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Practice BFor use with the lesson “Use the Converse of the Pythagorean Theorem”

Decide whether the numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute, or obtuse.

1. 5, 12, 13 2. Ï}

8 , 4, 6 3. 20, 21, 28

4. 15, 36, 39 5. Ï}

13 , 10, 12 6. 14, 48, 50

Graph points A, B, and C. Connect the points to form nABC. Decide whether nABC is right, acute, or obtuse.

7. A(23, 5), B(0, 22), C(4, 1) 8. A(28, 24), B(25, 22), C(21, 27)

x

y

2

2

x

y2

2

9. A(4, 1), B(7, 22), C(2, 24) 10. A(22, 2), B(6, 4), C(24, 10)

x

y2

2

x

y

2

2

11. A(0, 5), B(3, 6), C(5, 1) 12. A(22, 4), B(2, 0), C(5, 2)

x

y

1

1

x

y

2

2

In Exercises 13 and 14, copy and complete the statement with <, >, or 5 , if possible. If it is not possible, explain why.

J

K

L R

S

T

12

6

6 14

8

2 656

13. m∠J ? m∠R

14. m∠K 1 m∠L ? m∠S 1 m∠T

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The sides and classification of a triangle are given below. The length of the longest side is the integer given. What value(s) of x make the triangle?

15. x, x, 8; right 16. x, x, 12; obtuse

17. x, x, 6; acute 18. x, x 1 3, 15; obtuse

19. x, x 2 8, 40; right 20. x 1 2, x 1 3, 29; acute

In Exercises 21 and 22, use the diagram and the following information.

Roof The roof shown in the diagram at the right is shown from the front of the house.

The slope of the roof is 5 }

12 . The height of the roof is 15 feet.

21. What is the length from gutter to peak of the roof?

22. A row of shingles is 5 inches high. How many rows

5 in.

Shingle of shingles are needed for one side of the roof?

In Exercises 23–25, you will use two different methods for determining whether nABC is a right triangle.

23. Method 1 Find the slope of }

AC and the slope of }

BC .

x

y

1

1

B(23, 7)A(4, 6)

C(0, 3)

What do the slopes tell you about ∠ACB? Is nABC a right triangle? How do you know?

24. Method 2 Use the Distance Formula and the Converse of the Pythagorean Theorem to determine whether nABC is a right triangle.

25. Compare Which method would you use to determine whether a given triangle is right, acute, or obtuse? Explain.

Practice B continuedFor use with the lesson “Use the Converse of the Pythagorean Theorem”

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Lesson Apply the Pythag-orean Theorem, continued

x Pentagon Hexagon

x 5 1 45 54

x 5 2 90 108

x 5 3 135 162

x 5 4 180 216

x 5 5 225 270

x 5 6 270 324

x 5 7 315 378

x 5 8 360 432

x 5 9 405 486

x 5 10 450 540

c. P(triangle) 5 27x; P(square) 5 36x; P(pentagon) 5 45x; P(hexagon) 5 54x hexagon, Sample answer: the hexagon’s perimeter equation has the greatest slope.

5.

c

c

b

b

aa

Sample answer: Area of two squares is a2 1 b2. The two cuts must be of equal length c. From the diagram, the rearranged shape is a square with area c2. So, a2 1 b2 5 c2.

Lesson Use the Converse of the Pythagorean Theorem

Teaching Guide

1.

105

85 137

A

B C

2. no; 852 1 1052 Þ 1372

3.

85

132

157

4. yes; 852 1 1322 5 1572

Practice Level A

1. yes 2. yes 3. no 4. yes; right

5. yes; obtuse 6. yes; acute 7. no

8. yes; right 9. yes; obtuse

10.

x

y

1

1

B

A

C

11.

x

y

1

1

A

B C

right obtuse

12.

x

y

2

1

A B

C

13.

x

y

1

1

A

B

C

acute right

14.

x

y

1

1

AB

C

15.

x

y

1

1

A

B

C

obtuse acute

16. m∠ J 5 m∠ R

17. m∠ K 1 m∠ L 5 m∠ S 1 m∠ T

18. B 19. C 20. yes 21. yes

22. 1 }

2 ; 22; Because 1 1 }

2 2 (22) 5 21,

} AB ⊥

} BC .

So ∠ ABC is a right angle. Therefore n ABC is a right triangle by the definition of a right triangle.

23. (AB) 2 1 (BC) 2 5 20 1 20 5 40 5 (AC) 2, so by the Converse of the Pythagorean Theorem, n ABC is a right triangle.

Practice Level B

1. yes; right 2. yes; obtuse 3. yes; acute

4. yes; right 5. yes; obtuse 6. yes; right

7.

x

y

2

4

A

B

C

8.

x

y2

2

AB

C

acute obtuse

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Lesson Use the Converse of the Pythagorean Theorem, continued9.

x

y2

2

A

BC

10.

x

y

4

2

A

B

C

acute right

11.

x

y

1

1

AB

C

12.

x

y6

4

A

B

C

obtuse obtuse

13. > 14. < 15. x 5 4 Ï}

2 16. 6 < x < 6 Ï}

2

17. x > 3 Ï}

2 18. 6 < x < 9 19. x 5 32

20. x > 18 21. 39 ft 22. about 94 rows

23. 3 }

4 ; 2

4 } 3 ; Because 1 3 }

4 2 1 2

4 } 3 2 5 21,

} AC ⊥  } BC .

So ∠ACB is a right angle. Therefore nABC is a right triangle by the definition of a right triangle.

24. (AC)2 1 (BC)2 5 25 1 25 5 50 5 (AB)2, so by the Converse of the Pythagorean Theorem, nABC is a right triangle.

25. Start by finding the slopes to see if the triangle is a right triangle. If no two slopes lead to perpendicular line segments, then find the distances to determine whether the triangle is acute or obtuse.

Practice Level C

1. no 2. yes; obtuse 3. yes; right 4. yes; acute

5. yes; right 6. yes; obtuse

7.

x

y

1

1

A

B

C

8.

x

y

2

1

A

B

C

right acute

9.

x

y

1

1

A

B

C 10.

x

y

1

1

A

B

C

obtuse obtuse

11.

x

y

211

A

B

C

12.

x

y

4

2

A

B

C

acute acute

13. m∠ J < m∠ R

14. m∠ K 1 m∠ L > m∠ S 1 m∠T 15. D

16. Ï}

2 x 17. 8 Ï}

2 18. 5 < x < 5 Ï}

2

19. x > 15 Ï

}

2 }

2 20.

5 6 7 Ï}

23 }

2

21. 18 < x < 3 1 3 Ï

}

241 }

2 22. x >

23 1 Ï}

3929 }

2

23. acute 24. 182 ft 25. no 26. Since 82 1 82 < 122, n ABC is an obtuse triangle by Theorem 4. ∠ ACB is obtuse since it is opposite the longest side. Since vertical angles are congruent, ∠ ACB > ∠ DCE. So by substitution, ∠ DCE is obtuse.

Study Guide

1. right triangle 2. right triangle

3. not a right triangle 4. yes; right

5. yes; obtuse 6. yes; acute 7. yes; obtuse

8. yes; obtuse 9. yes; right

Interdisciplinary Application

1. 18.5 ft; yes 2. 15 ft; yes 3. Kitchen 1: sink and refrigerator; Kitchen 2: sink and refrigerator, stove and sink 4. not a right triangle; 42 1 5.52 Þ 92 5. right triangle; 62 1 2.52 5 6.52 6. It changes the perimeter by 0.2 feet and improves the sink to refrigerator distance by 0.4 feet; not a right triangle

Challenge Practice

1. x2 1 y2 5 9; x Þ 0 2. x 2 1 y2 > 9; x Þ 0

3. x 2 1 y2 < 9; x Þ 0 4. y 5 3 or y 5 23; x Þ 0

5. 23 < y < 3; x Þ 0 6. y > 3 and y < 23; x Þ 0

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