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Practice BFor use with the lesson “Use Similar Right Triangles”

Complete and solve the proportion.

1. x } 12

5 ? }

8 2.

15 }

x 5

x }

? 3.

9 }

x 5

x }

?

12

8x

x

2015

9

x

11

Find the value(s) of the variable(s).

4. a

612

5.

x

6 8

6. b

10

24

7.

w 1 9

8 18

8. y

z x

14

16

9. bc

a

32

24

Tell whether the triangle is a right triangle. If so, find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.

10.

14

122 85

11.

1519

6 19 12.

3018

6 34

Use the Geometric Mean Theorems to find AC and BD.

13.

30 40

A

D

B C

14. A

B D

C

610

15.

C D A

B

5

35

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GeometryChapter Resource Book 7-37

Lesson

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16. Complete the proof.

GIVEN: nXYZ is a right triangle with m∠XYZ 5 908.

ZWY

X

VU

} VW i } XY , } YU is an altitude of nXYZ.

PROVE: nYUZ , nVWZ

Statements Reasons

1. nXYZ is a right n with altitude } YU . 1. ?

2. nXYZ , nYUZ 2. ?

3. } VW i } XY 3. ?

4. ∠VWZ > ∠XYZ 4. ?

5. ∠Z > ∠Z 5. ?

6. ? 6. AA Similarity Postulate

7. nYUZ , nVWZ 7. ?

In Exercises 17–19, use the diagram.

17. Sketch the three similar triangles in the diagram.

J LM

K Label the vertices.

18. Write similarity statements for the three triangles.

19. Which segment’s length is the geometric mean of LM and JM?

20. Kite Design You are designing a diamond-shaped B

A

D

C

kite. You know that AB 5 38.4 centimeters, BC 5 72 centimeters, and AC 5 81.6 centimeters. You want to use a straight crossbar } BD . About how long should it be?

Practice B continuedFor use with the lesson “Use Similar Right Triangles”

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Lesson

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Lesson Use the Converse of the Pythagorean Theorem, continued 7. Given: AX 5 BX 5 AY 5 BY 5 7.5 in. DY 5 19.5 2 7.5 5 12 in.

Then, DX 5 Ï}}

(19.5)2 2 (7.5)2 5 18 in. because n ABD is isosceles. Because the pot is tangent to the planter at X, Y, and Z, CX 5 CY 5 CZ 5 5 in., the radius of the pot.

CD 5 DX 2 CX 5 18 2 5 5 13 in.

(CD)2 5 (CY)2 1 (DY)2

(CD)2 5 (CZ)2 1 (DZ)2

132 5 52 1 122

169 5 25 1 144

So, the screw locations hit the pot exactly where it touches the wall of the planter.

8. 0 < x < 3 1 Ï}

17 9. x > 15 1 2 Ï

}

30 } 5

10. x > 0 11. 22 Ï}

2 < x < 0 or 0 < x < 2 Ï}

2

12. x > 5 1 Ï

} 5 }

20 13. 5 < x <

40 1 10 Ï}

10 }

3

14. Quadrant II or III; Sample answer:

XY 5 Ï}

26 , XZ 5 Î}

106 1 }

4 ; YZ 5

Î}

83 1 }

4

So, n XYZ is not a right triangle by the Converse of the Pythagorean Theorem. The point Q needs to be above or below } PX . By Theorem 3, n XPQ is acute. So, Q has to be in either Quadrant II or III.

Lesson Use Similar Right Triangles

Teaching Guide

1. nBCD ~ nACB, nABD ~ nACB, nBCD ~ nABD 2. about 7.2 ft

3. about 12.1 ft, about 4.3 ft

Practice Level A

1. n ABC , n ADB , n BDC

2. n TSR , n TWS , n SWR 3.

DB

D

CBBACA

4.

W

S W

TSSRTR

5. 2.4 6. 7.1 7. C 8. x }

8 5

8 } 4 ; x 5 16

9. 12

} x 5

x } 3 ; x 5 6 10.

5 }

x 5

x } 4 ; x 5 2 Ï

} 5

11. x 5 8 12. y 5 4 13. z 5 2 14. no

15. yes; 14.4 16. yes; 5.4 17. 5 in.

Practice Level B

1. 12; 18 2. 5; 5 Ï}

3 3. 20; 6 Ï}

5 4. a 5 3

5. x 5 4 Ï}

3 6. b 5 25

} 6 7. w 5 3

8. x 5 12.25, y 5 3.75, z 5 7 Ï

}

15 }

4

9. a 5 42 2 }

3 , b 5 40, c 5 53

1 }

3 10. yes; 9.1

11. no 12. yes; 15.4 13. AC 5 50, BD 5 24

14. AC 5 2, BD 5 Ï}

15 15. AC 5 7, BD 5 Ï}

10

16. 1. Given 2. Theorem 5 3. Given 4. Corresponding Angles Postulate 5. Reflexive Property of Congruence 6. nXYZ , nVWZ 7. Transitive Property

17.

L

J

L

M

MJ

K

K

K

18. nLKJ , nKMJ, nLKJ , nLMK, nKMJ , nLMK 19. } KM 20. about 67.8 cm

Practice Level C

1. n FED , n FGE , n EGD; EG

2. n MQP , n MNQ , n QNP; NQ

3. n RST , n RUS , n SUT; RS

4. m 5 4 5. n 5 3 Ï}

6 6. k 5 4 7. w 5 6

8. y 5 3 9. a 5 14 10. yes; 11.5

11. yes; 17.7 12. no 13. AC 5 34, BD 5 240

} 17

14. AC 5 Ï}

30 , BD 5 15 Ï

}

2 }

2

15. AC 5 62 Ï

}

17 }

17 , BD 5 2 Ï

}

14

16. 1. Given; 2. Geometric Mean (Altitude) Theorem 6; 3. Given; 4. Substitution; 5. Solve for UZ; 6. Pythagorean Theorem; 7. Solve for YZ; 8. Substitution; 9. Simplify

17. Sample Answer:

Because AB

} CB

5 CB

} DB , (BC)2 5 (CB)2 5 AB p BD.

an

sw

er

s

A4GeometryChapter Resource Book

7.2

7.3