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
Transcript
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 andLaw 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.
Geometry – Rt. Triangle Trig ~3~ NJCTL.org
29cm31cm
Similarity in Right TrianglesClasswork
25. Find the geometric mean of the pairs. Write the answer is simplest radical form.a. 8 and 32b. 2 and 24c. 10 and 15d. 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 ~4~ NJCTL.org
30.
31.
32.
Homework
33. Find the geometric mean of the pairs. Write the answer is simplest radical form. a. 9 and 25b. 6 and 54c. 12 and 20d. 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 ~6~ NJCTL.org
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 eachleg?
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 thehypotenuse? 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 thelonger leg? What is the length of the shorter leg?
Find the lengths of the missing sides.
46.
Geometry – Rt. Triangle Trig ~7~ NJCTL.org
y
x
18cm60
47.
y
x18cm60
48.
yx
18cm
60
49.
yx
6cm
45
50.
y
x6cm45
51. Find the area of the triangle. Round to the nearest hundredth.
Geometry – Rt. Triangle Trig ~8~ NJCTL.org
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 thehypotenuse?
54. The length of the hypotenuse of a 45o-45o-90o triangle is 10 cm. What is the length of eachleg?
55. The length of the shorter leg of a 30o-60o-90o triangle is 13 cm. What is the length of thelonger 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 thehypotenuse? 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 thelonger leg? What is the length of the shorter leg?
Find the lengths of the missing sides.
58.
y
x
24cm60
59.
y
x
24cm60
60.
Geometry – Rt. Triangle Trig ~9~ NJCTL.org
yx
24cm
60
61.
y
x
10cm45
62.
y
x10cm45
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 longand 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.
Geometry – Rt. Triangle Trig ~10~ NJCTL.org
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 = ______ cosW = ______ tanW = _______.
sinB = ______ cosB = ______ tanB = _______.
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.
Geometry – Rt. Triangle Trig ~11~ NJCTL.org
73. Find the length of KU . Round to the nearest hundredth.
74. If the sin30 = .5, the cos60= ______________. Why?
If the cos25= .9063, the sin65=_____________. Why?
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 = ______ cosW = ______ tanW = _______.
sinB = ______ cosB = ______ tanB = _______.
Geometry – Rt. Triangle Trig ~12~ NJCTL.org
Evaluate. Round the nearest ten-thousandth.
77. sin45o
78. tan 77o
79. cos69o
80. tan33o
81. Find the length of RF. Round to the nearest hundredth.
82. Find the length of SC. Round to the nearest hundredth.
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
Geometry – Rt. Triangle Trig ~13~ NJCTL.org
Solve the triangle. Round to the nearest hundredth.
85.
m∠C = ______ AC = ______ BC = ______
86.
m∠D = ______ m∠F = ______ DF = ______
87.
m∠G = ______m∠I = ______ GH = ______
88.
Geometry – Rt. Triangle Trig ~14~ NJCTL.org
m∠J = ______ KL = ______ JL = ______
Homework
Solve the triangle. Round to the nearest hundredth.
89.
m∠C = ______ AC = ______ BC = ______
90.
m∠D = ______ m∠F = ______ DF = ______
91.
m∠G = ______m∠I = ______ GH = ______
92.
m∠J = ______ KL = ______ JL = ______
Geometry – Rt. Triangle Trig ~15~ NJCTL.org
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 38o. 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)Law of Sines and Law of Cosines
Class Work
99. Solve the triangle.
m∠P = ______ m∠Q = ______PR = ______
100. Solve the triangle.
Geometry – Rt. Triangle Trig ~16~ NJCTL.org
m∠S = ______ m∠T = ______ m∠U = ______
101. Solve the triangle.
F
E
D 26
19
42°
m∠E = ______ m∠F = ______ EF = ______
102. Solve the triangle.
z 65y
x25
30
103. Solve the triangle.
z
40
yx
1060
104. Show why sinAa
= sinCc
by drawing an auxiliary line from a vertex perpendicular to the
opposite side.
Geometry – Rt. Triangle Trig ~17~ NJCTL.org
b
a c
B
C A
Homework
105. Solve the triangle.
m∠P = ______ m∠Q = ______QR = ______
106. Solve the triangle.
m∠S = ______ m∠T = ______ m∠U = ______
107. Solve the triangle.
F
E
D 10
12
105°
m∠E = ______ m∠F = ______ DE = ______
108. Solve the triangle.
Geometry – Rt. Triangle Trig ~18~ NJCTL.org
z 58
yx
25
23
109. Solve the triangle.
z
45y
x
12
95
110. Show why sinBb
= sinCc
by drawing an auxiliary line from a vertex perpendicular to the
opposite side.
a c
bC
B
A
Area of an Oblique Triangle
Class Work
Find the area of the triangle for #’s 111-114
111.
Geometry – Rt. Triangle Trig ~19~ NJCTL.org
55°13
12
112.
55°
47°
12
8
113.
114.8
14 14
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).
Geometry – Rt. Triangle Trig ~20~ NJCTL.org
70°48°
18
14
100°43°16
22
b
a c
B
C A
Homework
Find the area of the triangle #’s 116-119
116.
30°23
18
117.
45°
37°
10
16
118.
119.
Geometry – Rt. Triangle Trig ~21~ NJCTL.org
8
10 10
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.
a c
bC
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 b. 9√13 c. √117 d. 3√13
2. Find the length of the missing side of the triangle. Write the answer in simplest radical form.
a. 4 √22 b. 16√22 c. 18.76 d. √352
3. Tell whether the lengths 7√2, 3√7, and 5 form the sides of an acute, right, obtuse, or not a triangle.
a. acute b. right c. obtuse d. not a triangle
4. Find the value of x. Write the answer in simplest radical form.
Geometry – Rt. Triangle Trig ~22~ NJCTL.org
a. 15√22
b. 7.5 c. 10√3 d. 9√13
5. Find the value of x. Write the answer in simplest radical form.
a. 4 √2 b. 8√2 c. 8√22
d. 4
6. Find the value of x. Write the answer in simplest radical form.
a. 2√2 b.4 √2 c. 4 √22
d. 2
7. In the triangle below, what is the side opposite to ∠J.
a. JR b. LJ c. JL d. LR
Geometry – Rt. Triangle Trig ~23~ NJCTL.org
8. What is the tanK?
a. 2 b. 12 c.
57 d.
75
9. Find the m∠V. Round the answer to the nearest hundredth.
a. 33.69o b. 56.31o c. 41.81o d. 48.19o
10. Find the length of DT . Round the answer to the nearest hundredth.
a. 8.31 b. 6.64 c. 6.26 d. 3.99
11. Find the m∠D. Round the answer to the nearest hundredth.
a. 50.71o b. 39.29o c. 35.10o d. 54.90o
Geometry – Rt. Triangle Trig ~24~ NJCTL.org
12. Find the m∠V. Round the answer to the nearest hundredth.
a. 30.18o b. 27.45o c. 32o d. 30.179o
13. Find the m∠C. Round the answer to the nearest hundredth.
a. 71.44o b. 54.70o c. 85.90o d. 39.40o
14. Find the geometric mean of 5 and 9. a. 3√5 b. 7 c. 5√3 d. 9√3
15. Solve for x.
x 9
16
a. 7 b. 12 c. 5√7 d. 25
16. In the figure, sin 47 = 15/x, which of the following equations is also true?
47°
15x
a. sin 43 = x/15 b. cos 43 = 15/x c. cos 47 = 15/x d. tan 47 = x/15
Geometry – Rt. Triangle Trig ~25~ 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 ~26~ NJCTL.org
Answer Key
1. hypotenuse – DL/sides - DU and UL 2. 153. 604. 285. 6√26. 6√77. 8√38. obtuse9. right10. Not a Triangle11. acute12. Perimeter = 46in/Area = 120in2
13. hypotenuse – ME/sides - MQ and QE 14. 6515. 3516. 3917. √14618. 5√719. 2√8620. obtuse21. Not a Triangle22. right23. acute24. Perimeter = (62+4 √30¿cm≈83.9cm/Area = 58√30cm2≈306.7 cm2
101. m∠E = 89o, m∠F = 49o, EF = 17.4102. x=85o, y=13.79, z=27.48103. x=80o, y=8.79, z=6.53104. sin C = h/a, h=a sinCsin 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 = 4.53108. x=99 o, y=54.26, z=63.19109. x=40 o,y=18.60,z=13.20110. sin C = h/b, h=b sinCsin B = h/c, h=c sinBb sin C = c sin B(sin C)/c = (sin B)/b 111. 63.89 sq units112. 46.95 sq units113. 53.66 sq units114. 93.64 sq units115. sin C = h/ah=a sin CA=(1/2)(base)(height)A=(1/2)b (a sinC)A=(1/2)ab sin C116. 103.5 sq units117. 79.22 sq units118. 34 sq units119. 173.33 sq units120. sin B = h/ch=c sin BA=(1/2)(base)(height)A=(1/2)a (c sinB)A=(1/2)ac sin B
Right Triangles & Trig Unit Review1. D2. A3. C4. C5. A6. B7. D8. A9. A10. C11. D
