3 APPLY
ASSIGNMENT GUIDE
BASIC
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
2. You can solve a right triangle if you are given the lengths of any two sides.
Day 1: pp. 570–572 Exs. 12–32
even, 34–40, 42–45
AVERAGE
Skill Check
Day 1: pp. 570–572 Exs. 12–32
even, 34–40, 42–45
✓
ADVANCED
Day 1: pp. 570–572 Exs. 12–32
even, 34–40, 42–46
BLOCK SCHEDULE WITH 9.5
pp. 570–572 Exs. 12–32 even,
34–40, 42–45
EXERCISE LEVELS
Level A: Easier
11–21
Level B: More Difficult
22–45
Level C: Most Difficult
46
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 12, 16, 24, 30, 38, 40. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 9
Resource Book, p. 98)
•
Transparency (p. 69)
1. Explain what is meant by solving a right triangle.
finding the lengths of the sides and the measures of the angles
Tell whether the statement is true or false.
true
3. You can solve a right triangle if you know only the measure of one acute angle.
false
CALCULATOR In Exercises 4–7, ™A is an acute angle. Use a calculator to
approximate the measure of ™A to the nearest tenth of a degree.
4. tan A = 0.7 35.0° 5. tan A = 5.4 79.5° 6. sin A = 0.9 64.2° 7. cos A = 0.1 84.3°
22. side lengths: 20, 21,
and 29; angle
measures: 90°, 43.6°,
and 46.4°
Solve the right triangle. Round decimals to the nearest tenth.
8.
23. side lengths: 7, 7, and
9.9; angle measures:
90°, 45°, and 45°
24. side lengths: 2, 6, and
6.3; angle measures:
90°, 71.6°, and 18.4°
9.
A
33
c
C
56
109
B
D
c = 65, m™A = 59.5°,
m™B = 30.5°
10. X
E
y
d
91
Z
x
4
60ⴗ
Y
F
d = 60, m™D = 33.4°,
m™E = 56.6°
x = 2, y = 3.5,
m™X = 30°
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 820.
25. side lengths: 4.5, 8,
and 9.2; angle
measures: 90°, 29.6°,
and 60.4°
26. side lengths: 4, 13.0,
and 13.6; angle
measures: 90°, 17.1°,
and 72.9°
27. side lengths: 6, 11.0,
and 12.5; angle
measures: 90°, 28.7°,
and 61.3°
q
FINDING MEASUREMENTS Use the diagram to find
the indicated measurement. Round your answer to
the nearest tenth.
11. QS
12. m™Q
73
48
13. m™S 41.1°
48.9°
T
S
55
CALCULATOR In Exercises 14–21, ™A is an acute angle. Use a calculator
to approximate the measure of ™A to the nearest tenth
of a degree.
14. tan A = 0.5 26.6° 15. tan A = 1.0
45° 16. sin A = 0.5 30° 17. sin A = 0.35 20.5°
18. cos A = 0.15
19. cos A = 0.64
20. tan A = 2.2
21. sin A = 0.11 6.3°
81.4°
50.2°
65.6°
SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimals to the
nearest tenth.
22–27. See margin.
22.
23.
A
24.
F
G
7
7
20
2
21
C
STUDENT HELP
D
B
E
J
6
H
12.5
S
HOMEWORK HELP
Example 1: Exs. 11–27,
34–37
Example 2: Exs. 28–33
Example 3: Exs. 38–41
25.
570
27.
P
8
K
570
26.
M
13.6
9.2
L
Chapter 9 Right Triangles and Trigonometry
N
4
q
R
6
T
SOLVING RIGHT TRIANGLES Solve the right triangle. Round decimals to
the nearest tenth. 28–33. See margin.
x = 10.9,
28.
29. s = 4.1, t = 11.3,
30. z = 13.8,
X
P
m™Y = 38°
m™T = 70°
U
q
4.5
R
26ⴗ
p
q
p = 4.0, q = 2.0,
m™P = 64°
C
T
D
52ⴗ
z
Y
51ⴗ
e
a
5
56ⴗ
s
d = 3.7,
32. e = 4.8,
m™F = 39°
31.
A
S
t
20ⴗ
12
33.
8.5
Z
x
L
3
4
E
M
m
34ⴗ
B
c
F
d
a = 7.4, c = 8.9, m™B = 34°
L
K
l = 5.9, m = 7.2, m™L = 56°
NATIONAL AQUARIUM Use the diagram
of one of the triangular windowpanes at the
National Aquarium in Baltimore, Maryland,
to find the indicated value.
? 1.9167
34. tan B ≈ 㛭㛭㛭
B
?
35. m™B ≈ 㛭㛭㛭
36 in.
?
36. AB ≈ 㛭㛭㛭
?
37. sin A ≈ 㛭㛭㛭
..
FOCUS ON
APPLICATIONS
RE
FE
L
AL I
38.
s
62.4°
77.8 in.
A
69 in.
COMMON ERROR
EXERCISE 22 Students may
want to use the sine ratio for this
problem. Caution them that they
cannot find sin A unless they know
the hypotenuse. Suggest that they
use the Pythagorean Theorem to
find the hypotenuse.
APPLICATION NOTE
EXERCISE 38 Elevation as
referred to in this exercise is the
vertical height from the ground.
Long’s Peak is in the Rocky
Mountain National Park.
40.
96.4 in.
4.76°
96.1 in.
8 in.
Not drawn to scale
C
0.4626
HIKING You are hiking up a mountain
peak. You begin hiking at a trailhead whose
elevation is about 9400 feet. The trail ends
near the summit at 14,255 feet. The horizontal
distance between these two points is about
17,625 feet. Estimate the angle of elevation
from the trailhead to the summit. about 15.4°
summit
14,255 ft
trailhead
9400 ft
horizontal distance 17,625 ft
RAMPS In Exercises 39–41, use the
information about wheelchair ramps.
BENCHMARKS
If you hike to the top
of a mountain you may find a
brass plate called a
benchmark. A benchmark
gives an official elevation for
the point that can be used by
surveyors as a reference for
surveying elevations of other
landmarks.
The Uniform Federal Accessibility
Standards specify that the ramp angle
used for a wheelchair ramp must be less
than or equal to 4.76°.
length of ramp
ramp angle
vertical
rise
horizontal distance
39. The length of one ramp is 20 feet. The vertical rise is 17 inches.
Estimate the ramp’s horizontal distance and its ramp angle.
about 239.4 in., or about 19 ft 11 in.; about 4.1°
ADDITIONAL PRACTICE
AND RETEACHING
40. You want to build a ramp with a vertical rise of 8 inches. You want to
minimize the horizontal distance taken up by the ramp. Draw a sketch
showing the approximate dimensions of your ramp. See margin.
41.
For Lesson 9.6:
Writing
Measure the horizontal distance and the vertical rise of a ramp
near your home or school. Find the ramp angle. Does the ramp meet the
specifications described above? Explain. Answers will vary.
9.6 Solving Right Triangles
571
• Practice Levels A, B, and C
(Chapter 9 Resource Book, p. 88)
• Reteaching with Practice
(Chapter 9 Resource Book, p. 91)
•
See Lesson 9.6 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
571
Test
Preparation
4 ASSESS
DAILY HOMEWORK QUIZ
Transparency Available
Use a calculator to approximate
the measure of acute ⬔A to the
nearest tenth of a degree.
1. sin A = 0.25 14.5°
2. cos A = 0.38 67.7°
Solve the right triangle. Round
decimals to the nearest tenth.
3. P
45. Sample answer: The
riser-to-tread ratio
affects the safety of
the stairway in several
ways. First, the deeper
the tread the more of a
person’s foot can fit on
the step. This makes a
person less likely to
fall. Also, the smaller
the angle of inclination
the less steep the
stairway. This makes
the stairs less tiring to
climb, and therefore,
safer.
★ Challenge
4
Q
side lengths: 4, 5, 6.4; angle
measures: 90°, 38.7°, 51.3°
4.
The horizontal part of a step is called the tread.
The vertical part is called the riser. The ratio of
the riser length to the tread length affects the safety
of a staircase. Traditionally, builders have used a
riser-to-tread ratio of about 8ᎏ14ᎏ inches : 9 inches.
A newly recommended ratio is 7 inches : 11 inches.
tread
riser
x⬚
42. Find the value of x for stairs built using the new riser-to-tread ratio. about 32.5
43. Find the value of x for stairs built using the old riser-to-tread ratio. about 42.5
44. Suppose you want to build a stairway that is less steep than either of the ones
in Exercises 42 and 43. Give an example of a riser-to-tread ratio that you
could use. Find the value of x for your stairway.
45.
Sample answer: 6 in.: 12 in.; about 26.6
Writing Explain how the riser-to-tread ratio that is used for a stairway
could affect the safety of the stairway. See margin.
C
PROOF Write a proof. See margin.
46.
GIVEN 䉴 ™A and ™B are acute angles.
R
5
MULTI-STEP PROBLEM In Exercises 42–45,
use the diagram and the information below.
S
EXTRA CHALLENGE
www.mcdougallittell.com
a
sin A
b
sin B
PROVE 䉴 ᎏᎏ = ᎏᎏ
a
b
c
A
B
Æ
( Hint: Draw an altitude from C to AB. Label it h.)
7
35°
T
U
side lengths: 7, 4.9, 8.5; angle
measures: 90°, 35°, 55°
MIXED REVIEW
Æ
46. Draw an altitude,
CD ,
Æ
from C to AB , and let
CD = h. In rt. ¤ACD,
h
b
sin A = }}. In rt.
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 9.6 are available in
blackline format in the Chapter 9
Resource Book, p. 95 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. WRITING Summarize how to
find the measures of the acute
angles of a right triangle for
which the sides are known.
Æ„
Æ„
47. AB 具3, 2典
Æ„
48. AC
h
¤BCD, sin B = }}.
a
49. DE
Thus, h = b • sin A
and h = a • sin B.
By the substitution
prop. of equality,
b • sin A = a • sin B.
Dividing both sides by
sin A • sin B gives
51. FH 具1, º2典
b
a
}} = }} , or
sin B
sin A
a
b
}} = }}.
sin A
sin B
具º1, º3典
Æ„
1
50. FG 具1, 0典
Æ„
B
D
具º2, 2典
Æ„
52. JK
y
C
A
x
1
具3, 1典
F
E
K
J
G
H
SOLVING PROPORTIONS Solve the proportion. (Review 8.1)
x
5
53. ᎏᎏ = ᎏᎏ
30
6
25
7
84
56. ᎏᎏ = ᎏᎏ 216
18
k
7
49
54. ᎏᎏ = ᎏᎏ
16
y
m
7
57. ᎏᎏ = ᎏᎏ
2
1
112
14
g
3
55. ᎏᎏ = ᎏ 12.6
10
42
4
8
58. ᎏᎏ = ᎏᎏ 22
11
t
CLASSIFYING TRIANGLES Decide whether the numbers can represent the
side lengths of a triangle. If they can, classify the triangle as right, acute, or
obtuse. (Review 9.3)
Sample answer: For one acute
angle, find the inverse sine of
the ratio of the side opposite the
angle to the hypotenuse. Then
subtract that angle from 90° to
get the other acute angle.
59. 18, 14, 2 no
60. 60, 228, 220 yes; acute 61. 8.5, 7.7, 3.6 yes; right
62. 250, 263, 80 yes; obtuse 63. 113, 15, 112 yes; right 64. 15, 75, 59 no
572
572
USING VECTORS Write the component form
of the vector. (Review 7.4 for 9.7)
Chapter 9 Right Triangles and Trigonometry