Right Triangles & Trigonometry Chapter Questions
1. What is the Pythagorean Theorem? What is the Pythagorean Theorem Converse?
2. How do you use the Pythagorean Theorem Converse to classify triangles by their angles?
3. Why does the altitude of a right triangle divide the triangle into two smaller triangles that
are similar to the original triangle and to each other?
4. What are the similarities and differences between a geometric mean and an arithmetic
mean?
5. Explain the difference of the side lengths of a 45°-45°-90° triangle and a 30°-60°-90°
triangle?
6. When do you use the sine, cosine or tangent trigonometric ratios to solve right triangles?
7. When do you use the inverse sine, cosine or tangent trigonometric ratios to solve right
triangles?
8. Explain why co-functions of complementary angles are equal (sine and cosine are
co-functions).
9. What of types of triangles can be solved using the Law of Sines and
Law of Cosines? What information must be given to use the Law of Sines? What
information must be given to use the Law of Cosines?
Geometry – Rt. Triangle Trig
~1~
NJCTL.org
Right Triangles & Trigonometry Chapter Problems
Pythagorean Theorem and its Converse
Class work
1. In the triangle above, the hypotenuse is side ______, the legs are sides _____ and ______.
Refer the diagram below to answer questions 2-7. Write the answer in simplest radical form.
2. If AB = 9 and BC = 12, then AC = _________.
3. If AB = 11 and AC = 61, then BC = _________.
4. If BC = 45 and AC = 53, then AB = __________.
5. If AC = 9 and AB = 3, then BC = _________.
6. If BC = 14 and AC = 8√7, then AB = __________.
7. If AB = 2√3 and BC = 6√5, then AC = _________.
Classify the sides as lengths of an acute, right, obtuse, or not a triangle.
8. 8, 11, 5
9. 20, 29, 21
10. 6, 2, 2√2
11. 9, 11√2, 7√5
12. Find the perimeter and area of the rectangle.
Geometry – Rt. Triangle Trig
~2~
NJCTL.org
Homework
13. In the triangle above the hypotenuse is side ______, the legs are sides _____ and ______.
Refer the diagram below to answer questions 14-19. Write the answer in simplest radical form.
14. If AB = 33 and BC = 56, then AC = _________.
15. If AB = 12 and AC = 37, then BC = _________.
16. If BC = 80 and AC = 89, then AB = __________.
17. If AC = 14 and AB = 5√2, then BC = _________.
18. If BC = 9 and AC = 16, then AB = __________.
19. If AB = 5√2 and BC = 7√6, then AC = _________.
Classify the sides as lengths of an acute, right, obtuse, or not a triangle.
20. 19, 7, 15
21. 9, 4, 3
22. 14, 14, 14√2
23. 3√11, 4√7, 5√6
24. Find the perimeter and area of the triangle.
31cm
29cm
Geometry – Rt. Triangle Trig
~3~
NJCTL.org
Similarity in Right Triangles
Classwork
25. Find the geometric mean of the pairs. Write the answer is simplest radical form.
a. 8 and 32
b. 2 and 24
c. 10 and 15
d. 14 and 18
26. GE is the altitude of triangle DEF. What are the three similar triangles?
Solve for x.
27.
28.
29.
Geometry – Rt. Triangle Trig
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NJCTL.org
30.
31.
32.
Homework
33. Find the geometric mean of the pairs. Write the answer is simplest radical form.
a. 9 and 25
b. 6 and 54
c. 12 and 20
d. 10 and 28
34. SQ is the altitude of triangle PQR. What are the three similar triangles?
Geometry – Rt. Triangle Trig
~5~
NJCTL.org
Solve for x.
35.
36.
37.
38.
39.
Geometry – Rt. Triangle Trig
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40.
Special Right Triangles
Class work
41. The length of each leg of an isosceles right triangle is 13 cm. What is the length of the hypotenuse?
42. The length of the hypotenuse of a 45o-45o-90o triangle is 16 cm. What is the length of each
leg?
43. The length of the shorter leg of a 30o-60o-90o triangle is 5 cm. What is the length of the longer leg? What
is the length of the hypotenuse?
44. The length of the longer leg of a 30o-60o-90o triangle is 12 cm. What is the length of the
hypotenuse? What is the length of the shorter leg?
45. The length of the hypotenuse of a 30o-60o-90o triangle is 12 cm. What is the length of the
longer leg? What is the length of the shorter leg?
Find the lengths of the missing sides.
46.
y
60
18cm
x
47.
x
60
18cm
y
Geometry – Rt. Triangle Trig
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NJCTL.org
48.
x
y
60
18cm
49.
x
45
y
6cm
50.
x
6cm
45
y
51. Find the area of the triangle. Round to the nearest hundredth.
52. A skateboarder constructs a ramp using plywood. The height of the ramp is 4 feet. If the ramp falls at a
60o angle, what is the length of the plywood?
Homework
53. The length of each leg of an isosceles right triangle is 9 cm. What is the length of the
hypotenuse?
54. The length of the hypotenuse of a 45o-45o-90o triangle is 10 cm. What is the length of each
leg?
Geometry – Rt. Triangle Trig
~8~
NJCTL.org
55. The length of the shorter leg of a 30o-60o-90o triangle is 13 cm. What is the length of the
longer leg? What is the length of the hypotenuse?
56. The length of the longer leg of a 30o-60o-90o triangle is 21 cm. What is the length of the
hypotenuse? What is the length of the shorter leg?
57. The length of the hypotenuse of a 30o-60o-90o triangle is 21 cm. What is the length of the
longer leg? What is the length of the shorter leg?
Find the lengths of the missing sides.
58.
y
60
24cm
x
59.
24cm
60
y
x
60.
x
60
y
24cm
61.
10cm
y
45
x
Geometry – Rt. Triangle Trig
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NJCTL.org
62.
x
10cm
45
y
63. Find the area of the triangle. Round to the nearest hundredth.
64. A skateboarder constructs a ramp using plywood. The length of the plywood is 7 feet long
and falls at a 40o angle. What is the height of the ramp?
Trigonometric Ratios
Class work
Refer to the triangle below to answer questions 65-66.
65. side opposite to ∠W is _______side adjacent to ∠W is _______
side opposite to∠B is _______side adjacent to ∠B is _______hypotenuse is _______.
66. sinW = ______
sinB = ______
cosW = ______
tanW = _______.
cosB = ______
tanB = _______.
Geometry – Rt. Triangle Trig
~10~
NJCTL.org
Evaluate. Round the nearest ten-thousandth.
67. sin90o
68. cos52o
69. tan25o
70. tan88o
71. Find the length of ̅̅̅̅
AF. Round to the nearest hundredth.
72. Find the length of ̅̅̅̅
SO. Round to the nearest hundredth.
̅̅̅̅. Round to the nearest hundredth.
73. Find the length of KU
74. If the sin30 = .5, the cos60= ______________. Why?
If the cos25= .9063, the sin65=_____________. Why?
Geometry – Rt. Triangle Trig
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NJCTL.org
Homework
Refer to the triangle below to answer questions 75-76.
75. side opposite to ∠W is _______side adjacent to ∠W is _______
side opposite to ∠B is _______side adjacent to ∠B is _______hypotenuse is _______.
76. sinW = ______
sinB = ______
cosW = ______
tanW = _______.
cosB = ______
tanB = _______.
Evaluate. Round the nearest ten-thousandth.
77. sin45o
78. tan 77o
79. cos69o
80. tan33o
̅̅̅̅. Round to the nearest hundredth.
81. Find the length of RF
̅̅̅. Round to the nearest hundredth.
82. Find the length of SC
Geometry – Rt. Triangle Trig
~12~
NJCTL.org
83. Find the length of ̅̅̅̅
BKBK. Round to the nearest hundredth.
84. If the sin60 = .8660, the cos30= ______________. Why?
If the cos65= .4226, the sin25=_____________. Why?
Solving Right Triangles
Class work
Solve the triangle. Round to the nearest hundredth.
85.
m∠C = ______ AC = ______
BC = ______
86.
m∠D = ______ m∠F = ______ DF = ______
Geometry – Rt. Triangle Trig
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NJCTL.org
87.
m∠G = ______ m∠I = ______ GH = ______
88.
m∠J = ______ KL = ______
JL = ______
Homework
Solve the triangle. Round to the nearest hundredth.
89.
m∠C = ______ AC = ______
BC = ______
90.
m∠D = ______ m∠F = ______ DF = ______
Geometry – Rt. Triangle Trig
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NJCTL.org
91.
m∠G = ______ m∠I = ______ GH = ______
92.
m∠J = ______ KL = ______
JL = ______
Angle of Elevation and Depression
Class work
93. Angela flies a kite at a 60o angle of elevation. The kite’s string is 275 feet. Angela’s arm is 4.5 feet off
the ground. How high is the kite off the ground? (Round to the nearest hundredth)
94. An airplane flies 6 miles above the ground. The distance from the start of the runway to the airplane is
15 miles. What is the angle of depression the airplane must use to land? (Round to the nearest hundredth)
95. You are looking at the top of a tree. The angle of elevation is 78o. The distance from the top of the tree
to your position is 125 feet. If you are 5.25 feet tall, how far are you from the base of the tree? (Round to the
nearest hundredth)
Homework
96. An airplane is 20 miles from the start of the runway. The angle of depression the airplane must use to
land is 40o. How high is the airplane above the ground? (Round to the nearest hundredth)
97. You are standing on a mountain that is 6842 feet high. You look down at your campsite at angle of 38 o.
If you are 5.6 feet tall, how far is the base of the mountain from the campsite? (Round to the nearest
hundredth)
98. You are looking at the top of a tree. The angle of elevation is 66o. The distance from the top of the tree
to your position is 108 feet. If you are 5.75 feet tall, how tall is the tree? (Round to the nearest hundredth)
Geometry – Rt. Triangle Trig
~15~
NJCTL.org
Law of Sines and Law of Cosines
Class Work
99. Solve the triangle.
m∠P = ______ m∠Q = ______ PR = ______
100. Solve the triangle.
m∠S = ______ m∠T = ______ m∠U = ______
101. Solve the triangle.
E
19
42°
26
D
F
m∠E = ______ m∠F = ______ EF = ______
102. Solve the triangle.
z
65
y
x
30
25
Geometry – Rt. Triangle Trig
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NJCTL.org
103. Solve the triangle.
x
y
z
40
60
10
104. Show why
𝑠𝑖𝑛𝐴
𝑎
=
𝑠𝑖𝑛𝐶
𝑐
by drawing an auxiliary line from a vertex perpendicular to the
opposite side.
B
c
a
C
A
b
Homework
105. Solve the triangle.
m∠P = ______ m∠Q = ______ QR = ______
106. Solve the triangle.
m∠S = ______ m∠T = ______ m∠U = ______
Geometry – Rt. Triangle Trig
~17~
NJCTL.org
107. Solve the triangle.
E
12
105°
F
10
D
m∠E = ______ m∠F = ______ DE = ______
108. Solve the triangle.
58
z
25
x
23
y
109. Solve the triangle.
45
y
x
12
95
z
110. Show why
𝑠𝑖𝑛𝐵
𝑏
=
𝑠𝑖𝑛𝐶
𝑐
by drawing an auxiliary line from a vertex perpendicular to the
opposite side.
B
a
C
c
b
A
Geometry – Rt. Triangle Trig
~18~
NJCTL.org
Area of an Oblique Triangle
Class Work
Find the area of the triangle for #’s 111-114
111.
12
55°
13
112.
8
55°
12
47°
113.
14
14
114.
8
14
48°
70°
18
Geometry – Rt. Triangle Trig
~19~
NJCTL.org
115. Derive the formula A=1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side. Given: a, b and m∠C Find: A=1/2 ab sin(C).
B
c
a
C
A
b
Homework
Find the area of the triangle #’s 116-119
116.
18
23
30°
117.
45°
16
37°
10
118.
10
10
8
Geometry – Rt. Triangle Trig
~20~
NJCTL.org
119.
22
100°
43°
16
120. Derive the formula A=1/2 ac sin(B) for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.
B
a
C
c
b
A
Right Triangles & Trigonometry Unit Review
Multiple Choice - Choose the correct answer for each question. No partial credit will be given.
1.
Find the length of the missing side of the triangle. Write the answer in simplest radical form.
a. 117
2.
c. √117
d. 3√13
Find the length of the missing side of the triangle. Write the answer in simplest radical form.
a. 4√22
3.
b. 9√13
b. 16√22
c. 18.76
d. √352
Tell whether the lengths 7√2, 3√7, and 5 form the sides of an acute, right, obtuse, or
not a triangle.
a. acute
Geometry – Rt. Triangle Trig
b. right
c. obtuse
~21~
d. not a triangle
NJCTL.org
4.
Find the value of x. Write the answer in simplest radical form.
a.
5.
15√2
2
b. 8√2
c.
8√2
2
d. 4
Find the value of x. Write the answer in simplest radical form.
a. 2√2
7.
d. 9√13
Find the value of x. Write the answer in simplest radical form.
a. 4√2
6.
c. 10√3
b. 7.5
b. 4√2
c.
4√2
2
d. 2
In the triangle below, what is the side opposite to ∠J.
a. JR
Geometry – Rt. Triangle Trig
b. LJ
c. JL
~22~
d. LR
NJCTL.org
8.
What is the tanK?
a. 2
9.
c.
2
5
7
d.
7
5
b. 56.31o
c. 41.81o
d. 48.19o
Find the length of ̅̅̅̅
DT. Round the answer to the nearest hundredth.
a. 8.31
11.
1
Find the m∠V. Round the answer to the nearest hundredth.
a. 33.69o
10.
b.
b. 6.64
c. 6.26
d. 3.99
Find the m∠D. Round the answer to the nearest hundredth.
a. 50.71o
Geometry – Rt. Triangle Trig
b. 39.29o
c. 35.10o
~23~
d. 54.90o
NJCTL.org
12.
Find the m∠V. Round the answer to the nearest hundredth.
a. 30.18o
13.
b. 27.45o
c. 32o
d. 30.179o
Find the m∠C. Round the answer to the nearest hundredth.
a. 71.44o
b. 54.70o
14. Find the geometric mean of 5 and 9.
a. 3√5
b. 7
c. 85.90o
d. 39.40o
c. 5√3
d. 9√3
c. 5√7
d. 25
15. Solve for x.
16
9
x
a. 7
b. 12
16. In the figure, sin 47 = 15/x, which of the following equations is also true?
15
x
47°
a. sin 43 = x/15
Geometry – Rt. Triangle Trig
b. cos 43 = 15/x
c. cos 47 = 15/x
~24~
d. tan 47 = x/15
NJCTL.org
Short Constructed Response - Write the answer for each question. No partial credit will be given.
17. Tell whether the lengths 7√3, 3√11, and 4√5 form the sides of an acute, right, obtuse or not a
triangle.
18.
Michael is flying a kite at an angle of 45o. The kite's string is 202 feet long and Amy arm is
4 feet off the ground. How high is the kite off the ground?
19.
Find the area of the triangle.
Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.
20. Solve the triangle and find the area. Round to the nearest hundredth.
Geometry – Rt. Triangle Trig
~25~
NJCTL.org
Answer Key
1. hypotenuse – DL/sides - DU and UL
2. 15
3. 60
4. 28
5. 6√2
6. 6√7
7. 8√3
8. obtuse
9. right
10. Not a Triangle
11. acute
12. Perimeter = 46in/Area = 120in2
13. hypotenuse – ME/sides - MQ and QE
14. 65
15. 35
16. 39
17. √146
18. 5√7
19. 2√86
20. obtuse
21. Not a Triangle
22. right
23. acute
24. Perimeter = (62+4√30)cm≈83.9cm
/Area = 58√30cm2≈306.7 cm2
37. 4√6
38. 4
39. 30
40. 420/2
41. 13√2cm
42. 8√2cm
43.LL = 5√3cm H=8√3𝑐𝑚
44. 𝐻 = 8√3cm SL=4√3
45. LL = 6√3, SL = 6
46. x = 18√3𝑐𝑚, y = 36cm
47. x = 9𝑐𝑚, y = 9√3cm
48. 𝑥 = 6√3𝑐𝑚, y=12√3𝑐𝑚
49. x=6cm, y=6√2𝑐𝑚
50. x=3√2, y=3√2
51. 292.72ft2
52. 8ft
53. 9√2 cm
54. 5√2 cm
55. H=26cm LL=13√3 cm
56. SL=7√3 cm H=14√3 cm
57. LL=(21√3)/2cm SL=10.5cm
58. x=24√3 𝑐𝑚 y=48
59. x=12√3cm, y=12cm
60. x=16√3 𝑐𝑚, y=8√3 𝑐𝑚
61. x=10, y=10√2 cm
62. x=5√2 cm, y=5√2 cm
63. 187.06 ft2
64. 6.58ft
65. 10,24,24,10,26
66. 5/3,12/13,5/12,12/13,5/13,12/5
67. 1.0000
68. 0.6157
69. 0.4663
70. 28.6363
71. 8.07
72. 14.83
73. 16.26
74. 0.5, 0.9063 because co-functions of
complementary angels are equal
75. 14, 48, 48, 14, 50
76. 7/25, 24/25, 7/24, 24/25, 7/25, 24/7
77. 0.7072
78. 4.3315
79. 0.3584
80. 0.6494
81. 22.28
82. 23.66
25.
a. 16
b. 4√3
c. 5√6
d. 6√7
26. FEGEDGFDE
27. 4.8
28. 2√10
29. 4 or 16
30. 4√6
31. 7.2
1
32. 53
33.
a. 15
b. 18
c. 4√15
d. 2√70
34. RQPQSPRSQ
35. 5.4
36. x=2
Geometry – Rt. Triangle Trig
~26~
NJCTL.org
83. 2.20
84. 0.8660, 0.4226 because co-functions of
complementary angels are equal
85. m∠C = 63o, BC = 6.62, AC = 14.59
86. m∠D = 59.74o, m∠F = 30.26o, DF =
13.89
87. m∠G = 39.52o, m∠I = 50.48o, GH = 8.49
88. m∠J = 22o, KL = 6.74, JL = 16.69
89. m∠C = 57o, BC = 5.84, AC = 10.73
90. m∠D = 60.26o, m∠F = 29.74o, DF = 16.12
91. m∠G = 53.13o, m∠I = 36.87o, GH = 9
92. m∠J = 16, KL = 5.51, JL = 19.23
93. 242.66 ft
94. 23.58o
95. 25.99ft
96. 12.86ft
97. 8764.37ft
98. 104.41ft
99. m∠P = 57.41o, m∠Q = 85.59o, PR = 8.28
100. m∠S = 126.22o, m∠T = 31.47o, m∠U =
22.31o
101. m∠E = 89o, m∠F = 49o, EF = 17.4
102. x=85o, y=13.79, z=27.48
103. x=80o, y=8.79, z=6.53
104. sin C = h/a, h=a sinC
sin A = h/c, h=c sinA
a sin C = c sin A
(sin C)/c = (sin A)/a
105. m∠P = 90.02o, m∠Q = 57.98o, QR =
9.44
106. m∠S = 39.57o, m∠T = 121.86o, m∠U =
18.57o
107. m∠E = 53.6o, m∠F = 21.4o, DE = 4.53
108. x=99 o, y=54.26, z=63.19
109. x=40 o,y=18.60,z=13.20
110. sin C = h/b, h=b sinC
sin B = h/c, h=c sinB
b sin C = c sin B
(sin C)/c = (sin B)/b
111. 63.89 sq units
112. 46.95 sq units
113. 53.66 sq units
114. 93.64 sq units
115. sin C = h/a
h=a sin C
A=(1/2)(base)(height)
A=(1/2)b (a sinC)
A=(1/2)ab sin C
116. 103.5 sq units
117. 79.22 sq units
118. 34 sq units
119. 173.33 sq units
120. sin B = h/c
h=c sin B
A=(1/2)(base)(height)
A=(1/2)a (c sinB)
A=(1/2)ac sin B
Right Triangles & Trig Unit Review
1. D
2. A
3. C
4. C
5. A
6. B
7. D
8. A
9. A
10. C
11. D
Geometry – Rt. Triangle Trig
12. A
13. C
14. A
15. B
16. B
17. acute
18. 146.83 feet
19. 22√26
20. BC = 9.54, m∠C = 20, m∠B=126.97,
A=22.87 sq units
~27~
NJCTL.org