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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.
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 2
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
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 3
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.
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 4
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%
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 5
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
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 6
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
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 7
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
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 8
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
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=
𝑑
𝑒
5. 𝑠𝑖𝑛𝐷 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=
𝑑
𝑒
6. 𝑐𝑜𝑠𝐷 =𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=
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
sin , cos , tan5 5 3
A A A
12. 1 3 3
sin , cos , tan2 2 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.
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 9
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.
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 10
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.
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 11
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.
Chapter 8 – Right Triangle Trigonometry Answer Key
CK-12 Geometry Concepts 12
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| |