Geometry 235
Study Guide: Chapter 8
Name:
Date:
Hour:
Section I: Find the missing side length of the following right triangles. Please show all work and leave all
answers in reduced radical form is necessary.
1. A
D
2.
H
3.
2 3
11
3
B
C
4
E
AC =
G
F
5
I
DF =
4.
5.
L
GI =
26
N
2
O
Q
6.
3 2
61
11
55
K
J
P
M
JK =
MN =
R
6
QR =
Section II: Determine if the following side lengths would make an acute, obtuse, or right triangle.
7. 18, 24, 30
7.
8.
8.
4, 4, 10
9. 3.5, 2.7, 4.3
9.
10. 4, 12, 13
10.
11. 20, 21, 29
11.
Section III: Using the given triangles below, set up the following trigonometric ratios. Leave your answers as
S
fractions in lowest terms.
S


12.
Sin =

13.
Sin =
y
x
Cos =

R


z
Q
17
15
Cos =

Tan =
T
8
Tan =
U

Section IV: Solve the following trigonometric ratios for the given variable. Round your answers to the nearest
hundredth if necessary.
14. sin 82 
x
10
15. cos 59 
x=
15
m
m=
17. cos30 
6
y
d=
18. tan 45 
y=
d
5
16. tan 28 
x
5
19. sin 60 
x=
w
9
w=
Section V: Find the missing value of the triangles below by setting up and solving a trigonometric ratio.
D
20.
45
23
Q
4
N
21.
J
22.
L
6
25
32
H
P
A
QA =
23. B
HL =
25
R
PC =
K
24.
C
O
25.
60
30
5
M
45
F
RF =
2 23
G
KG =
E
EI =
I
Section VI: Solve the following trigonometric ratios for the variable. Round your answer to the nearest
hundredth if necessary.
26. sin x 
7
10
27. tan m 
x=
15
21
28. cos w 
m=
29. tan r 
15
10
30. sin m 
r=
5
8
w=
5
19
31. cos d 
m=
5
10
d=
Section VII: Solve the following right triangles by finding all missing values. Round your answers to the
nearest hundredth if necessary.
A
32.
R
33.
45
7
S
8
B
Q
C
12
AC =
QR =
mA =
SR =
mC =
mR =
S
34.
L
35.
10
25
Q
20
R
M
64
N
SQ =
LM =
RS =
MN =
mR =
mN =
Section VIII: Solve each of the following word problems. Show all work and diagrams needed to solve the
problem, and round your answers to the nearest tenth, if necessary.
36. A television station is going to construct an 11-foot tower for a new antenna. The tower will be supported
by three cables, each attached to the top of the tower and to points on the roof of the building that are 10
feet from the base of the tower. Find the total length of the three cables.
37. A construction worker is making sure one corner of the foundation of a house is a right angle. To do this,
the worker makes a mark 7 feet from the corner along one wall and another mark 6 feet from the same
corner along the other wall. The worker then measures the distance between the two marks and finds the
distance to be 12 feet. Is the corner a right angle?
38. A water slide makes an angle of about 13 with the ground. The slide extends horizontally about 65.1
meters. What is the height of the slide in meters to the nearest tenth?
39. A mountain climber has a rope 189 feet long attached to his belt being held by his partner down on the
ground. The angle of elevation from the ground to the climber is 55°. Estimate the climber's height above
the ground. Round your answer to the nearest integer.
40. A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 14 feet and
a horizontal length of 22 feet. What is the angle of the ramp? Round your answer to the nearest integer, if
necessary.
Name: Worked Out Solution Key
Date:
Hour:
Geometry 235
Study Guide: Chapter 8
Section I: Find the missing side length of the following right triangles. Please show all work and leave all
answers in reduced radical form as necessary.
1. A
D
2.
H
3.
2 3
11
3
B
C
4
E
G
F
5

22  2 3
2
52  11  DF 2
32  4 2  AC 2
9  16  AC 2
36  DF
2
4  12  GI 2
DF =
5.
L
6
26
N
O
Q
6.
3 2
11
K
J
11  JK  61
2
2
121  JK  3721
JK  3600
2
55
M
2
26  NM 2  55
P
2
26  NM 2  55
2
3 2 
NM  29
60
MN =
29
2
R
6 2
2
 QR  6
9  2  QR 2  36
18  QR 2  36
NM 2  29
2
JK  60
JK =
QR 2  18
QR  18  3 2
QR = 3 2
Section II: Determine if the following side lengths would make an acute, obtuse, or right triangle.
7.
Right 
8.
Obtuse 
9.
Acute 
2
2
2
10. 4, 12, 13 4  12 ? 13  160  169  Obtuse 
10.
Obtuse 
11. 20, 21, 29 202  212 ? 292  841 841  Right 
11.
Right 
7. 18, 24, 30
182  242 ? 302  900  900  Right 
2
8.
4, 4, 10
I
16  GI 2
4  GI
4
GI =
61
2
 GI 2
25  11  DF
5  AC
4.
2
4  4  3  GI 2
6  DF
5

2
25  AC 2
AC =
2
2
4  10 ? 42  4 10 ? 16  14  16  Obtuse 
2
2
2
9. 3.5, 2.7, 4.3 2.7  3.5 ? 4.3  19.54  18.49  Acute 
Section III: Using the given triangles below, set up the following trigonometric ratios. Leave your answers as
S
fractions in lowest terms.
S


z
12.
Sin =
y
x
x
Cos =
z

z
R
Q
Tan =

x
y
z
y
x
x
y

13.
Sin =
Cos =
8

z
T

15
15
17
15
y

8
17
Tan =
17
15
17
8
17
15
8
U
8
Section IV: Solve the following trigonometric ratios for the given variable. Round your answers to the nearest
hundredth if necessary.
x
15
d
14. sin 82 
15. cos 59 
16. tan 28 
10
m
5
m cos 59  15
10sin82  x
5 tan 28  d
15
9.90  x
1.19  d
m
cos 59
m  29.12
x=
9.90
d=
x
5
5 tan 45  x
5 x
6
y
y cos30  6
6
cos30
y  6.93
y
7.72
19. sin 60 
5
x=
1.19
w
9
9sin 60  w
7.79  w
18. tan 45 
17. cos30 
y=
29.12
m=
7.79
w=
Section V: Find the missing value of the triangles below by setting up and solving a trigonometric ratio.
20.
21.
D
45
23
Q
4
N
L
J
22.
6
25
H
A
23
cos 25 
x
x cos 25  23
x
sin 45 
4
4sin 45  x
25.38
6
x
x sin 32  6
sin 32 
2.83  x
23
x
cos 25
x  25.38
QA =
32
P
HL =
2.83
6
sin 32
x  4.62
x
PC =
4.62
C
25
23. B
R
K
24.
60
2 23
sin 30 
G
F
45
5
x
x sin 45  5
E
x
2 23
sin 45 
2 23 sin 30  x
4.80  x
x  12.5
12.5
RF =
5
M
30
x
cos 60 
25
25cos 60  x
O
25.
KG =
I
5
sin 45
x  7.07
x
4.80
7.07
EI =
Section VI: Solve the following trigonometric ratios for the variable. Round your answer to the nearest
hundredth if necessary.
15
21
15
m  tan 1
21
m  35.54
7
10
7
x  sin 1
10
x  44.43
x=
44.43
m=
28. cos w 
35.54
w=
5
19
5
m  sin 1
19
m  6.76
15
10
15
r   tan 1
10
r   56.31
56.31
m=
51.32
5
10
5
d   cos 1
10
d   60
30. sin m 
29. tan r 
r=
5
8
5
w  cos 1
8
w  51.32
27. tan m 
26. sin x 
31. cos d 
6.76
d=
60
Section VII: Solve the following right triangles by finding all missing values. Round your answers to the
nearest hundredth if necessary.
A
32.
Find AC :
7
7 2  122  AC 2
B
C
12
AC =
193
mA =
59.74
mC =
30.26
49  144  AC 2
193  AC 2
193  AC
Find mA :
tan A 
12
7
12
7
mA  59.74
mA  tan 1
Find mC :
tan C 
12
7
7
12
mC  30.26
mC  tan 1
33.
R
45
S
Find mR :
8
Q
90  45  mR  180
135  mR  180
mR  45
Find QR :
QR
tan 45 
8
8 tan 45  QR
8  QR
Find SR :
8
SR
8  cos 45  SR
cos 45 
11.31  SR
QR =
8
SR =
11.31
mR =
45
S
34.
90  25  mR  180
25
Q
Find mR :
20
R
115  mR  180
mR  55
SQ =
18.13
RS =
8.45
mR =
Find mN :
10
M
90  64  mN  180
64
N
LM =
4.88
MN =
11.13
mN =
SQ
20
20 cos 25  SQ
cos 25 
18.13  SQ
Find SR :
SR
20
20sin 25  SR
sin 25 
8.45  SR
55
L
35.
Find SQ :
26
154  mN  180
mN  26
Find LM :
10
tan 64 
LM
LM tan 64  10
10
LM 
tan 64
LM  4.88
Find MN :
10
MN
MN sin 64  10
sin 64 
10
sin 64
RS  11.13
MN 
Section VIII: Solve each of the following word problems. Show all work and diagrams needed to solve the
problem, and round your answers to the nearest tenth, if necessary.
36. A television station is going to construct an 11-foot tower for a new antenna. The tower will be supported
by three cables, each attached to the top of the tower and to points on the roof of the building that are 10
feet from the base of the tower. Find the total length of the three cables.
112  102  x 2
Length of Cable
11 ft
121  100  x 2
x
10 ft
221  x 2
Need 3lengths of cable :
3  221  44.6 ft
221  x
37. A construction worker is making sure one corner of the foundation of a house is a right angle. To do this,
the worker makes a mark 7 feet from the corner along one wall and another mark 6 feet from the same
corner along the other wall. The worker then measures the distance between the two marks and finds the
distance to be 12 feet. Is the corner a right angle?
62  7 2  122 

36  49  144   Not a Right Angle
85  144 

38. A water slide makes an angle of about 13 with the ground. The slide extends horizontally about 65.1
meters. What is the height of the slide in meters to the nearest tenth?
x 

65.1 

65.1tan13  x   Height is Aprox.15.0m
15.0  x 



tan13 
x
13
65.1m
39. A mountain climber has a rope 189 feet long attached to his belt being held by his partner down on the
ground. The angle of elevation from the ground to the climber is 55°. Estimate the climber's height above
the ground. Round your answer to the nearest integer.
x
189 ft
55
sin55  189 

189sin55  x   Height is Aprox.155 ft
155  x 
40. A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 14 feet and
a horizontal length of 22 feet. What is the angle of the ramp? Round your answer to the nearest integer, if
necessary.

14
tan x 

22

14 ft
1 14 
x  tan
  Angleis Aprox.32
22 
x
x  32.47 
22 ft

