legs is equal to the square of the length In a right triangle, the sum of the squares of the of the hypotenuse. The length of the hypotenuse is 15 lengths of the legs is equal to the square of the length and the lengths of the legs are 9 and x. of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 12 and 16.

eSolutions Manual - Powered by Cognero Page 5 8-2 The Pythagorean Theorem and Its Converse

11. 12.

SOLUTION: SOLUTION: In a right triangle, the sum of the squares of the In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 5 of the hypotenuse. The length of the hypotenuse is 66 and the lengths of the legs are 2 and x. and the lengths of the legs are 33 and x.

eSolutions Manual - Powered by Cognero Page 6 8-2 The Pythagorean Theorem and Its Converse

13.

SOLUTION: 14. In a right triangle, the sum of the squares of the SOLUTION: lengths of the legs is equal to the square of the length In a right triangle, the sum of the squares of the of the hypotenuse. The length of the hypotenuse is x lengths of the legs is equal to the square of the length and the lengths of the legs are . of the hypotenuse. The length of the hypotenuse is

and the lengths of the legs are .

eSolutions Manual - Powered by Cognero Page 7 8-2 The Pythagorean Theorem and Its Converse

PERSEVERANCE Use a Pythagorean Triple to find x.

16. 15. SOLUTION: SOLUTION: Find the greatest common factor of 14 and 48. Find the greatest common factors of 16 and 30.

The GCF of 14 and 48 is 2. Divide this value out. The GCF of 16 and 30 is 2. Divide this value out.

Check to see if 7 and 24 are part of a Pythagorean Check to see if 8 and 15 are part of a Pythagorean triple. triple.

We have one Pythagorean triple 7-24-25. The We have one Pythagorean triple 8-15-17. The multiples of this triple also will be Pythagorean triple. multiples of this triple also will be Pythagorean triple. So, x = 2(25) = 50. So, x = 2(17) = 34. Use the Pythagorean Theorem to check it. Use the Pythagorean Theorem to check it.

eSolutions Manual - Powered by Cognero Page 8 8-2 The Pythagorean Theorem and Its Converse

17. 18.

SOLUTION: SOLUTION: 74 is the hypotenuse, so it is the greatest value in the 78 is the hypotenuse, so it is the greatest value in the Pythagorean triple. Pythagorean triple.

Find the greatest common factor of 74 and 24. Find the greatest common factor of 78 and 72. 78 = 2 × 3 × 13 72 = 2 × 2 × 2 × 3 × 3

The GCF of 24 and 74 is 2. Divide this value out. The GCF of 78 and 72 is 6. Divide this value out. 78 ÷ 6 = 13 72 ÷ 6 = 12

Check to see if 12 and 37 are part of a Pythagorean Check to see if 13 and 12 are part of a Pythagorean triple with 37 as the largest value. triple with 13 as the largest value. 132 – 122 = 169 – 144 = 25 = 52

We have one Pythagorean triple 12-35-37. The We have one Pythagorean triple 5-12-13. The multiples of this triple also will be Pythagorean triple. multiples of this triple also will be Pythagorean triple. So, x = 2(35) = 70. So, x = 6(5) = 30. Use the Pythagorean Theorem to check it. Use the Pythagorean Theorem to check it.

eSolutions Manual - Powered by Cognero Page 9 8-2 The Pythagorean Theorem and Its Converse

19. BASKETBALL The support for a basketball goal 20. DRIVING The street that Khaliah usually uses to forms a right triangle as shown. What is the length x get to school is under construction. She has been of the horizontal portion of the support? taking the detour shown. If the construction starts at the where Khaliah leaves her route and ends at the point where she re-enters her normal route, about how long is the stretch of road under construction?

SOLUTION: In a right triangle, the sum of the squares of the SOLUTION: lengths of the legs is equal to the square of the length Let x be the length of the road that is under of the hypotenuse. The length of the hypotenuse is construction. The road under construction and the and the lengths of the legs are detour form a right triangle. . The length of the hypotenuse is x and the lengths of the legs are 0.8 and 1.8.

Therefore, a stretch of about 2 miles is under construction.

Therefore, the horizontal of the support is about 3 ft.

eSolutions Manual - Powered by Cognero Page 10 8-2 The Pythagorean Theorem and Its Converse

Determine whether each set of numbers can be 22. 10, 12, 23 the measures of the sides of a triangle. If so, SOLUTION: classify the triangle as acute, obtuse, or right. Justify your answer. By the triangle inequality theorem, the sum of the 21. 7, 15, 21 lengths of any two sides should be greater than the length of the third side. SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Since the sum of the lengths of two sides is less than that of the third side, the set of numbers cannot be measures of a triangle. Therefore, the set of numbers can be measures of a triangle. ANSWER: No; 23 > 10 + 12 Now, classify the triangle by comparing the square of 23. 4.5, 20, 20.5 the longest side to the sum of the squares of the other two sides. SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side.

Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an obtuse Therefore, the set of numbers can be measures of a triangle. triangle.

ANSWER: Classify the triangle by comparing the square of the Yes; obtuse longest side to the sum of the squares of the other two sides.

Therefore, by the converse of Pythagorean Theorem, a triangle with the given measures will be a right triangle.

eSolutions Manual - Powered by Cognero Page 11 8-2 The Pythagorean Theorem and Its Converse

24. 44, 46, 91 26. 4 , 12, 14

SOLUTION: SOLUTION: By the triangle inequality theorem, the sum of the By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the lengths of any two sides should be greater than the length of the third side. length of the third side.

Therefore, the set of numbers can be measures of a Since the sum of the lengths of two sides is less than triangle. that of the third side, the set of numbers cannot be measures of a triangle. Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other ANSWER: two sides. No; 91 > 44 + 46

25. 4.2, 6.4, 7.6

SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the Therefore, by Pythagorean Inequality Theorem, a length of the third side. triangle with the given measures will be an obtuse triangle.

ANSWER: Yes; obtuse Therefore, the set of numbers can be measures of a triangle.

Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.

Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an acute triangle.

eSolutions Manual - Powered by Cognero Page 12 8-2 The Pythagorean Theorem and Its Converse

Find x.

28. 27. SOLUTION: SOLUTION: The triangle with the side lengths 9, 12, and x form a right triangle.

In a right triangle, the sum of the squares of the The segment of length 16 units is divided to two lengths of the legs is equal to the square of the length congruent segments. So, the length of each segment of the hypotenuse. The length of the hypotenuse is x will be 8 units. Then we have a right triangle with the and the lengths of the legs are 9 and 12. sides 15, 8, and x.

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 15.

eSolutions Manual - Powered by Cognero Page 13 8-2 The Pythagorean Theorem and Its Converse

COORDINATE GEOMETRY Determine whether is an acute, right, or obtuse triangle for the given vertices. Explain. 30. X(–3, –2), Y(–1, 0), Z(0, –1)

SOLUTION: 29.

SOLUTION:

We have a right triangle with the sides 14, 10, and x. Use the to find the length of each In a right triangle, the sum of the squares of the side of the triangle. lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 14 and the lengths of the legs are 10 and x.

Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.

Therefore, by the converse of Pythagorean Theorem, a triangle with the given measures will be a right triangle.

eSolutions Manual - Powered by Cognero Page 14 8-2 The Pythagorean Theorem and Its Converse

31. X(–7, –3), Y(–2, –5), Z(–4, –1) 32. X(1, 2), Y(4, 6), Z(6, 6)

SOLUTION: SOLUTION:

Use the distance formula to find the length of each Use the distance formula to find the length of each side of the triangle. side of the triangle.

Classify the triangle by comparing the square of the Classify the triangle by comparing the square of the longest side to the sum of the squares of the other longest side to the sum of the squares of the other two sides. two sides.

Therefore, by the Pythagorean Inequality Theorem, a Therefore, by the Pythagorean Inequality Theorem, a triangle with the given measures will be an acute triangle with the given measures will be an obtuse triangle. triangle.

ANSWER: ANSWER: acute; , , ; obtuse; XY = 5, YZ = 2, ;

eSolutions Manual - Powered by Cognero Page 15 8-2 The Pythagorean Theorem and Its Converse

33. X(3, 1), Y(3, 7), Z(11, 1) 34. JOGGING Brett jogs in the park three a

SOLUTION: week. Usually, he takes a -mile path that cuts through the park. Today, the path is closed, so he is taking the orange route shown. How much farther will he jog on his alternate route than he would have if he had followed his normal path?

Use the distance formula to find the length of each side of the triangle.

SOLUTION: The normal jogging route and the detour form a right triangle. One leg of the right triangle is 0.45 mi. and let x be the other leg. The hypotenuse of the right Classify the triangle by comparing the square of the longest side to the sum of the squares of the other triangle is . In a right triangle, the sum of the two sides. squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Therefore, by the converse of Pythagorean Theorem, a triangle with the given measures will be a right triangle.

ANSWER: right; XY = 6, YZ = 10, XZ = 8; 62 + 82 = 102 So, the total distance that he runs in the alternate route is 0.45 + 0. 6 = 1.05 mi. instead of his normal distance 0.75 mi. Therefore, he will be jogging an extra distance of 0.3 miles in his alternate route.

35. PROOF Write a paragraph proof of Theorem 8.5.

SOLUTION: Theorem 8.5 states that if the sum of the squares of the lengths of the shortest sides of a triangle are eSolutions Manual - Powered by Cognero Page 16 8-2 The Pythagorean Theorem and Its Converse

equal to the square of the length of the longest side, then the triangle is a right triangle. Use the hypothesis to create a given statement, namely that c2 = a2 + b2

for triangle ABC. You can accomplish this proof by constructing another triangle (Triangle DEF) that is congruent to triangle ABC, using SSS triangle Proof: Draw on with equal to a. theorem. Show that triangle DEF is a At D, draw line . Locate point F on m so that right triangle, using the Pythagorean theorem. DF = b. Draw and call its measure x. Because therefore, any triangle congruent to a right triangle is a right triangle, a2 + b2 = x2. But a2 + b2 = but also be a right triangle. c2, so x2 = c2 or x = c. Thus, by SSS.

Given: with sides of measure a, b, and c, This means . Therefore, C must be a right , making a right triangle. where c2 = a2 + b2 Prove: is a right triangle. PROOF Write a two-column proof for each theorem. 36. Theorem 8.6

SOLUTION:

You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two , a right Proof: Draw on line with measure equal to a. angle, relationship between sides. Use the properties At D, draw line . Locate point F on m so that that you have learned about right , acute DF = b. Draw and call its measure x. Because angles, Pythagorean Theorem,angle relationships and is a right triangle, a2 + b2 = x2. But a2 + b2 = equivalent expressions in algebra to walk through the proof. 2 2 2 c , so x = c or x = c. Thus, by SSS. This means . Therefore, C must be a Given: In , c2 < a2 + b2 where c is the length right angle, making a right triangle. of the longest side. In , R is a . ANSWER: Prove: is an acute triangle. Given: with sides of measure a, b, and c, where c2 = a2 + b2 Prove: is a right triangle.

Proof: Statements (Reasons) 1. In , c2 < a2 + b2 where c is the length of the longest side. In , R is a right angle. (Given) eSolutions Manual - Powered by Cognero Page 17 8-2 The Pythagorean Theorem and Its Converse

2. a2 + b2 = x2 (Pythagorean Theorem) prove. Here, you are given two triangles, relationship between angles.Use the properties that you have 2 2 3. c < x (Substitution Property) learned about triangles, angles and equivalent 4. c < x (A property of square roots) expressions in algebra to walk through the proof. 5. m R = ( of a right angle) 6. m C < m R (Converse of the Hinge Theorem) Given: In , c2 > a2 + b2, where c is the length 7. m C < (Substitution Property) of the longest side. 8. C is an acute angle. (Definition of an acute Prove: is an obtuse triangle. angle) 9. is an acute triangle. (Definition of an acute triangle)

ANSWER: Given: In , c2 < a2 + b2 where c is the length of the longest side. In , R is a right angle. Prove: is an acute triangle. Statements (Reasons) 1. In , c2 > a2 + b2, where c is the length of the longest side. In , R is a right angle. (Given) 2. a2 + b2 = x2 (Pythagorean Theorem) Proof: 2 2 Statements (Reasons) 3. c > x (Substitution Property) 4. c > x (A property of square roots) 1. In , c2 < a2 + b2 where c is the length of 5. m R = (Definition of a right angle) the longest 6. m C > m R (Converse of the ) side. In , R is a right angle. (Given) 7. m C > (Substitution Property of ) 2 2 2 2. a + b = x (Pythagorean Theorem) 8. C is an obtuse angle. (Definition of an obtuse 3. c2 < x2 (Substitution Property) angle) 4. c < x (A property of square roots) 9. is an obtuse triangle. (Definition of an 5. m R = (Definition of a right angle) obtuse triangle) 6. m C < m R (Converse of the Hinge Theorem) ANSWER: 7. m C < (Substitution Property) 2 2 2 8. C is an acute angle. (Definition of an acute Given: In , c > a + b , where c is the length angle) of the longest side. 9. is an acute triangle. (Definition of an acute Prove: is an obtuse triangle. triangle)

37. Theorem 8.7

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to eSolutions Manual - Powered by Cognero Page 18 8-2 The Pythagorean Theorem and Its Converse

Statements (Reasons) SENSE-MAKING Find the and 1. In , c2 > a2 + b2, where c is the length of of each figure. the longest side. In , R is a right angle. (Given) 2. a2 + b2 = x2 (Pythagorean Theorem) 3. c2 > x2 (Substitution Property) 38. 4. c > x (A property of square roots) SOLUTION: 5. m R = (Definition of a right angle) The area of a triangle is given by the formula 6. m C > m R (Converse of the Hinge Theorem) 7. m C > (Substitution Property of Equality) where b is the and h is the height of 8. C is an obtuse angle. (Definition of an obtuse the triangle. Since the triangle is a right triangle the angle) base and the height are the legs of the triangle. So, 9. is an obtuse triangle. (Definition of an obtuse triangle)

The perimeter is the sum of the lengths of the three sides. Use the Pythagorean Theorem to find the length of the hypotenuse of the triangle. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

The length of the hypotenuse is 20 units. Therefore, the perimeter is 12 + 16 + 20 = 48 units.

ANSWER: P = 48 units; A = 96 units2

39.

SOLUTION: eSolutions Manual - Powered by Cognero Page 19 8-2 The Pythagorean Theorem and Its Converse

SOLUTION: The given figure can be divided as a right triangle and a as shown.

The area of a triangle is given by the formula where b is the base and h is the height of the triangle. The to the base of an isosceles The total are of the figure is the sum of the of triangle bisects the base. So, we have two right the right triangle and the rectangle. triangles with one of the legs equal to 5 units and the

hypotenuse is 13 units each. Use the Pythagorean The area of a triangle is given by the formula Theorem to find the length of the common leg of the triangles. where b is the base and h is the height of In a right triangle, the sum of the squares of the the triangle. Since the triangle is a right triangle the lengths of the legs is equal to the square of the length base and the height are the legs of the triangle. So, of the hypotenuse.

The area of a rectangle of length and width w is given by the formula A = × w. So,

Therefore, the total area is 24 + 32 = 56 sq. units.

The perimeter is the sum of the lengths of the four boundaries. Use the Pythagorean Theorem to find the The altitude is 12 units. length of the hypotenuse of the triangle. In a right triangle, the sum of the squares of the Therefore, lengths of the legs is equal to the square of the length of the hypotenuse.

The perimeter is the sum of the lengths of the three sides. Therefore, the perimeter is 13 + 13 + 10 = 36 units.

ANSWER: P = 36 units; A = 60 units2

The hypotenuse is 10 units. Therefore, the perimeter is 4 + 8 + 10 + 10 = 32 units

ANSWER: 40. P = 32 units; A = 56 units2 eSolutions Manual - Powered by Cognero Page 20 8-2 The Pythagorean Theorem and Its Converse

41. ALGEBRA The sides of a triangle have lengths x, x 42. ALGEBRA The sides of a triangle have lengths 2x, + 5, and 25. If the length of the longest side is 25, 8, and 12. If the length of the longest side is 2x, what what value of x makes the triangle a right triangle? values of x make the triangle acute?

SOLUTION: SOLUTION: By the converse of the Pythagorean Theorem, if the By the triangle inequality theorem, the sum of the square of the longest side of a triangle is the sum of lengths of any two sides should be greater than the squares of the other two sides then the triangle is a length of the third side. right triangle.

So, the value of x should be between 2 and 10.

By the Pythagorean Inequality Theorem, if the square of the longest side of a triangle is less than the sum of squares of the other two sides then the triangle is an acute triangle.

Use the Quadratic Formula to find the roots of the .

Therefore, for the triangle to be acute,

ANSWER: Since x is a length, it cannot be negative. Therefore, x = 15. 43. TELEVISION The screen aspect , or the ratio ANSWER: of the width to the length, of a high-definition 15 television is 16:9. The of a television is given by the distance across the screen. If an HDTV is 41 wide, what is its screen size? Refer to the photo on page 547.

eSolutions Manual - Powered by Cognero Page 21 8-2 The Pythagorean Theorem and Its Converse

distance allotted in a slide design is 14 feet, approximately how long should the slide be?

SOLUTION: Use the ratio to find the length of the television. Let x be the length of the television. Then, SOLUTION: Use the ratio to find the vertical distance. Let x be the vertical distance. Then,

Solve the proportion for x. Solve the proportion for x.

The two adjacent sides and the diagonal of the television form a right triangle. The vertical distance, the horizontal distance, and the In a right triangle, the sum of the squares of the slide form a right triangle. lengths of the legs is equal to the square of the length In a right triangle, the sum of the squares of the of the hypotenuse. lengths of the legs is equal to the square of the length of the hypotenuse.

Therefore, the screen size is about 47 inches. Therefore, the slide will be about 16 ft long. ANSWER: ANSWER: about 16 ft 44. PLAYGROUND According to the Handbook for Public Playground Safety, the ratio of the vertical distance to the horizontal distance covered by a slide should not be more than about 4 to 7. If the horizontal eSolutions Manual - Powered by Cognero Page 22 8-2 The Pythagorean Theorem and Its Converse

Find x.

46. 45. SOLUTION: SOLUTION: In a right triangle, the sum of the squares of the In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is x of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are x – 3 and 9. and the lengths of the legs are x – 4 and 8. Solve for x. Solve for x.

eSolutions Manual - Powered by Cognero Page 23 8-2 The Pythagorean Theorem and Its Converse

47. c. Verbal Make a conjecture about the ratio of the SOLUTION: hypotenuse to a leg of an isosceles right triangle. In a right triangle, the sum of the squares of the SOLUTION: lengths of the legs is equal to the square of the length a. Let AB = AC. Use the Pythagorean Theorem to of the hypotenuse. The length of the hypotenuse is x find BC. + 1 and the lengths of the legs are x and . Solve for x.

ANSWER: Let MN = PM. Use the Pythagorean Theorem to find NP.

48. MULTIPLE REPRESENTATIONS In this problem, you will investigate special right triangles. a. Geometric Draw three different isosceles right triangles that have whole-number side lengths. Label the triangles ABC, MNP, and XYZ with the right angle located at A, M, and X, respectively. Label the leg lengths of each side and find the length of the hypotenuse in simplest radical form.

b. Tabular Copy and complete the table below.

eSolutions Manual - Powered by Cognero Page 24 8-2 The Pythagorean Theorem and Its Converse

Let ZX = XY. Use the Pythagorean Theorem to find ZY.

b.

b. Complete the table below with the side lengths calculated for each triangle. Then, compute the given .

c. Sample answer: The ratio of the hypotenuse to a leg of an isosceles right triangle is .

49. CHALLENGE Find the value of x in the figure.

c. Summarize any observations made based on the patterns in the table. The ratio of the hypotenuse to a leg of an isosceles right triangle is .

ANSWER: SOLUTION: a. Label the vertices to make the sides easier to identify. Find each side length, one at a , until you get to the value of x.

By the Pythagorean Theorem, in a right triangle, the sum of the squares of the lengths of the legs is equal eSolutions Manual - Powered by Cognero Page 25 8-2 The Pythagorean Theorem and Its Converse

to the square of the length of the hypotenuse. Use the theorem to find the lengths of the required segments.

50. ARGUMENTS True or false? Any two right triangles with the same hypotenuse have the same area. Explain your reasoning.

SOLUTION: A good approach to this problem is to come up with a couple different sets of three numbers that form a right triangle, but BC is the hypotenuse of the right triangle BDC. each have a hypotenuse of 5. You could pick a leg length ( less that 5) and then solve for the other leg. Leave any non- in radical form, for accuracy.

For example:

Find FC.

We have BF = BC – FC = 17.4 – 12.7 = 4.7. Then, find the area of a right triangle with legs of 2 and Then, find the area of a right triangle with legs of 3 Use the value of BF to find x in the right triangle and 4. BEF.

eSolutions Manual - Powered by Cognero Page 26 8-2 The Pythagorean Theorem and Its Converse

Thus, the triangle will be a right Triangle. If you double or halve the side lengths, all three sides of the new triangles are proportional to the sides of the Since these areas are not equivalent, the statement is False. original triangle. Using the Side- Side-Side Theorem, you know that both of the new triangles are ANSWER: similar to the original triangle, so they are both right. False; sample answer: A right triangle with legs measuring 3 in. ANSWER: and 4 in. has a hypotenuse of 5 in. and an area of

or 6 in2. A right triangle with legs measuring 2 in. and

also has a hypotenuse of 5 in., but its area is

in 2, which is not equivalent to 6 in2. Right; sample answer: If you double or halve the side 51. OPEN-ENDED Draw a right triangle with side lengths, all three sides of the new triangles are lengths that form a Pythagorean triple. If you double proportional to the sides of the original triangle. Using the length of each side, is the resulting triangle acute, the Side- Side-Side Similarity Theorem, you know right, or obtuse? if you halve the length of each that both of the new triangles are similar to the side? Explain. original triangle, so they are both right. SOLUTION:

If you double the side lengths, you get 6, 8, and 10. Determine the type of triangle by using the Converse of the Pythagorean Theorem:

If you halve the side lengths, you get 1.5, 2, and 2.5. Determine the type of triangle by using the Converse of the Pythagorean Theorem: eSolutions Manual - Powered by Cognero Page 27 8-2 The Pythagorean Theorem and Its Converse

52. WRITING IN MATH Research incommensurable 53. Jessica has three wooden rods with the lengths magnitudes, and describe how this phrase relates to shown below. She wants to place them together, if the use of irrational numbers in geometry. Include one possible, to create a triangular picture frame.

example of an used in geometry.

SOLUTION: Incommensurable magnitudes are magnitudes of the same kind that do not have a common unit of measure. Irrational numbers were invented to describe geometric relationships, such as ratios of incommensurable magnitudes that cannot be described using rational numbers. For example, to express the measures of the sides of a square with an area of 2 square units, the irrational number √2 is needed. Which of the following is the best description of the frame Jessica can create?

A The frame will be an acute triangle. B The frame will be a right triangle. C The frame will be an obtuse triangle. D The rods cannot be placed together to form a triangle. ANSWER: Sample answer: Incommensurable magnitudes are magnitudes of the same kind that do not have a

common unit of measure. Irrational numbers were

invented to describe geometric relationships, such as ratios of incommensurable magnitudes that cannot be SOLUTION: described using rational numbers. For example, to Let a = 8, b = 13, and c = 16. express the measures of the sides of a square with an area of 2 square units, the irrational number √2 is 82 + 132 ? 162 needed. 64 + 169 ? 256 233 < 256 Since the square of the hypotenuse is greater than the sum of the squares of the two legs, the triangle will be obtuse. Choice C is correct.

eSolutions Manual - Powered by Cognero Page 28 8-2 The Pythagorean Theorem and Its Converse

54. Dontrell’s school has a rectangular lawn with the 55. A square park has a diagonal walkway from one shown. Students often walk along the corner to another. If the walkway is 120 meters long, diagonal of the lawn as a shortcut. what is the approximate length of each side of the park in meters?

SOLUTION: Sketch a square and label it with the given information. Let x represent the length of a side of To the nearest tenth of a meter, how much shorter is the square. it to walk directly from J to L rather than from J to K to L?

A 4.9 m B 9.4 m C 10.0 m D 17.1 m

SOLUTION: The distance from J to K to L = JK + KL = 16 + 6 = 22 m. The distance directly from J to L = . Set up an equation for the triangle formed by the two Then, subtract these two to determine how sides of the square and the diagonal using the much longer one is than the other: Pythagorean Theorem. Solve for x. . Thus, the correct choice is A.

eSolutions Manual - Powered by Cognero Page 29 8-2 The Pythagorean Theorem and Its Converse

56. Lin wants to build a triangular vegetable bed with 57. Which of the following is the best estimate of the sides that are 7 feet, 8 feet, and 10 feet long. Which perimeter of ΔRST? of the following best describes the vegetable bed?

A It will be an acute triangle. B It will be a right triangle. C It will be an obtuse triangle. D Lin cannot form a triangle with these side lengths.

SOLUTION: A 30.0 First, use the Triangle Inequality Theorem to B 27.9 determine if the three lengths can make a triangle. C 17.0 D 10.9

Yes, these three side lengths can form SOLUTION: a triangle. To determine the perimeter of ΔRST, first find TS, using the Pythagorean Theorem. Use the Pythagorean Inequality Theorem to determine if the triangle is acute, right, or obtuse.

Therefore, the triangle formed is an acute triangle. The correct choice is A.

ANSWER: A The perimeter of ΔRST = 12+5+10.9 = 27.9. The correct choice is B.

eSolutions Manual - Powered by Cognero Page 30 8-2 The Pythagorean Theorem and Its Converse

58. MULTI-STEP A gardener wishes to make a 59. A has a perimeter of 20 units. The length of triangular garden. He has fence segments of length 8 one diagonal for the rhombus is 8. feet, 14 feet, 15 feet, 17 feet, and 20 feet. a. Find the side length of the rhombus. a. What combination of fence lengths will make a right triangle? b. Find the length of the other diagonal.

b. What combination of fence lengths will make an SOLUTION: acute triangle? a. In a rhombus, all four sides are the same length. Since the perimeter is 20 units, each side must be 5 c. What combinations of fence lengths will make an units. obtuse triangle? b. The of a rhombus bisect each other and SOLUTION: are also . Since one diagonal is 8, when a. Recognize that 8, 15, 17 is a Pythagorean Triple, bisected, each part is 4. So half of the other diagonal so the fence lengths 8 feet, 15 feet, and 17 feet will is the third side in a right triangle whose hypotenuse is form a right triangle. 5 and one leg is 4. So the other leg is 3, and the length of the diagonal is 6 units. b. To form an acute triangle, the square of the longest side has to be less than the sum of the ANSWER: squares of the two shorter sides. One possibility is 14 5 units; 6 units feet, 15 feet, and 17 feet. 142 + 152 > 172 196 + 225 > 289 421 > 289

c. To form an obtuse triangle, the square of the longest side has to be greater than the sum of the squares of the two shorter sides. One possibility is 8 feet, 14 feet, and 20 feet. 82 + 142 > 202 64 + 196 > 400 260 < 400

ANSWER: 8, 15, 17; 14, 15, 17; 8, 14, 20