Download 5-7 The Pythagorean Theorem (pp. 348–355)

Answers
5-7 The Pythagorean Theorem (pp. 348–355)
EXERCISES
EXAMPLES
■
Find the value of x. Give your answer in
simplest radical form.
a2 + b2 = c2
62 + 32 = x2
45 = x 2
x = 3 √
5
Ý
È
Î
■
Pyth. Thm.
Substitution
Simplify.
Find the positive
square root
and simplify.
Find the missing side length. Tell if the sides
form a Pythagorean triple. Explain.
a2 + b2 = c2
a + (1.6) 2 = 2 2
a 2 = 1.44
£°È
a = 1.2
2
Ó
47. x = 2 √
10
48. x = 2 √33
Pyth. Thm.
Substitution
Solve for a 2.
Find the positive
square root.
The side lengths do not form a Pythagorean
triple because 1.2 and 1.6 are not whole
numbers.
49. 6; the lengths do not form a
Pythagorean triple because 4.5
and 7.5 are not whole numbers.
Find the value of x. Give your answer in simplest
radical form.
47.
48.
Ó
£{
Ý
50. 40; the lengths do form a
Pythagorean triple because
they are nonzero whole numbers
that satisfy the equation
a 2 + b 2 = c 2.
n
È
Ý
Find the missing side length. Tell if the sides form a
Pythagorean triple. Explain.
Ý
Ý
49.
50.
{°x
Ó{
Ç°x
51. triangle; obtuse
52. not a triangle
53. triangle; right
ÎÓ
54. triangle; acute
2
55. x = 26 √
Tell if the measures can be the side lengths of a
triangle. If so, classify the triangle as acute, obtuse,
or right.
2
56. x = 6 √
51. 9, 12, 16
52. 11, 14, 27
57. x = 32
53. 1.5, 3.6, 3.9
54. 2, 3.7, 4.1
3
58. x = 24; y = 24 √
3 ; y = 12
59. x = 6 √
√
3
14 28 √3
60. x =
;y=
3
3
61. 21 ft 3 in.
_
5-8 Applying Special Right Triangles (pp. 356–362)
EXERCISES
EXAMPLES
Find the values of the variables. Give your
answers in simplest radical form.
Find the values of the variables. Give your answers in
simplest radical form.
■
55.
{xÂ
£™
Ý
This is a 45°-45°-90° triangle.
2
x = 19 √
Hyp. = leg √2
62. 15 ft 7 in.
56.
ÓÈ
£ÓÊÊ
{xÂ
{xÂ
_
Ý
Ý
Ý
■
This is a 45°-45°-90° triangle.
£x
{xÂ
{xÂ
Ý
15 = x
_
√2
15 √
2
_
=x
2
■
Þ
Ý
ÈäÂ
ÎäÂ
15 = x √
2
Hyp. = leg √
2
y = 11 √
3
58.
£ÈÊÊȖе
ÓÊ
Ê
{xÂ
Þ
ÎäÂ
Ý
{nÊÊ
{xÂ
ÈäÂ
Ý
Divide both sides by √
2.
Þ
59.
60.
ÎäÂ
Ý
ÈÊÊ
Rationalize the denominator.
£{
Ý
Þ
ÈäÂ
This is a 30°-60°-90° triangle.
22 = 2x Hyp. = 2(shorter leg)
ÓÓ
11 = x
57.
Divide both sides by 2.
Longer leg = (shorter leg) √
3
Find the value of each variable. Round to the
nearest inch.
61.
62.
Ã
Ã
ÎäÊvÌ
£nÊvÌ
Ã
…
£nÊvÌ
ÈäÂ
£nÊvÌ
Ã
Study Guide: Review
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Study Guide: Review
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Answers
8-2 Trigonometric Ratios (pp. 525–532)
Find each length. Round to the nearest
hundredth.
EF
AB
Since the opp. leg and
hyp. are involved,
use a sine ratio.
Î{Â
4.2
AB = _
tan 34°
AB ≈ 6.23 in.
15. 1.31 cm
16. m∠C = 68°; AB ≈ 4.82; AC ≈
1.95
näÂ
££Ê“
7
13. PR
6
Ç°Óʓ
*
17. m∠H ≈ 53°; m∠G ≈ 37°; HG ≈
5.86
+
әÂ
18. m∠S = 40°; RS ≈ 42.43; RT ≈
27.27
19. m∠Q ≈ 41°; m∠N ≈ 49°; QN ≈
13.11
,
{°Óʈ˜°
4.2
tan 34° = _
AB
AB tan 34° = 4.2
14. 10.32 cm
Find each length. Round to the nearest hundredth.
12. UV 1
n°£ÊV“
ÇxÂ
EF
sin 75° = _
8.1
EF = 8.1(sin 75°)
EF ≈ 7.82 cm
■
13. 6.30 m
EXERCISES
EXAMPLES
■
12. 11.17 m
14. XY
15. JL
9
{ÇÂ
Since the opp. and adj.
legs are involved, use
a tangent ratio.
ÎÎÂ
£Ó°ÎÊV“
8
£°{ÊV“
<
8-3 Solving Right Triangles (pp. 534–541)
EXERCISES
EXAMPLE
■
Find the unknown measures in LMN.
Round lengths to the nearest hundredth and
angle measures to the nearest degree.
Find the unknown measures. Round lengths to
the nearest hundredth and angle measures to the
nearest degree.
16. ÓÓÂ
È£Â
x°Ó
n°x
17.
The acute angles of a right triangle are
complementary. So m∠N = 90° - 61° = 29°.
MN
sin L = _
LN
8.5
sin 61° = _
LN
8.5 ≈ 9.72
LN = _
sin 61°
MN
tan L = _
LM
8.5
tan 61° = _
LM
8.5 ≈ 4.71
LM = _
tan 61°
ΰx
Write a trig. ratio.
Substitute the given
values.
18.
*
19.
-
™°™
Solve for LN.
Write a trig. ratio.
{°Ç
n°È
ÎÓ°x
,
xäÂ
/
+
Substitute the given
values.
Solve for LM.
Study Guide: Review
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Study Guide: Review
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Answers
8-4 Angles of Elevation and Depression (pp. 544–549)
20. angle of depression
21. angle of elevation
EXERCISES
EXAMPLES
22. 36 ft
■
23. 458 m
24. 22°
25. 31.4
A pilot in a plane spots a forest fire on the
ground at an angle of depression of 71°.
The plane’s altitude is 3000 ft. What is the
horizontal distance from the plane to the fire?
Round to the nearest foot.
3000
tan 71° = _
* Ç£Â
XF
3000
_
XF =
tan 71°
ÎäääÊvÌ
XF ≈ 1033 ft
Classify each angle as an angle of elevation or angle
of depression.
£
Ç£Â
Ó
8
■
A diver is swimming at a depth of 63 ft
below sea level. He sees a buoy floating at
sea level at an angle of elevation of 47°.
How far must the diver swim so that he is
directly beneath the buoy? Round to the
nearest foot.
63
tan 47° = _
XD
63
ÈÎÊvÌ
XD = _
tan 47°
XD ≈ 59 ft
{ÇÂ
8
20. ∠1
21. ∠2
22. When the angle of elevation to the sun is 82°,
a monument casts a shadow that is 5.1 ft long.
What is the height of the monument to the
nearest foot?
23. A ranger in a lookout tower spots a fire in the
distance. The angle of depression to the fire is
4°, and the lookout tower is 32 m tall. What is
the horizontal distance to the fire? Round to the
nearest meter.
8-5 Law of Sines and Law of Cosines (pp. 551–558)
EXERCISES
EXAMPLES
Find each measure. Round lengths to the
nearest tenth and angle measures to the
nearest degree.
■
m∠B
Find each measure. Round lengths to the nearest
tenth and angle measures to the nearest degree.
{
24. m∠Z
9
8
{äÂ
n
Ç
È
nnÂ
sin C
sin B = _
_
Law of Sines
AB
AC
sin B = _
sin 88°
_
Substitute the given values.
6
8
6 sin 88°
Multiply both sides by 6.
sin B = _
8
6
sin
88°
m∠B = sin -1 _ ≈ 49°
8
(
574
Chapter 8
£È
£ÎäÂ
ÓÎÂ
Chapter 8 Right Triangles and Trigonometry
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25. MN
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