8.1 Pythagorean Theorem and Pythagorean Triples

Chapter 8 – Right Triangle Trigonometry Answer Key

CK-12 Geometry Concepts 1

8.1 Pythagorean Theorem and Pythagorean Triples

Answers

1. 505

2. 9 5

3. 799

4. 12

5. 10

6. 10 14

7. 26

8. 3 41

9. 2 2x y

10. 9 2

11. Yes

12. No

13. No

14. Yes

15. Yes

16. No

17. 𝑎2 + 2𝑎𝑏 + 𝑏2

18. 𝑐2 + 4 (1

2) 𝑎𝑏 = 𝑐2 + 2𝑎𝑏

19. 𝑎2 + 2𝑎𝑏 + 𝑏2 = 𝑐2 + 2𝑎, simplify to get the Pythagorean Theorem.

20. 1

2(𝑎 + 𝑏)(𝑎 + 𝑏) =

1

2(𝑎2 + 2𝑎𝑏 + 𝑏2)

21. 2 (1

2) 𝑎𝑏 +

1

2𝑐 = 𝑎𝑏 +

1

2𝑐

22. 1

2(𝑎2 + 2𝑎𝑏 + 𝑏2) = 𝑎𝑏 +

1

2𝑐, simplify to get the Pythagorean Theorem.



8.2 Applications of the Pythagorean Theorem

Answers

1. 124.9 u2

2. 289.97 u2

3. 72.0 u2

4. 4 5

5. 493

6. 5 10

7. 20.6” x 36.6”

8. 25.2” x 33.6”

9. √3

4𝑠2

10. 16√3

11. a) c = 15

b) 12 < c < 5

c) 15 < c < 21

12. a) a = 7

b) 7 < a < 24

c) 1 < c < 7

13. obtuse

14. right

15. acute

16. acute



17. right

18. obtuse

19. obtuse

20. acute

21. obtuse

22. One way is to use the distance formula to find the distances of all three sides and then

use the converse of the Pythagorean Theorem. The second way would be to find the

slope of all three sides and determine if two sides are perpendicular.



8.3 Inscribed Similar Triangles

Answers

1. ∆𝐾𝑀𝐿~∆𝐽𝑀𝐿~∆𝐽𝐾𝐿

2. 6 3

3. 6 7

4. 3 21

5. 16√2

6. 15√7

7. 2√35

8. 14√6

9. 20√10

10. 2√102

11. 𝑥 = 12√5

12. 𝑦 = 5√5

13. 𝑧 = 9√2

14. 𝑥 = 4

15. 𝑦 = √465

16. 𝑧 = 14√5

17. 32 8 41

, 10.25, 2 41 12.815 5

x y z

18. x = 9, y = 3 34

19. 9 481 81

9.87, , 4020 40

x y z

20. 6.1%



21. 10.4%

22. 9.4%

23. ratios are 3

1 and

9

3, which both reduce to the common ratio 3. Yes, this is true for the next

pair of terms since 27

9 also reduces to 3.

24. geometric mean; geometric mean

25. 10

26. 20

27. 1



8.4 45-45-90 Right Triangles

Answers

1. 4 2

2. 2x

3. 15 2

4. 11 2

5. 12

6. 4 6

7. 90 2 or 127.3’

8. 2

2

s

9. 𝑎 = 2√2, 𝑏 = 2

10. 𝑐 = 6√2, 𝑑 = 12

11. 𝑒 = 𝑓 = 13√2

12. 𝑞 = 14, 𝑝 = 14√2

13. 𝑥 = 𝑤 = 9√2

14. 𝑠 = 2√2, 𝑡 = 4

15. 𝑓 = 𝑔 =15√2

2 𝑜𝑟 7.5√2



8.5 30-60-90 Right Triangles

Answers

1. 𝑏 = 5√3 , ℎ = 10

2. 𝑏 = 𝑥√3, ℎ = 2𝑥

3. 12

4. 10 3

5. 𝑔 = 10√3, ℎ = 20

6. 𝑘 = 12, 𝑗 = 12√3

7. 𝑥 = 11√3, 𝑦 = 22√3

8. m = 9, n = 18

9. 𝑡 = 3√3, 𝑠 = 9

10. 𝑎 = 9√3, 𝑏 = 18√3

11. 𝑞 = 6√3, 𝑝 = 18

12. 𝑣 = 15, 𝑤 = 10√3

13. 3

2√3 in

14. 25

4√3 𝑓𝑡2

15. 27

2√3 𝑖𝑛2

16. 12

17. 3960 ft



8.6 Sine, Cosine, and Tangent

Answers

1. 𝑡𝑎𝑛𝐷 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡=

𝑑

𝑓

2. 𝑠𝑖𝑛𝐹 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=

𝑓

𝑒

3. 𝑡𝑎𝑛𝐹 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡=

𝑓

𝑑

4. 𝑐𝑜𝑠𝐹 =𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡


𝑑

𝑒

5. 𝑠𝑖𝑛𝐷 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒


𝑑

𝑒

6. 𝑐𝑜𝑠𝐷 =𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡


𝑓

𝑒

7. cos D = sin F and sin D = cos F

8. reciprocals

9. 2 2

sin , cos , tan 12 2

A A A

10. 1 2 2 2

sin , cos , tan3 3 4

A A A

11. 4 3 4


A A A

12. 1 3 3


A A A

13. 8 15 8

sin , cos , tan17 17 15

A A A

14. Because the legs of a right triangle can never be longer than the hypotenuse, making

the sine and cosine ratio always less than one.

15. The legs of a 45-45-90 triangle are always the same, making the tangent ratio x

x, which

will always equal one.

16. The tangent will increase as the angle increases. When the angle approaches 90, the

tangent gets larger very quickly.



8.7 Trigonometric Ratios with a Calculator

Answers

1. sin 24 ≈ 0.4067

2. cos 45 ≈ 0.7071

3. tan 88 ≈ 28.6363

4. sin 43 ≈ 0.6820

5. tan 12 ≈ 0.2126

6. cos 79 ≈ 0.1908

7. sin 82 ≈ 0.9903

8. x ≈ 9.4, y ≈ 17.7

9. x ≈ 14.1, y ≈ 19.4

10. x ≈ 20.8, y ≈ 22.3

11. x ≈ 19.3, y ≈ 5.2

12. sin 80 ≈ 0.9849, cos 10 ≈ 0.9849

13. The sine of an angle is the same as the cosine of the other acute angle in the right

triangle.

14. 90

15. They are reciprocals of each other.



8.8 Trigonometry Word Problems

Answers

1. x ≈ 435.9 ft.

2. x = 56 meters

3. 25.3 ft

4. 42.9 ft

5. 94.6 ft

6. 49 ft

7. 14 miles

8. 47.6

9. 1.6

10. 44.0

11. 192

11𝑓𝑡 ≈ 17 𝑓𝑡 5 𝑖𝑛; 54°

12. 51°

13. Tommy used 𝐴

𝑂 instead of

𝑂

𝐴 for his tangent ratio.

14. Tommy used the correct ratio in his equation here, but he used the incorrect angle

measure he found previously which caused his answer to be incorrect. This illustrates

the benefit of using given information whenever possible.

15. Tommy could have used Pythagorean Theorem to find the hypotenuse instead of a

trigonometric ratio.



8.9 Inverse Trigonometric Ratios

Answers

1. 𝑚∠𝐴 = sin−1 (10

18) = 33.7°

2. 𝑚∠𝐴 = tan−1 (9

15) = 31.0°

3. 𝑚∠𝐴 = cos−1 (32

45) = 44.7°

4. 𝑚∠𝐴 = tan−1 (23

28) = 39.4°

5. 𝑚∠𝐴 = cos−1 (11

16) = 46.6°

6. 𝑚∠𝐴 = sin−1 (6

10) = 36.9°

7. 𝑚∠𝐴 = sin−1(0.5684) = 34.6°

8. 𝑚∠𝐴 = cos−1(0.1234) = 82.9°

9. 𝑚∠𝐴 = tan−1(2.78) = 70.2°

10. mA = 38, BC ≈ 9.38, AC ≈ 15.23

11. mA = 18.4, mB = 71.6, AB ≈ 12.6

12. mA ≈ 45.6, mC ≈ 44.4, AB ≈ 7.14

13. mA = 60, BC = 12, AC = 12 3

14. mA ≈ 48.2, mB ≈ 41.8, CB ≈ 15.7

15. mB = 50, AB ≈ 49.8, AC ≈ 38.1

16. You would use a trig ratio when given a side and an angle and the Pythagorean

Theorem if you are given two sides and no angles.



8.10 Laws of Sines and Cosines

Answers

1. mB = 84, a = 10.9, b = 13.4

2. mB = 47, a = 16.4, c = 11.8

3. mA = 38.8, mC = 39.2, c = 16.2

4. b = 8.5, mA = 96.1, mC = 55.9

5. mA = 25.7, mB = 36.6, mC = 117.7

6. mA = 81, mB = 55.4, mC = 43.6

7. b = 11.8, mA = 42, mC = 57

8. b = 8.0, mB = 25.2, mC = 39.8

9. mA = 33.6, mB = 50.7, mC = 95.7

10. mC = 95, AC = 3.2, AB = 16.6

11. BC = 33.7, mC = 39.3, mB = 76.7

12. mA = 42, BC = 34.9, AC = 22.0

13. 𝑚∠𝐵 = 105°, 𝑚∠𝐶 = 55°, 𝐴𝐶 = 14.1

14. 𝑚∠𝐵 = 35°, 𝐴𝐵 = 12, 𝐵𝐶 = 5

15. Yes, BC would still be 5 units (see

isosceles triangle in diagram); the

measures of ∠𝐶 are supplementary

as shown. A

B

CC

512

20° 55°125° 55°

70°35°

8.4

14.1| |

8.1 Pythagorean Theorem and Pythagorean Triples

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