Chapter 8: Right Triangles and Trigonometry

Chapter 8: Right Triangles and Trigonometry
Right Triangles and
Trigonometry
8
•
Solve problems using the
geometric mean, the Pythagorean
Theorem, and its converse.
•
Use trigonometric ratios to solve
right triangle problems.
•
Solve triangles using the Law of
Sines and the Law of Cosines.
Key Vocabulary
trigonometric ratio (p. 456)
Law of Sines (p. 471)
Law of Cosines (p. 479)
Real-World Link
Bridges The William H. Natcher bridge across the Ohio
River has a cable-stayed design. The cables form right
triangles with the supports and the length of the bridge.
Right Triangles and Trigonometry Make this Foldable to help
you organize your notes. Begin with seven sheets of grid paper.
IN
the top right corner to
the bottom edge to form
a square.
3 Staple the sheets
along the fold in four
places.
430 Chapter 8 Right Triangles and Trigonometry
David Sailors/CORBIS
2 Fold the rectangular
part in half.
4 Label each sheet with
a lesson number and
the rectangular part
with the chapter title.
3IGHT5RIANGLES
1 Stack the sheets. Fold
GET READY for Chapter 8
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at geometryonline.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Solve each proportion. Round to the
nearest hundredth. (Lesson 7-1)
3
8
12
1. _
=_
2. _c = _
a
4
5
f
6
d
3. _
=_
=_
20
5
10
3
6
4
_1
4. _
=_
y= z
3
1 in.
3.5
The proportion _
=_
5. MINIATURES
x
12 in.
relates the height of a miniature chair to
the height of a real chair. Solve the
proportion. (Lesson 7-1)
EXAMPLE 1
a
31
Solve the proportion _
=_
. Round to the
5
30
nearest hundredth if necessary.
31
a
_
=_
30
Write the proportion.
5
5a = 30(31) Find the cross products.
5a = 930
Simplify.
a = 186
Find the measure of the hypotenuse of
each right triangle having legs with the
given measures. Round to the nearest
hundredth. (Extend 1-3)
6. 5 and 12
7. 6 and 8
8. 15 and 15
9. 14 and 27
10. PAINTING A ladder is propped against a
wall as shown. To the nearest tenth, what
is the length of the ladder? (Extend 1-3)
Divide each side by 5.
EXAMPLE 2
Find the measure of the hypotenuse of the
right triangle having legs with the measures
10 and 24. Round to the nearest hundredth
if necessary.
a2 + b2 = c2
Pythagorean Theorem
102 + 242 = c2
Substitution
100 + 576 = c2
Evaluate the exponents.
676 = c2
FT
Simplify.
√
676 = √c2
26 = c
FT
11. The measure of one angle in a right
triangle is three times the measure of the
second angle. Find the measures of each
angle of the triangle. Find x. (Lesson 4-2)
Take the square root of each side.
Simplify.
EXAMPLE 3
3x˚
(x 10)˚
Find x.
x˚
x + 3x + x - 10 = 180
5x = 190
x = 38
Chapter 8 Get Ready for Chapter 8
431
8-1
Geometric Mean
Main Ideas
• Find the geometric
mean between two
numbers.
• Solve problems
involving
relationships between
parts of a right
triangle and the
altitude to its
hypotenuse.
When you look at a painting, you
should stand at a distance that allows
you to see all of the details in the
painting. The distance that creates the
best view is the geometric mean of the
distance from the top of the painting to
eye level and the distance from the
bottom of the painting to eye level.
New Vocabulary
geometric mean
Geometric Mean The geometric mean between two numbers is the
positive square root of their product.
Geometric Mean
You may wish to review
square roots and
simplifying radicals
on pp. 790–791.
For two positive numbers a and b, the geometric mean is the positive
number x where the proportion a : x = x : b is true. This proportion can
a
_x
2
be written using fractions as _
x = b or with cross products as x = ab
or x = √
ab .
EXAMPLE
Geometric Mean
Find the geometric mean between each pair of numbers.
b. 6 and 15
a. 4 and 9
_4 = _x
Definition of geometric
mean
x
_6 = _
Definition of geometric
mean
x2 = 36
Cross products
x2 = 90
Cross products
x
9
x
15
Take the positive square
root of each side.
x = √
90
x= 6
Simplify.
x = 3 √
10 Simplify.
x ≈ 9.5
1A. 5 and 45
432 Chapter 8 Right Triangles and Trigonometry
Robert Brenner/PhotoEdit
Take the positive square root
of each side.
x = √
36
1B. 8 and 10
Use a calculator.
Altitude of a −−−
Triangle Consider right triangle XYZ
Z
with altitude WZ drawn from the right angle Z to the
−−
hypotenuse XY. A special relationship exists for the
three right triangles, XYZ, XZW, and ZYW.
X
Y
W
GEOMETRY SOFTWARE LAB
Right Triangles Formed by the Altitude
Use The Geometer’s
Sketchpad to draw a right
triangle XYZ with right
angle Z. Draw the altitude
−−
ZW from the right angle to
the hypotenuse.
Z
THINK AND DISCUSS
1. Find the measures of ∠X,
Y
W
X
∠XZY, ∠Y, ∠XWZ, ∠XZW,
∠YWZ, and ∠YZW.
2. What is the relationship between
m∠X and m∠YZW? between
m∠Y and m∠XZW?
3. Drag point Z to another position. Describe the relationship between the
measures of ∠X and ∠YZW and between m∠Y and m∠XZW.
MAKE A CONJECTURE
4. How are XYZ, XZW, and ZYW related?
The Geometry Software Lab suggests the following theorem.
8.1
If the altitude is drawn from the vertex of the right angle
of a right triangle to its hypotenuse, then the two triangles
formed are similar to the given triangle and to each other.
Altitudes of a
Right Triangle
The altitude drawn to
the hypotenuse
originates from the
right angle. The other
two altitudes of a right
triangle are the legs.
Y
X
Example: XYZ ∼ XWY ∼ YWZ
W
Z
You will prove Theorem 8.1 in Exercise 38.
By Theorem 8.1, since XWY ∼ YWZ, the corresponding sides are
−−−
−−−
XW = _
YW . Notice that XW
proportional. Thus, _
and ZW are segments
YW
ZW
of the hypotenuse of the largest triangle.
8.2
The measure of an altitude drawn from the vertex of the
right angle of a right triangle to its hypotenuse is the
geometric mean between the measures of the two
segments of the hypotenuse.
Y
X
W
Z
Example: YW is the geometric mean of XW and ZW.
You will prove Theorem 8.2 in Exercise 39.
Lesson 8-1 Geometric Mean
433
EXAMPLE
Altitude and Segments of the Hypotenuse
In PQR, RS = 3 and QS = 14. Find PS.
RS
PS
_
=_
PS
QS
x
_3 = _
x
14
x2 = 42
P
Q
Theorem 8.2
S
R
RS = 3, QS = 14, and PS = x
Cross products
x ≈ 6.5
Use a calculator to take the positive square root of each side.
Square Roots
Since these numbers
represent measures,
you can ignore the
negative square root
value.
2. Refer to PQR above. If RS = 0.8 and QS = 2.2, find PS.
ARCHITECTURE Mr. Martinez is designing a walkway to pass over a
train. To find the train height, he holds a carpenter’s square at eye level
and sights along the edges from the street to the top of the train. If Mr.
Martinez’s eye level is 5.5 feet above the street and he is 8.75 feet from
the train, find the train’s height. Round to the nearest tenth.
−−
Z
Draw a diagram. Let YX be the altitude
drawn from the right angle of WYZ.
WX = _
YX
_
Theorem 8.2
ZX
YX
5.5
8.75
_
=_
8.75
ZX
WX = 5.5 and YX = 8.75
5.5ZX = 76.5625
ZX ≈ 13.9
Cross products
Y
X
5.5 ft
Divide each side by 5.5.
8.75 ft
W
The elevated train is 5.5 + 13.9 or about 19.4 feet high.
3. Makayla is using a carpenter’s square to sight the top of a waterfall. If
her eye level is 5 feet from the ground and she is a horizontal distance of
28 feet from the waterfall, find the height of the waterfall to the nearest
tenth.
Personal Tutor at geometryonline.com
The altitude to the hypotenuse of a right triangle determines another
relationship between the segments.
8.3
If the altitude is drawn from the vertex of the right angle
of a right triangle to its hypotenuse, then the measure of a
leg of the triangle is the geometric mean between the
X
measures of the hypotenuse and the segment of the
hypotenuse adjacent to that leg.
XZ
XZ
YZ
XY
Example: _
=_
and _
=_
XY
XW
YZ
WZ
You will prove Theorem 8.3 in Exercise 40.
434 Chapter 8 Right Triangles and Trigonometry
Y
W
Z
EXAMPLE
Hypotenuse and Segment of Hypotenuse
R
Find x and y in PQR.
−−
−−−
PQ and RQ are legs of right triangle PQR. Use Theorem 8.3
to write a proportion for each leg and then solve.
PQ
PR = _
_
PQ
PS
_6 = _y
y
2
y2 = 12
4
S
2
RQ
PR = _
_
RQ
SR
_6 = _x
x
4
PS = 2, PQ = y, PR = 6
x2 = 24
Cross products
P
y
Q
RS = 4, RQ = x, PR = 6
Cross products
y = √
12 Take the square root.
x = √
24 Take the square root.
y = 2 √
3 Simplify.
x = 2 √
6 Simplify.
y ≈ 3.5
x ≈ 4.9
Use a calculator.
x
Use a calculator.
B 2
14
x
4. Find x and y in ABC.
y
A
Example 1
(p. 432)
Example 2
(p. 434)
C
Find the geometric mean between each pair of numbers.
2. 36 and 49
3. 6 and 8
4. 2 √2 and 3 √
2
1. 9 and 4
Find the measure of the altitude drawn to the hypotenuse.
6.
5. A
E
2
D
6
Example 3
(p. 434)
Example 4
(p. 435)
G
B
C
H
16
F
12
7. DANCES Danielle is making a banner for the
dance committee. The banner is to be as high
as the wall of the gymnasium. To find the
height of the wall, Danielle held a book up to
her eyes so that the top and bottom of the wall
were in line with the bottom edge and binding
of the cover. If Danielle’s eye level is 5 feet off
the ground and she is standing 12 feet from the
wall, how high is the wall?
Find x and y.
8.
x
B
Extra Examples at geometryonline.com
9.
C
8
D
C
y
y
3
A
B
x
2 √3
D
2
A
Lesson 8-1 Geometric Mean
435
HOMEWORK
HELP
For
See
Exercises Examples
10–17
1
18–23
2
24–25
3
26–31
4
Find the geometric mean between each pair of numbers.
45 and √
80 13. √
28 and √
1372
10. 5 and 6
11. 24 and 25
12. √
8 √
3
6 √
3
15. _ and _
3
14. _
and 1
5
5
2 √
2
5 √
2
16. _ and _
5
6
6
13
5
17. _
and _
7
Find the measure of the altitude drawn to the hypotenuse.
19. F
20. J
18. B
5
7
8 M
16
K
12
D
H
9
12
A
E
C
21.
L
G
22. V 2W
Q
23. Z
X
13
10
P
21
U
R
S 7
T
2.5
Y
25.
24. CONSTRUCTION The slope of
4
_
the roof shown below is . A
3
builder wants to put a support
brace from point C perpendicular
−−
to AP. Find the length of the brace.
N
ROADS City planners want to build
a road to connect points A and B.
Find out how long this road will
need to be.
"
P
X
5 yd
!
4 yd
MI
3 yd
A
C
B
Find x, y, and z.
26.
8
27.
28.
6
x
y
x
z
x
8
z
z
3
MI
y
5
y
29.
30.
x
y
z
10
4
15
31.
z
y
36
6x
12
x
z
8
y
x
The geometric mean and one extreme are given. Find the other extreme.
17 is the geometric mean between a and b. Find b if a = 7.
32. √
is the geometric mean between x and y. Find x if y = √
33. √12
3.
436 Chapter 8 Right Triangles and Trigonometry
Determine whether each statement is always, sometimes, or never true.
34. The geometric mean for consecutive positive integers is the average of
the two numbers.
35. The geometric mean for two perfect squares is a positive integer.
36. The geometric mean for two positive integers is another integer.
37. The measure of the altitude of a triangle is the geometric mean between
the measures of the segments of the side opposite the initial vertex.
PROOF Write a proof for each theorem.
38. Theorem 8.1
39. Theorem 8.2
40. Theorem 8.3
41. RESEARCH Use the Internet or other resource to write a brief description
of the golden ratio, which is also known as the divine proportion, golden
mean, or golden section.
EXTRA
PRACTICE
See pages 815, 835.
Self-Check Quiz at
geometryonline.com
H.O.T. Problems
42. PATTERNS The spiral of the state shell of
Texas, the lightning whelk, can be modeled
by a geometric mean. Consider the sequence
−−− −− −−− −−− −− −− −−−
of segments OA, OB, OC, OD, OE, OF, OG,
−−− −−
−−
OH, OI, and OJ. The length of each of these
segments is the geometric mean between
the lengths of the preceding segment and
the succeeding segment. Explain this
relationship. (Hint: Consider FGH.)
G
C
J
F
B
O
H
D
A
E
I
43. OPEN ENDED Find two pairs of numbers with a geometric mean of 12.
44. REASONING Draw and label a right triangle with an altitude drawn
from the right angle. From your drawing, explain the meaning of the
hypotenuse and the segment of the hypotenuse adjacent to that leg in
Theorem 8.3.
45. FIND THE ERROR RST is a right isosceles triangle. Holly and Ian are
−−
finding the measure of altitude SU. Who is correct? Explain your reasoning.
Holly
Ian
RS _
_
= SU
SU
RU _
_
=
SU RT
9.9 _
x
_
x = 14
x2 = 138.5
138.5
x = √
x = 11.8
S
SU TU
_7 = _x
x 7
x
R
x = 49
x=7
7
U
9.9
T
7
C
46. CHALLENGE Find the exact value of DE, given
AD = 12 and BD = 4.
A
47.
D
E
B
Writing in Math
Describe how the geometric mean can be used
to view paintings. Include an explanation of what happens when
you are too far or too close to a painting.
Lesson 8-1 Geometric Mean
437
49. REVIEW What are the solutions for the
quadratic equation x2 + 9x = 36?
48. What are the values of x and y?
8 cm
6 cm
y cm
10 cm
F -3, -12
H 3, -12
G 3, 12
J -3, 12
x cm
50. REVIEW Tulia borrowed $300 at 15%
simple interest for two years. If she
makes no payments either year, how
much interest will she owe at the end
of the two-year period?
A 4 and 6
B 2.5 and 7.5
C 3.6 and 6.4
D 3 and 7
A $90.00
C $30.00
B $45.00
D $22.50
51. The measures of the sides of a triangle are 20, 24, and 30. Find the measures
of the segments formed where the bisector of the smallest angle meets the
opposite side. (Lesson 7-5)
A
For Exercises 52 and 53, use ABC. (Lesson 7-4)
G
52. If AG = 4, GB = 6, and BH = 8, find BC.
B
53. If AB = 12, BC = 14, and HC = 4, find AG.
H
C
Use the Exterior Angle Inequality Theorem to list all
angles that satisfy the stated condition. (Lesson 5-2)
7
8
54. measures less than m∠8
3
1
4
2
5
55. measures greater than m∠1
56. measures less than m∠7
6
57. measures greater than m∠6
Write an equation in slope-intercept form for the line that satisfies the
given conditions. (Lesson 3-4)
58. m = 2, y-intercept = 4
59. passes through (2, 6) and (-1, 0)
60. m = -4, passes through (-2, -3)
61. x-intercept is 2, y-intercept = -8
PREREQUISITE SKILL Use the Pythagorean Theorem to find the length
of the hypotenuse of each right triangle. (Lesson 1-4)
62.
63.
5 ft
12 ft
64.
3 cm
5 in.
4 cm
438 Chapter 8 Right Triangles and Trigonometry
3 in.
EXPLORE
8-2
Geometry Lab
The Pythagorean Theorem
In Chapter 1, you learned that the Pythagorean Theorem relates the measures
of the legs and the hypotenuse of a right triangle. Ancient cultures used the
Pythagorean Theorem before it was officially named in 1909.
ACTIVITY
Use paper folding to develop the Pythagorean Theorem.
Step 1 On a piece of
patty paper, make a mark
along one side so that the
two resulting segments
are not congruent. Label
one as a and the other
as b.
a
Step 2 Copy these
measures on the other
sides in the order shown
at the right. Fold the
paper to divide the
square into four
sections. Label the area
of each section.
a
b
b ab
b2
Step 3 On another
sheet of patty paper,
mark the same lengths
a and b on the sides in
the different pattern
shown at the right.
Step 4 Use your
straightedge and pencil
to connect the marks as
shown at the right. Let c
represent the length of
each hypotenuse.
b
b
a2
ab
a
b
a
b
Step 5 Label the area
of each section, which
is _12ab for each triangle
and c2 for the square.
1
2 ab
c
c
1
2 ab
c2
c
c
1
2 ab
a
Step 6 Place the squares side by side and
color the corresponding regions that have
the same area. For example, ab = _12ab + _12 ab.
a
b
b
a
b
c
c
1
2 ab
a
c
c
a
1
2 ab
ab
b2
c2
=
a2
1
2 ab
ab
1
2 ab
1
2 ab
The parts that are not shaded tell us that
a2 + b2 = c2.
ANALYZE THE RESULTS
1. Use a ruler to find actual measures for a, b, and c. Do these
measures confirm that a2 + b2 = c2?
2. Repeat the activity with different a and b values. What do you
notice?
3. Explain why the drawing at the right is an illustration of the
Pythagorean Theorem.
4. CHALLENGE Use a geometric diagram to show that for any positive
numbers a and b, a + b > √
a2 + b2.
Explore 8-2 Geometry Lab: The Pythagorean Theorem
439
8-2
The Pythagorean Theorem
and Its Converse
Main Ideas
• Use the Pythagorean
Theorem.
• Use the converse of
the Pythagorean
Theorem.
New Vocabulary
Pythagorean triple
The Talmadge Memorial Bridge over
the Savannah River, in Georgia, has two
soaring towers of suspension cables.
Note the right triangles being formed
by the roadway, the perpendicular
tower, and the suspension cables. The
Pythagorean Theorem can be used to
find measures in any right triangle.
The Pythagorean Theorem In Lesson 1-3, you used the Pythagorean
Theorem to find the distance between two points by finding the length of
the hypotenuse when given the lengths of the two legs of a right triangle.
You can also find the measure of any side of a right triangle given the
other two measures.
8.4
Pythagorean Theorem
In a right triangle, the sum of the squares of the
measures of the legs equals the square of the
measure of the hypotenuse.
B
c
Symbols: a2 + b2 = c2
A
b
a
C
The geometric mean can be used to prove the Pythagorean Theorem.
Proof
Pythagorean Theorem
Given:
ABC with right angle at C
Prove:
a2
+
b2
=
C
c2
a
B
b
h
y D
x
A
c
Proof:
Draw right triangle ABC so C is the right angle. Then draw the altitude
−−
from C to AB. Let AB = c, AC = b, BC = a, AD = x, DB = y, and CD = h.
440 Chapter 8 Right Triangles and Trigonometry
Alexandra Michaels/Getty Images
Two geometric means now exist.
_c = _a
a
a2
y
= cy
and
_c = _b
and
b2
x
b
= cx Cross products
Add the equations.
a2 + b2 = cy + cx
a2 + b2 = c(y + x) Factor.
a2 + b2 = c2
Since c = y + x, substitute c for (y + x).
You can use the Pythagorean Theorem to find the length of the hypotenuse or
a leg of a right triangle if the other two sides are known.
Find the Length of the Hypotenuse
Real-World Link
Due to the curvature of
Earth, the distance
between two points is
often expressed as
degree distance using
latitude and longitude.
This measurement
closely approximates
the distance on a plane.
Source: NASA
GEOGRAPHY California’s NASA Dryden is
located at about 117 degrees longitude and
34 degrees latitude. NASA Ames, also in
California, is located at about 122 degrees
longitude and 37 degrees latitude. Use the
lines of longitude and latitude to find the
degree distance to the nearest tenth
between NASA Dryden and NASA Ames.
The change in longitude between the two
locations is |117-122| or 5 degrees. Let this
distance be a.
38˚
NASA
Ames
36˚
NASA
Dryden
34˚
122˚
120˚
118˚
116˚
The change in latitude is |37 - 34| or 3 degrees latitude. Let this distance
be b.
Use the Pythagorean Theorem to find the distance in degrees from NASA
Dryden to NASA Ames, represented by c.
a2 + b2 = c2
52 + 32 = c2
25 + 9 = c2
34 = c2
Pythagorean Theorem
a = 5, b = 3
Simplify.
Add.
√
34 = c Take the positive square root of each side.
5.8 ≈ c Use a calculator.
The degree distance between NASA Dryden and NASA Ames is about
5.8 degrees.
1. GEOGRAPHY Houston, Texas, is located at about 30 degrees latitude and
about 95 degrees longitude. Raleigh, North Carolina, is located at about
36 degrees latitude and about 79 degrees longitude. Find the degree
distance to the nearest tenth.
Personal Tutor at geometryonline.com
Lesson 8-2 The Pythagorean Theorem and Its Converse
StockTrek/Getty Images
441
EXAMPLE
Find the Length of a Leg
Find x.
(XY)2
X
(YZ)2
= (XZ)2 Pythagorean Theorem
x2 = 142 XY = 7, XZ = 14
+
72 +
49 + x2 = 196
x2 = 147
x = √
147
x = 7 √
3
x ≈ 12.1
2. Find x.
14 in.
7 in.
Simplify.
Y
x in.
Z
Subtract 49 from each side.
Take the square root of each side.
Simplify.
Use a calculator.
12.1 cm
x cm
16.2 cm
Converse of the Pythagorean Theorem The converse of the Pythagorean
Theorem can help you determine whether three measures of the sides of a
triangle are those of a right triangle.
8.5
Converse of the Pythagorean Theorem
If the sum of the squares of the measures of two sides of a
triangle equals the square of the measure of the longest side,
then the triangle is a right triangle.
Symbols: If a2 + b2 = c2, then ABC is a right triangle.
B
c
A
a
b
C
You will prove Theorem 8.5 in Exercise 30.
EXAMPLE
Distance
Formula
When using the
Distance Formula, be
sure to follow the
order of operations
carefully. Perform the
operation inside the
parentheses first,
square each term, and
then add.
Verify a Triangle is a Right Triangle
COORDINATE GEOMETRY Verify that PQR is a
right triangle.
Q (–3, 6)
y
R (5, 5)
Use the Distance Formula to determine the lengths
of the sides.
P (3, 2)
(-3 - 3)2 + (6 - 2)2 x1 = 3, y1 = 2, x2 = -3, y2 = 6
PQ = √
= √
(-6)2 + 42
Subtract.
= √
52
Simplify.
[5 - (-3)]2 + (5 - 6)2 x1 = -3, y1 = 6, x2 = 5, y2 = 5
√
= √
82 + (-1)2
Subtract.
QR =
= √
65
PR =
(5 - 3)2 + (5 - 2)2
√
Simplify.
x1 = 3, y1 = 2, x2 = 5, y2 = 5
= √
22 + 32
Subtract.
= √
13
Simplify.
442 Chapter 8 Right Triangles and Trigonometry
O
x
By the converse of the Pythagorean Theorem, if the sum of the squares of
the measures of two sides of a triangle equals the square of the measure of
the longest side, then the triangle is a right triangle.
PQ2 + PR2 = QR2
(√
52 )
2
)
- (√
13 ) (√65
52 + 13 65
65 = 65
2
Converse of the Pythagorean Theorem
2
PQ = √
52 , PR = √
13 , QR = √
65
Simplify.
Add.
Since the sum of the squares of two sides equals the square of the longest
side, PQR is a right triangle.
3. Verify that ABC with vertices A(2, -3), B(3, 0), and C(5, -1) is a
right triangle.
A Pythagorean triple is three whole numbers that satisfy the equation
a 2 + b 2 = c 2, where c is the greatest number. One common Pythagorean triple
is 3-4-5. If the measures of the sides of any right triangle are whole numbers,
the measures form a Pythagorean triple.
EXAMPLE
Pythagorean Triples
Determine whether each set of measures can be the sides of a right
triangle. Then state whether they form a Pythagorean triple.
a. 8, 15, 16
Since the measure of the longest side is 16, 16 must be c, and a or b are 8
and 15, respectively.
a2 + b2 = c2
82 + 152 162
64 + 225 256
289 ≠ 256
Pythagorean Theorem
a = 8, b = 15, c = 16
Simplify.
Add.
Since 289 ≠ 256, segments with these measures cannot form a right
triangle. Therefore, they do not form a Pythagorean triple.
√3
√6
3
b. _, _, and _
5
Comparing
Numbers
If you cannot quickly
identify the greatest
number, use a
calculator to find
decimal values for
each number and
compare.
5
5
a2 + b2 = c2
2
2
(_) + (_) (_)
√
3
5
√
6
5
Pythagorean Theorem
2
√
√3
6
3
3
a=_
,b=_
,c=_
5
5
5
5
3
6
9
_
+_
_
Simplify.
25
25
25
9
9
_
= _
Add.
25
25
9
9
Since _
=_
, segments with these measures form a right triangle.
25
25
However, the three numbers are not whole numbers. Therefore, they do
not form a Pythagorean triple.
4A. 20, 48, and 52
Extra Examples at geometryonline.com
√
√2
√
3
5
4B. _, _, and _
7
7
7
Lesson 8-2 The Pythagorean Theorem and Its Converse
443
Examples 1 and 2
(pp. 441–442)
Find x.
1.
2.
3.
6
x
4
7
x
10
37.5
5
7
x
20
4. COMPUTERS Computer displays are usually
measured along the diagonal of the screen.
A 14-inch display has a diagonal that measures
14 inches. If the height of the screen is 8 inches,
how wide is the screen?
Example 3
(p. 442)
Example 4
(p. 443)
HOMEWORK
HELP
For
See
Exercises Examples
9–14
1, 2
15–18
3
19–26
4
14 in.
8 in.
5. COORDINATE GEOMETRY Determine whether
JKL with vertices J(-2, 2), K(-1, 6), and L(3, 5)
is a right triangle. Explain.
Determine whether each set of numbers can be the measures of the sides of a
right triangle. Then state whether they form a Pythagorean triple.
6. 15, 36, 39
7. √
40 , 20, 21
8. √
44 , 8, √
108
Find x.
9.
10.
8
x
11.
8
8
4
28
x
14
12.
20
13.
40
14.
33
x
x
x
25
25
15
32
x
COORDINATE GEOMETRY Determine whether QRS is a right triangle for the
given vertices. Explain.
15. Q(1, 0), R(1, 6), S(9, 0)
16. Q(3, 2), R(0, 6), S(6, 6)
17. Q(-4, 6), R(2, 11), S(4, -1)
18. Q(-9, -2), R(-4, -4), S(-6, -9)
Determine whether each set of numbers can be the measures of the sides of a
right triangle. Then state whether they form a Pythagorean triple.
19. 8, 15, 17
20. 7, 24, 25
21. 20, 21, 31
22. 37, 12, 34
√
74
1 ,_
1, _
23. _
5
7
35
√3
√
2 35
24. _, _, _
2
3
36
3 _
25. _
, 4, 1
4 5
6 _
10
26. _
, 8, _
7 7
7
27. GARDENING Scott wants to plant flowers in a triangular plot. He has three
lengths of plastic garden edging that measure 20 inches, 21 inches, and
29 inches. Discuss whether these pieces form a right triangle. Explain.
444 Chapter 8 Right Triangles and Trigonometry
Getty Images
28. NAVIGATION A fishing trawler off the coast of Alaska was ordered by the U.S.
Coast Guard to change course. They were to travel 6 miles west and then sail
12 miles south to miss a large iceberg before continuing on the original
course. How many miles out of the way did the trawler travel?
y
29. PROOF Use the Pythagorean Theorem and the
figure at the right to prove the Distance Formula.
A(x1, y1)
d
B (x2, y2)
C (x1, y2)
30. PROOF Write a paragraph proof of Theorem 8.5.
x
O
31. Find the value of x in the figure shown.
X
Real-World Career
Military
All branches of the
military use navigation.
Some of the jobs using
navigation include
radar/sonar operators,
boat operators, airplane
navigators, and space
operations officers.
GEOGRAPHY For Exercises 32 and 33, use
the following information.
Denver is located at about 105° longitude
and 40° latitude. San Francisco is located
at about 122° longitude and 38° latitude.
Las Vegas is located at about 115° longitude
and 36° latitude. Using the lines of longitude
and latitude, find each degree distance.
32. San Francisco to Denver
33. Las Vegas to Denver
34. SAILING The mast of a sailboat is
supported by wires called shrouds.
What is the total length of wire
needed to form these shrouds?
12 ft
125°
120°
115°
110°
105°
100°
40°
San Francisco
35°
Denver
Las Vegas
30°
35. LANDSCAPING Six congruent
square stones are arranged in an
L-shaped walkway through a
garden. If x = 15 inches, then find
the area of the L-shaped walkway.
3 ft
For more information,
go to
26 ft
shrouds
x
geometryonline.com.
9 ft
EXTRA
PRACTICE
See pages 815, 835.
Self-Check Quiz at
geometryonline.com
H.O.T. Problems
36. PAINTING A painter sets a ladder up to reach the bottom
of a second-story window 16 feet above the ground.
The base of the ladder is 12 feet from the house. While
the painter mixes the paint, a neighbor’s dog bumps
the ladder, which moves the base 2 feet farther away
from the house. How far up the side of the house does
the ladder reach?
37. FIND THE ERROR Maria and Colin
are determining whether 5-12-13
is a Pythagorean triple. Who is
correct? Explain your reasoning.
Colin
?
13 + 5 2 = 12 2
?
169 + 25 = 144
193 =/ 144
no
2
x
2 ft
12 ft
Maria
5 + 122 = 132
25 + 144 = 169
169 = 169
yes
2
Lesson 8-2 The Pythagorean Theorem and Its Converse
Phil Mislinski/Getty Images
16 ft
445
38. OPEN ENDED Draw a pair of similar right triangles. Are the measures of the
sides of each triangle a Pythagorean triple? Explain.
39. REASONING True or false? Any two right triangles with the same
hypotenuse have the same area. Explain your reasoning.
40. CHALLENGE The figure at the right is a rectangular
prism with AB = 8, BC = 6, and BF = 8. Find HB.
(
%
&
$
41.
-
!
Writing in Math
"
About how many flags is Miko going
to place?
B 75
43. A rectangle has an area of 25 square
inches. If the dimensions of the
rectangle are doubled, what will be
the area of the new rectangle?
F 12.5 in 2
G 50 in 2
FT
FT
C 67
D 45
H 100 in 2
J 625 in 2
44. REVIEW Which equation is equivalent
to 5(3 - 2x) = 7 - 2(1 - 4x)?
A 18x = 10
B 2x = -10
C 2x = 10
D 10x = -10
Find the geometric mean between each pair of numbers. (Lesson 8-1)
45. 3 and 12
46. 9 and 12
47. 11 and 7
48. 6 and 9
49. GARDENS A park has a garden plot shaped like a triangle. It is bordered
by a path. The triangle formed by the outside edge of the path is
similar to the triangular garden. The perimeter of the outside edge of
the path is 53 feet, the longest edge is 20 feet. The longest edge of the
garden plot is 12 feet. What is the perimeter of the garden? (Lesson 7-5)
50. Could the sides of a triangle have the lengths 12, 13, and 25? Explain. (Lesson 5-4)
PREREQUISITE SKILL Simplify each expression by rationalizing the
denominator. (Pages 790–791)
7
51. _
√
3
#
Explain how right triangles are used to build suspension
bridges. Which parts of the right triangle are formed by the cables?
42. Miko is going to rope off an area of
the park for an upcoming concert. He
is going to place a plastic flag for
every three feet of rope.
A 82
'
18
52. _
√
2
√
14
53. _
446 Chapter 8 Right Triangles and Trigonometry
√
2
3 √
11
54. _
√
3
24
55. _
√
2
EXPLORE
8-3
Geometry Lab
Patterns in Special
Right Triangles
Triangles with angles 45°-45°-90° measuring or 30°-60°-90° are called
special right triangles. There are patterns in the measures of the sides of
these triangles.
ACTIVITY 1
Identify patterns in 45°-45°-90° triangles.
Step 1 Draw a square with sides 4 centimeters
long. Label the vertices A, B, C, and D.
−−
Step 2 Draw the diagonal AC.
Step 3 Use a protractor to measure ∠CAB and
∠ACB.
!
"
$
#
Step 4 Use the Pythagorean Theorem to find
AC. Write in simplest form.
ANALYZE THE RESULTS
1. Repeat the activity for squares with sides 6 centimeters long and
8 centimeters long.
2. MAKE A CONJECTURE What is the length of the hypotenuse of a
45°-45°-90° triangle with legs that are n units long?
ACTIVITY 2
Identify patterns in 30°-60°-90° triangles.
Step 1 Construct an equilateral triangle with
sides 2 inches long. Label the vertices
F, G, and H.
−−
Step 2 Find the midpoint of FH and label it J.
−−
Draw median GJ.
'
&
Step 3 Use a protractor to measure ∠FGJ, ∠F,
and ∠GJF.
(
*
Step 4 Use the Pythagorean Theorem to find GJ.
Write in simplest form.
ANALYZE THE RESULTS
3. Repeat the activity to complete a table
like the one at the right.
4. MAKE A CONJECTURE What are the lengths
of the long leg and the hypotenuse of a
30°-60°-90° triangle with a short leg n
units long?
FG
FJ
GJ
2 in.
4 in.
5 in.
Explore 8-3 Geometry Lab: Patterns in Special Right Triangles
447
8-3
Special Right Triangles
Main Ideas
• Use properties of
45°-45°-90° triangles.
• Use properties of
30°-60°-90° triangles.
Many quilt patterns use half square
triangles to create a design. The
pinwheel design was created with eight
half square triangles rotated around the
center. The measures of the angles in
the half square triangles are 45°, 45°,
and 90°.
Properties of 45°-45°-90° Triangles Facts about 45°-45°-90° triangles are
used to solve many geometry problems. The Pythagorean Theorem
allows us to discover special relationships that exist among the sides of
a 45°-45°-90° triangle.
Draw a diagonal of a square. The two triangles formed
are isosceles right triangles. Let x represent the measure
of each side and let d represent the measure of the
hypotenuse.
d2 = x2 + x2
Pythagorean Theorem
d2 = 2x2
2
d = √2x
Add.
x
x
d
Take the positive square root of each side.
d = √
2 · √
x2 Factor.
d = x √
2
Simplify.
This algebraic proof verifies that the length of the hypotenuse of any
2 times the length of its leg. The ratio of the sides
45°-45°-90° triangle is √
2.
is 1 : 1 : √
8.6
In a 45°-45°-90° triangle, the length of the hypotenuse is √
2
times the length of a leg.
N
ƒ
N •
N
ƒ
You can use this relationship to find the measure of the hypotenuse of a
45°-45°-90° triangle given the measure of a leg of the triangle.
448 Chapter 8 Right Triangles and Trigonometry
Pin Wheel Quilt. American, 19th century. Private Collection/Bridgeman Art Library
EXAMPLE
Find the Measure of the Hypotenuse
WALLPAPER TILING Assume that the length of one of the
legs of the 45°-45°-90° triangles in the wallpaper in the
figure is 4 inches. What is the length of the diagonal of
the entire wallpaper square?
The length of each leg of the 45°-45°-90° triangle is 4 inches.
The length of the hypotenuse is √
2 times as long as a leg. So,
the length of the hypotenuse of one of the triangles is 4 √2.
There are four 45°-45°-90° triangles along the diagonal of the square. So,
2 ) or 16 √2 inches.
the length of the diagonal of the square is 4(4 √
1. The length of the leg of a 45°-45°-90° triangle is 7 centimeters.
What is the length of the hypotenuse?
EXAMPLE
Find the Measure of the Legs
C
Find x.
Rationalizing
Denominators
To rationalize a
denominator, multiply
the fraction by 1 in the
form of a radical over
itself so that the
product in the
denominator is a
rational number.
xm
The length of the hypotenuse of a 45°-45°-90° triangle
2 times the length of a leg of the triangle.
is √
xm
45˚
45˚
B
A
6m
AB = (AC) √2
6 = x √
2
6
_
=x
6
_
·
√
2
√
2
√
2
_
=x
√
2
√
6
2
_
=x
2
AB = 6, AC = x
Divide each side by √
2.
Rationalize the denominator.
Multiply.
3 √
2=x
Divide.
4.24 ≈ x
Use a calculator.
2. Refer to ABC. Suppose BA = 5m. Find x.
Properties of 30°-60°-90° Triangles There is also a special relationship
among the measures of the sides of a 30°-60°-90° triangle.
When an altitude is drawn from any vertex of an
equilateral triangle, two congruent 30°-60°-90°
−−−
−−−
triangles are formed. LM and KM are congruent
segments, so let LM = x and KM = x. By the Segment
Addition Postulate, LM + KM = KL. Thus, KL = 2x.
Since JKL is an equilateral triangle, KL = JL = JK.
Therefore, JL = 2x and JK = 2x.
Extra Examples at geometryonline.com
J
30˚ 30˚
2x
L
60˚
x
2x
a
M
Lesson 8-3 Special Right Triangles
x
60˚
K
449
Let a represent the measure of the altitude. Use the Pythagorean Theorem
to find a.
(JM)2 + (LM)2 = (JL)2
a2 + x2 = (2x)2
a2 + x2 = 4x2
a2 = 3x2
a = √
3x2
a = √
3 · √
x2
a = x √
3
Pythagorean Theorem
JM = a, LM = x, JL = 2x
Simplify.
Subtract x2 from each side.
Take the positive square root of each side.
Factor.
Simplify.
So, in a 30°-60°-90° triangle, the measures of the sides are x, x √
3 , and 2x.
The ratio of the sides is 1: √3 : 2.
The relationship of the side measures leads to Theorem 8.7.
8.7
30°-60°-90°
Triangle
In a 30°-60°-90° triangle, the length of the hypotenuse is twice the
length of the shorter leg, and the length of the longer leg is √
3
times the length of the shorter leg.
30˚
n√3
2n
The shorter leg is
opposite the 30° angle,
and the longer leg is
opposite the 60° angle.
60˚
EXAMPLE
n
30°-60°-90° Triangles
Find the missing measures.
a. If BC = 14 inches, find AC.
−−
−−
−−
AC is the longer leg, AB is the shorter leg, and BC
is the hypotenuse.
1
AB = _
(BC)
B
60˚
2
1
=_
(14) or 7
2
BC = 14
A
AC = √
3 (AB)
= √3(7) or 7 √
3
AB = 7
≈ 12.12
AC is 7 √3 or about 12.12 inches.
b. If AC = 8 inches, find BC.
AC = √3(AB)
8 = √3(AB)
8
_
= AB
√
3
√
8
3
_ = AB
3
BC = 2AB
( )
8 √3
=2 _
3
√
16
3
=_
3
≈ 9.24
16 √
3
3
BC is _ or about 9.24 inches.
3. Refer to ABC. Suppose AC = 12 in. Find BC.
450 Chapter 8 Right Triangles and Trigonometry
C
EXAMPLE
Checking
Reasonableness
of Results
To check the
coordinates of P in
Example 4, use a
protractor to
−−
draw DP such that
m∠CDP = 30. Then
from the graph, you
can estimate the
coordinates of P.
Special Triangles in a Coordinate Plane
COORDINATE GEOMETRY Triangle PCD is a 30°-60°-90° triangle with right
−−
angle C. CD is the longer leg with endpoints C(3, 2) and D(9, 2). Locate
point P in Quadrant I.
−−−
−−
y
CD lies on a horizontal gridline. Since PC
−−−
will be perpendicular to CD, it lies on a
−−−
vertical gridline. Find the length of CD.
CD = |9 - 3| = 6
−−−
−−
CD is the longer leg. PC is the shorter leg.
C (3, 2)
So, CD = √3(PC). Use CD to find PC.
CD = √
3 (PC)
6 = √
3 (PC) CD = 6
6
_
= PC
6
_
·
√
3
√3
√3
_
x
O
P
Divide each side by √
3.
60˚
x
= PC
D (9, 2)
2x
Rationalize the denominator.
√3
6 √3
_
= PC
3
30˚
x √3 ⫽ 6
C
D
Multiply.
2 √
3 = PC
Simplify.
Point P has the same x-coordinate as C. P is located 2 √3 units above C.
So, the coordinates of P are (3, 2 + 2 √3) or about (3, 5.46).
−−
4. Triangle RST is a 30°-60°-90° triangle with right angle RST. ST is the
shorter leg with endpoints S(1, 1) and T(4, 1). Locate point R in
Quadrant I.
Personal Tutor at geometryonline.com
Example 1
(p. 449)
1. SOFTBALL Find the distance from home plate to
second base if the bases are 90 feet apart.
2nd Base
90 ft
3rd
Base
45˚ 45˚
d
90 ft
1st
Base
45˚ 45˚
90 ft
90 ft
Home Plate
Example 2
(p. 449)
Find x and y.
2.
45˚
3.
y
3
x
y
30˚
8
x
B
Example 3
(p. 450)
Find the missing measures.
4. If c = 8, find a and b.
5. If b = 18, find a and c.
a
60˚
C
c
b
30˚
Lesson 8-3 Special Right Triangles
A
451
Example 4
(p. 451)
HOMEWORK
HELP
For
See
Exercises Examples
8–10,
2
18, 20
11–17, 19,
3
21–23
24–27
4
28–32
1
−−
Triangle ABD is a 30°-60°-90° triangle with right angle B and with AB as
the shorter leg. Graph A and B, and locate point D in Quadrant I.
6. A(8, 0), B(8, 3)
7. A(6, 6), B(2, 6)
Find x and y.
8.
x˚ y
9.6
x
9.
y˚
10.
5
y
17
45˚
x
11.
12.
x
60˚
13.
x
18
60˚
y
60˚
12
y
11 30˚
60˚
y
x
For Exercises 14 and 15, use the figure at the right.
14. If a = 10 √3, find CE and y.
15. If x = 7 √
3 , find a, CE, y, and b.
B
x
E
60˚
c
y
a
C
b
16. The length of an altitude of an equilateral triangle is
12 feet. Find the length of a side of the triangle.
30˚
A
17. The perimeter of an equilateral triangle is 45 centimeters. Find the
length of an altitude of the triangle.
18. The length of a diagonal of a square is 22 √2 millimeters. Find the
perimeter of the square.
19. The altitude of an equilateral triangle is 7.4 meters long. Find the
perimeter of the triangle.
20. The diagonals of a rectangle are 12 inches long and intersect at an
angle of 60°. Find the perimeter of the rectangle.
21. The sum of the squares of the measures of the sides of a square
is 256. Find the measure of a diagonal of the square.
−− −−
22. Find x, y, z, and the perimeter
23. If PQ SR, find a, b, c, and d.
of trapezoid ABCD.
y
$
P (a, b) Q (8√3, d )
#
6√3
!
X
ƒ
ƒ
Y
Z
"
60˚
S ( 0, 0)
T (c, 0) U (e, 0)
24. PAB is a 45°-45°-90° triangle with right angle B. Find the
coordinates of P in Quadrant I for A(-3, 1) and B(4, 1).
25. PGH is a 45°-45°-90° triangle with m∠P = 90°. Find the
coordinates of P in Quadrant I for G(4,-1) and H(4, 5).
452 Chapter 8 Right Triangles and Trigonometry
45˚
R (f, 0)
x
−−−
26. PCD is a 30°-60°-90° triangle with right angle C and CD the longer
leg. Find the coordinates of P in Quadrant III for C(-3, -6) and D(-3, 7).
−−−
27. PCD is a 30°-60°-90° triangle with m∠C = 30 and hypotenuse CD.
−−−
Find the coordinates of P for C(2, -5) and D(10, -5) if P lies above CD.
Real-World Link
Triangle Tiling Buildings
in Federation Square in
Melbourne, Australia,
feature a tiling pattern
called a pinwheel tiling.
The sides of each right
triangle are in the ratio
1 : 2 : √
5.
Source:
www.federationsquare.
com.au
TRIANGLE TILING For Exercises 28–31, use the following information.
Triangle tiling refers to the process of taking many copies of a single
triangle and laying them next to each other to fill an area. For example,
the pattern shown is composed of tiles like the one outlined.
28. How many 30°-60°-90° triangles are
used to create the basic pattern, which
resembles a circle?
29. Which angle of the 30°-60°-90° triangle
is being rotated to make the basic shape?
30. Explain why there are no gaps in the
basic pattern.
31. Use grid paper to cut out 30°-60°-90°
triangles. Color the same pattern on each
triangle. Create one basic figure that would be
D
part of a wallpaper tiling.
32. BASEBALL The diagram at the right shows some
dimensions of U.S. Cellular Field in Chicago,
−−
Illinois. BD is a segment from home plate to
−−
dead center field, and AE is a segment from
the left field foul-ball pole to the right field
foul-ball pole. If the center fielder is standing
at C, how far is he from home plate?
33. Find x, y, and z.
EXTRA
PRACTICE
See pages 815, 835.
x
B
D
C
B
36. In regular hexagon UVWXYZ, each
side is 12 centimeters long. Find WY.
X
W
Y
V
30˚
30˚
x
H.O.T. Problems
347 ft
F
Self-Check Quiz at
geometryonline.com
30˚ 30˚
45˚
G
A
35. Each triangle in the figure is a
30°-60°-90° triangle. Find x.
4
347 ft
H
45˚
60˚
x
E
y
E
45˚
34. If BD = 8 √3 and m∠DHB = 60°,
find BH.
z
8
C
A
12
U
Z
37. OPEN ENDED Draw a rectangle that has a diagonal twice as long as its width.
Then write an equation to find the length of the rectangle.
38. CHALLENGE Given figure ABCD, with
−− −−−
AB DC, m∠B = 60°, m∠D = 45°, BC = 8,
and AB = 24, find the perimeter.
A
B
60˚
45˚
D
C
Lesson 8-3 Special Right Triangles
John Gollings/Courtesy Federation Square
453
39.
Writing in Math Refer to the information about quilting on page
448. Describe why quilters use the term half square triangles to describe
45°-45°-90° triangles. Explain why 45°-45°-90° triangles are used in
this pattern instead of 30°-60°-90° triangles.
41. Look at the right triangle below.
Which of the following could be
the triangle’s dimensions?
40. A ladder is propped against a
building at a 30° angle.
X
FT
ƒ
What is the length of the ladder?
X
A 5 ft
C 10 √
3 ft
F 9
H 18 √
2
B 10 ft
D 20 ft
G 9 √
2
J
36
Determine whether each set of measures contains the sides of a right triangle.
Then state whether they form a Pythagorean triple. (Lesson 7-2)
42. 3, 4, 5
43. 9, 40, 41
44. 20, 21, 31
45. 20, 48, 52
46. 7, 24, 25
47. 12, 34, 37
Find x, y, and z. (Lesson 7-1)
48.
x
y
49.
50.
10
12
4
z
z
z
8
y
x
15
x
y
5
Write an inequality relating each pair of angles. (Lesson 5-5)
51. m∠ALK, m∠ALN
52. m∠ALK, m∠NLO
53. m∠OLK, m∠NLO
54. m∠KLO, m∠ALN
A
8.5
4.7
4
K
N
L
6
6
4
8.5
O
55. SCALE MODELS Taipa wants to build a scale model of the Canadian
Horseshoe Falls at Niagara Falls. The height is 52 meters. If she wants
the model to be 80 centimeters tall, what scale factor will she use? (Lesson 7-1)
PREREQUISITE SKILL Solve each equation. (Pages 781–782)
x
56. 5 = _
x
57. _
= 0.14
10
58. 0.5 = _
7
60. _
n = 0.25
m
61. 9 = _
24
62. _
x = 0.4
3
9
0.8
454 Chapter 8 Right Triangles and Trigonometry
k
13
59. 0.2 = _
g
35
63. _
y = 0.07
6.5
EXPLORE
8-4
Graphing Calculator Lab
Trigonometry
You have investigated the patterns in the measures of special right
triangles. The study of the patterns in all right triangles is called
trigonometry. You can use the Cabri Junior application on a TI-83/84 Plus
to investigate these patterns.
ACTIVITY
Step 1 Use the line tool on the F2 menu to draw a line.
Label the points on the line A and B.
Step 2 Press F3 and choose the Perpendicular tool to
create a perpendicular line through point B.
Draw and label a point C on the perpendicular.
Step 3 Use the segment tool on the F2 menu to
−−
draw AC.
−−
−−
Step 4 Find and label the measures of BC and AC
using the Distance and Length tool under
Measure on the F5 menu. Use the Angle tool
for the measure of ∠A.
BC
Step 6 Calculate and display the ratio _
using the
AC
Calculate tool on the F5 menu. Label the ratio
as A/B.
Step 7 Press CLEAR . Then use the arrow keys to move
the cursor close to point B. When the arrow is
clear, press and hold the ALPHA key. Drag B and
observe the ratio.
ANALYZE THE RESULTS
1. Discuss the effect of dragging point B on BC, AC, m∠A, and the
BC
ratio _
.
AC
AB
BC
2. Use the calculate tool to find the ratios _
and _
. Then drag B and
AB
AC
observe the ratios.
3. MAKE A CONJECTURE The sine, cosine, and tangent functions are
trigonometric functions based on angle measures. Make a note of
m∠A. Exit Cabri Jr. and use SIN , COS and TAN on the calculator to
find sine, cosine and tangent for m∠A. Compare the results to the
ratios you found in the activity. Make a conjecture about the
definitions of sine, cosine, and tangent.
Explore 8-4 Graphing Calculator Lab: Trigonometry
455
8-4
Trigonometry
Main Ideas
• Find trigonometric
ratios using right
triangles.
• Solve problems using
trigonometric ratios.
New Vocabulary
trigonometry
trigonometric ratio
sine
cosine
tangent
The branch of mathematics known as
trigonometry was developed for use by
astronomers and surveyors. Surveyors
use an instrument called a theodolite
(thee AH duh lite) to measure angles. It
consists of a telescope mounted on a
vertical axis and a horizontal axis. After
measuring the angles, surveyors apply
trigonometry to calculate distance or
height.
Trigonometric Ratios The word trigonometry comes from two Greek
terms, trigon, meaning triangle, and metron, meaning measure. The study
of trigonometry involves triangle measurement. A ratio of the lengths of
sides of a right triangle is called a trigonometric ratio. The three most
common trigonometric ratios are sine, cosine, and tangent.
Trigonometric Ratios
Words
Symbols
leg opposite ∠A
sine of ∠A = __
BC
sin A = _
leg opposite ∠B
sine of ∠B = __
AC
sin B = _
hypotenuse
hypotenuse
AB
AC
cos A = _
leg adjacent to ∠B
cosine of ∠B = __
BC
cos B = _
hypotenuse
A
AB
BC
tan A = _
leg opposite ∠B
tangent of ∠B = __
leg adjacent to ∠B
AC
tan B = _
BC
hypotenuse
leg
opposite
⬔A
leg opposite ⬔B
C
B
AB
leg opposite ∠A
tangent of ∠A = __
leg adjacent to ∠A
B
AB
leg adjacent to ∠A
cosine of ∠A = __
hypotenuse
Models
hypotenuse
A
leg adjacent to ⬔A
leg
adjacent to
⬔B
C
B leg
AC
hypotenuse
A
leg adjacent to ⬔A
and opposite ⬔B
opposite
⬔A and
adjacent to
C ⬔B
Trigonometric ratios are related to the acute angles of a right triangle,
not the right angle.
456 Chapter 8 Right Triangles and Trigonometry
Arthur Thevenart/CORBI
Reading Math
Memory Hint SOH-CAHTOA is a mnemonic device
for learning the ratios for
sine, cosine, and tangent
using the first letter of
each word in the ratios.
opp
hyp
adj
cos A = _
hyp
opp
tan A = _
adj
EXAMPLE
T
Find Sine, Cosine, and Tangent Ratios
3
4
Find sin R, cos R, tan R, sin S, cos S, and tan S.
Express each ratio as a fraction and as a decimal.
S
sin R = __
opposite leg
hypotenuse
ST
=_
RS
4 or 0.8
=_
5
cos R = _
adjacent leg
hypotenuse
RT
=_
RS
3
=_
or 0.6
5
tan R = __
opposite leg
hypotenuse
RT
_
=
RS
3
or 0.6
=_
5
cos S = _
adjacent leg
hypotenuse
ST
_
=
RS
4 or 0.8
=_
5
tan S = __
sin A = _
sin S = __
R
5
opposite leg
adjacent leg
ST
=_
RT
4 or 1.−
=_
3
3
opposite leg
adjacent leg
RT
_
=
ST
3
= _ or 0.75
4
K
13
1. Find sin J, cos J, tan J, sin K, cos K, and tan K.
Express each ratio as a fraction and as a decimal.
J
5
12
L
You can use paper folding to investigate trigonometric ratios in similar
right triangles.
GEOMETRY LAB
A
Trigonometric Ratios
• Fold a rectangular piece of paper along a diagonal from A to C. Then cut
along the fold to form right triangle ABC. Write the name of each angle on
the inside of the triangle.
A
B
B
−−
• Fold the triangle so that there are two segments perpendicular to BA. Label
−− −− −− −− −−
points D, E, F, and G as shown. Use a ruler to measure AC, AB, BC, AF, AG,
−− −− −−
−−
FG, AD, AE, and DE to the nearest millimeter.
ANALYZE THE RESULTS
1. What is true of AED, AGF, and ABC?
2. Copy the table. Write the ratio of the side lengths for each ratio. Then calculate a
C
C
A
ED
G
B
F
C
value for each ratio to the nearest ten-thousandth.
In AED
In AGF
In ABC
sin A
cos A
tan A
3. Study the table. Write a sentence about the patterns you observe.
4. What is true about m∠A in each triangle?
Extra Examples at geometryonline.com
Lesson 8-4 Trigonometry
457
As the Geometry Lab suggests, the value of a trigonometric ratio depends only
on the measure of the angle. It does not depend on the size of the triangle.
EXAMPLE
Graphing
Calculator
Be sure your calculator
is in degree mode
rather than radian
mode. Your calculator
may require you to
input the angle before
using the trigonometric
key.
Use a Calculator to Evaluate Expressions
Use a calculator to find cos 39° to the nearest ten-thousandth.
KEYSTROKES:
COS 39 ENTER
cos 39° ≈ 0.7771
2. sin 67°
Use Trigonometric Ratios You can use trigonometric ratios to find the
missing measures of a right triangle if you know the measures of two sides of
a triangle or the measure of one side and one acute angle.
EXAMPLE
Use Trigonometric Ratios to Find a Length
SURVEYING Dakota is standing on the ground 97 yards from the base of
a cliff. Using a theodolite, he noted that the angle formed by the
ground and the line of sight to the top of the cliff is 56°. Find the
height of the cliff to the nearest yard.
Let x be the height of the cliff in yards.
leg opposite
x
tan 56° = _
tan = _
leg adjacent
x yd
97
97 tan 56° = x
Multiply each side by 97.
56˚
Use a calculator to find x.
KEYSTROKES:
97 yd
97 TAN 56 ENTER 143.8084139
The cliff is about 144 yards high.
3. MEASUREMENT Jonathan is standing 15 yards from a roller coaster. The
angle formed by the ground to the top of the roller coaster is 71°. How tall
is the roller coaster?
When solving equations like 3x = -27, you use the inverse of multiplication
to find x. In trigonometry, you can find the measure of the angle by using the
inverse of sine, cosine, or tangent.
Given equation
To find the angle
sin A = x
A=
sin-1
(x)
A equals the inverse sine of x.
A=
cos-1
(y)
A equals the inverse cosine of y.
A=
tan-1
(z)
A equals the inverse tangent of z.
cos A = y
tan A = z
458 Chapter 8 Right Triangles and Trigonometry
Read as
EXAMPLE
Calculators
The second functions
of the 3). , #/3 ,
and 4!. keys are
usually the inverses.
Use Trigonometric Ratios to Find an Angle Measure
COORDINATE GEOMETRY Find m∠A in right
triangle ABC for A(1, 2), B(6, 2), and C(5, 4).
y
Explore You know the coordinates of the vertices of
a right triangle and that ∠C is the right
angle. You need to find the measure of one
of the angles.
Plan
Use the Distance Formula to find the
measure of each side. Then use one of the
trigonometric ratios to write an equation.
Use the inverse to find m∠A.
Solve
AB =
(6 - 1)2 + (2 - 2)2
√
BC =
A(1, 2)
O
B(6, 2)
x
(5 - 6)2 + (4 - 2)2
√
= √
1 + 4 or √
5
= √
25 + 0 or 5
AC =
C(5, 4)
(5 - 1)2 + (4 - 2)2
√
= √
16 + 4
or 2 √
= √20
5
Use the cosine ratio.
You can use
trigonometry
to help you
come closer to locating
the hidden treasure.
Visit geometryonline.com.
AC
cos A = _
leg adjacent
hypotenuse
cos = _
AB
2 √
5
5
cos A = _
AC = 2 √
5 and AB = 5
( )
2 √
5
A = cos-1 _ Solve for A.
5
Use a calculator to find m∠A.
KEYSTROKES:
ND [COS-1] 2 ND [ √ ] 5
5 %.4%2
m∠A ≈ 26.56505118
The measure of ∠A is about 26.6.
Check
Use the sine ratio to check the answer.
BC
sin A = _
sin = _
AB
leg opposite
hypotenuse
√
5
5
sin A = _ BC = √5 and AB = 5
KEYSTROKES:
ND [SIN-1] ND [ √ ] 5 5 %.4%2
m∠A ≈ 26.56505118
The answer is correct.
4. Find m∠P in right PQR for P(2, -1), Q(4, 3), and R(8, 1).
Personal Tutor at geometryonline.com
Lesson 8-4 Trigonometry
459
Example 1
(p. 457)
Example 2
(p. 458)
Example 3
(p. 458)
Use ABC to find sin A, cos A, tan A, sin B, cos B,
and tan B. Express each ratio as a fraction and as a
decimal to the nearest hundredth.
1. a = 14, b = 48, and c = 50
2. a = 8, b = 15, and c = 17
B
c
a
b
C
A
Use a calculator to find each value. Round to the nearest
ten-thousandth.
4. cos 60°
5. cos 33°
3. sin 57°
6. tan 30°
7. tan 45°
8. sin 85°
9. SURVEYING Maureen is standing on horizontal
ground level with the base of the CN Tower in
Toronto, Ontario. The angle formed by the ground
and the line segment from her position to the top
of the tower is 31.2°. She knows that the height
of the tower to the top of the antennae is about
1815 feet. Find her distance from the CN Tower
to the nearest foot.
Find the measure of each angle to the nearest tenth of a degree.
11. sin B = 0.6307
10. tan A = 1.4176
Example 4
(p. 459)
HOMEWORK
HELP
For
See
Exercises Examples
14–17
1
18–23
2
24, 25
3
26–28
4
COORDINATE GEOMETRY Find the measure of the angle to the nearest tenth
in each right triangle ABC.
12. ∠A in ABC, for A(6, 0), B(-4, 2), and C(0, 6)
13. ∠B in ABC, for A(3, -3), B(7, 5), and C(7, -3)
P
Use PQR with right angle R to find sin P, cos P, tan P, 1
sin Q, cos Q, and tan Q. Express each ratio as a fraction,
R
and as a decimal to the nearest hundredth.
14. p = 12, q = 35, and r = 37
15. p = √6, q = 2 √3, and r = 3 √2
3 √3
2
3
16. p = _
, q = _, and r = 3
2
17. p = 2 √3, q = √
15 , and r = 3 √3
Use a calculator to find each value. Round to the nearest ten-thousandth.
19. tan 42.8°
20. cos 77°
18. sin 6°
21. sin 85.9°
22. tan 12.7°
23. cos 22.5°
24. AVIATION A plane is one mile above sea level
when it begins to climb at a constant angle of
3° for the next 60 ground miles. About how
far above sea level is the plane after its climb?
460 Chapter 8 Right Triangles and Trigonometry
David R. Frazier/Photo Researchers
3˚
60 mi
2
Q
0
25. MONUMENTS At 351 feet tall, the Jefferson Davis Monument in Fairview,
Kentucky, is the largest concrete obelisk in the world. Pedro is looking at
the top of the monument at an angle of 75°. How far away from the
monument is he standing?
COORDINATE GEOMETRY Find the measure of each angle to the
nearest tenth in each right triangle.
26. ∠J in JCL for J(2, 2), C(2, -2), and L(7, -2)
27. ∠C in BCD for B(-1, -5), C(-6, -5), and D(-1, 2)
28. ∠X in XYZ for X(-5, 0), Y(7, 0), and Z(0, √
35 )
Real-World Link
The Jefferson Davis
Monument in Fairview,
Kentucky, is the fourth
tallest monument in
the United States.
The walls are seven
feet thick at the base,
tapering to two feet
thick at the top.
Source: parks.ky.gov
Use the figure to find each trigonometric
ratio. Express answers as a fraction and
as a decimal rounded to the nearest
A
ten-thousandth.
29. sin A
32. sin x°
35. cos B
C
5√26
x˚
25
30. tan B
33. cos x°
36. sin y°
y˚
5 √26
D
B
1
31. cos A
34. tan A
37. tan x°
Find the measure of each angle to the nearest tenth of a degree.
39. cos C = 0.2493
40. tan E = 9.4618
38. sin B = 0.7245
41. sin A = 0.4567
42. cos D = 0.1212
43. tan F = 0.4279
Find x. Round to the nearest tenth.
45.
44.
x
46.
17
24˚
x
62˚
12
60
19
x˚
47.
48.
34
x
17˚
49.
6.6
x
18
31˚
15
x˚
SAFETY For Exercises 50 and 51, use the following
information.
To guard against a fall, a ladder should make an angle
of 75° or less with the ground.
50. What is the maximum height that a 20-foot ladder
can reach safely?
51. How far from the building is the base of the ladder
at the maximum height?
EXTRA
PRACTICE
Find x and y. Round to the nearest tenth.
53.
52.
x˚
x
See pages 816, 835.
24
36
Self-Check Quiz at
geometryonline.com
y˚
55˚
12
75˚
54.
C
y
x
24
47˚
A
32˚
y
D
Lesson 8-4 Trigonometry
Gibson Stock Photography
B
461
H.O.T. Problems
55. OPEN ENDED Draw a right triangle and label the measure of one acute
angle and the measure of the side opposite that angle. Then solve for the
remaining measures.
56. CHALLENGE Use the figure at the right to find sin x°.
A
10
57. REASONING Explain the difference between
x
x
-1 _
tan A = _
y and tan
y = A.
()
58.
x˚
D
Writing in Math
B
8
10
Refer to the information on
theodolites on page 456. Explain how surveyors
determine angle measures. Include the kind of
information one obtains from a theodolite.
20
59. In the figure, if cos x = _
, what are
29
sin x and tan x?
8
C
60. REVIEW What is the solution set of the
quadratic equation x2 + 4x - 2 = 0?
F {-2, 2}
G {-2 + √
6 , -2 - √
6}
X
H {-2 + √
2 , -2 - √
2}
J no real solution
29
29
A sin x = _
and tan x = _
21
21
20
21 and tan x = _
B sin x = _
29
21
61. REVIEW Which of the following has the
same value as 9-15 × 93?
29
21
and tan x = _
C sin x = _
20
21
21 and tan x = _
21
D sin x = _
29
A 9-45
C 9-12
B 9-18
D 9-5
20
Find each measure. (Lesson 8-3)
B
62. If a = 4, find b and c.
a
63. If b = 3, find a and c.
C
60˚
c
b
30˚
A
Determine whether each set of measures can be the sides of a right triangle.
Then state whether they form a Pythagorean triple. (Lesson 8-2)
64. 4, 5, 6
65. 5, 12, 13
66. 9, 12, 15
67. 8, 12, 16
68. TELEVISION During a 30-minute television program, the ratio of minutes of
commercials to minutes of the actual show is 4 : 11. How many minutes
are spent on commercials? (Lesson 7-1)
s
t
PREREQUISITE SKILL Find each angle measure if h k. (Lesson 3-2)
1
69. m∠15
70. m∠7
71. m∠3
72. m∠12
73. m∠11
74. m∠4
462 Chapter 8 Right Triangles and Trigonometry
30˚
4
7
6
5
3
2
8
9
10
11
12
117˚
15 14
h
k
APTER
CH
8
Mid-Chapter Quiz
Lessons 8-1 through 8-4
Find the measure of the altitude drawn to the
hypotenuse. (Lesson 8-1)
1.
2. X
B
5
9
A
21
7
C
6. WOODWORKING Ginger made a small
square table for her workshop with a
diagonal that measures 55 inches. What
are the measures of the sides? Recall that a
square has right angles at the corners and
congruent sides. (Lesson 8-3)
Y
Z
3. Determine whether ABC with vertices
A(2, 1), B(4, 0), and C(5, 7) is a right triangle.
Explain. (Lesson 8-2)
4. MULTIPLE CHOICE To get from your campsite to
a trail head, you must take the path shown
below to avoid walking through a pond.
M
Find x and y. (Lesson 8-3)
7.
8.
y
45˚
x
30˚
x
9. MULTIPLE CHOICE In the right triangle, what is
AB if BC = 6? (Lesson 8-3)
A
TRAILHEAD
M
y
6
3
4x˚
2x˚
CAMPSITE
C
About how many meters would be saved
if it were possible to walk through the
pond? (Lesson 8-2)
A 55.0
C 24.7
B 39.2
D 15.8
5. DOG WALKING A man is walking his dog
on level ground in a straight line with the
dog’s favorite tree. The angle from the
man’s present position to the top of a nearby
telephone pole is 45º. The angle from the tree
to the top of the telephone pole is 60º. If the
telephone pole is 50 feet tall, about how far is
the man with the dog from the tree? (Lesson 8-3)
FT
ƒ
ƒ
B
F 12 units
H 4 √
3 units
G 6 √
2 units
J 2 √
3 units
Find x to the nearest tenth. (Lesson 8-4)
10.
11.
x
53
16
˚
32
12.
13.
2
x˚
10
x˚
x
5
9.7
17˚
14. GARDENING The lengths of the sides of a
triangular garden are 32 feet, 24 feet, and
40 feet. What are the measures of the angles
formed on each side of the garden? (Lesson 8-4)
Find the measure of each angle to the nearest
tenth of a degree. (Lesson 8-4)
15. sin T = 0.5299
16. cos W = 0.0175
Chapter 8 Mid-Chapter Quiz
463
8-5
Angles of Elevation
and Depression
Main Ideas
• Solve problems
involving angles
of elevation.
• Solve problems
involving angles
of depression.
A pilot is getting ready to take off from Mountain Valley airport.
She looks up at the peak of a mountain immediately in front of her.
The pilot must estimate the speed needed and the angle formed by
a line along the runway and a line from the plane to the peak of the
mountain to clear the mountain.
New Vocabulary
Angles of Elevation An angle of
angle of elevation
angle of depression
elevation is the angle between the
line of sight and the horizontal
when an observer looks upward.
A
•
B
line of sight
D angle of
elevation
EXAMPLE
C
Angle of Elevation
AVIATION The peak of Goose Bay Mountain is 400 meters higher than
the end of a local airstrip. The peak rises above a point 2025 meters
from the end of the airstrip. A plane takes off from the end of the
runway in the direction of the mountain at an angle that is kept
constant until the peak has been cleared. If the pilot wants to clear
the mountain by 50 meters, what should the angle of elevation be
for the takeoff to the nearest tenth of a degree?
Make a drawing.
50 m D
B
400 m
2025 m
A
C
Since CB is 400 meters and BD is 50 meters, CD is 450 meters.
Let x represent m∠DAC.
opposite
CD
tan x° = _
tan = _
AC
450
tan x° = _
2025
adjacent
CD = 450, AC = 2025
( 2025 )
450
x = tan-1 _
Solve for x.
x ≈ 12.5
Use a calculator.
The angle of elevation for the takeoff should be more than 12.5°.
1. SHADOWS Find the angle of elevation of the Sun when a 7.6-meter
flagpole casts a 18.2-meter shadow. Round to the nearest tenth of
a degree.
464 Chapter 8 Right Triangles and Trigonometry
Extra Examples at geometryonline.com
Angles of Depression An angle of
angle of depression
B
depression is the angle between the
line of sight when an observer looks
downward and the horizontal.
A
line of sight
C
D
Angle of Depression
The tailgate of a moving van is 3.5 feet
above the ground. A loading ramp is
attached to the rear of the van at an
incline of 10°. Which is closest to the
length of the ramp?
A 3.6 ft
C 19.8 ft
B 12.2 ft
D 20.2 ft
10˚
3.5 ft
sin 10° ≈ 0.17
cos 10° ≈ 0.98
tan 10° ≈ 0.18
Read the Test Item
The angle of depression between the ramp and the horizontal is 10°.
Use trigonometry to find the length of the ramp.
Solve the Test Item
10˚
A
D
3.5 ft
C
10˚
B
The ground and the horizontal level with the back of the van are parallel.
Therefore, m∠DAB = m∠ABC since they are alternate interior angles.
opposite
3.5
sin 10° = _
sin = _
AB
AB sin 10° = 3.5
3.5
AB = _
Check Results
Before moving
on to the next
question, check
the reasonableness
of your answer.
Analyze your result
to determine that
it makes sense.
sin 10°
3.5
AB ≈ _
0.17
AB ≈ 20.2
hypotenuse
Multiply each side by AB.
Divide each side by sin 10°.
sin 10° ≈ 0.17
Divide.
The ramp is about 20.2 feet long. So the correct answer is choice D.
2. HIKING Ayana is hiking in a national park.
A forest ranger is standing in a fire tower
that overlooks a meadow. She sees Ayana
at an angle of depression measuring 38°.
If Ayana is 50 feet away from the base of
the tower, which is closest to the height
of the fire tower?
F 30.8 ft
H 39.4 ft
G 39.1 ft
J 63.5 ft
ƒ
FT
sin 38° ≈ 0.62
cos 38° ≈ 0.79
tan 38° ≈ 0.78
Personal Tutor at geometryonline.com
Lesson 8-5 Angles of Elevation and Depression
465
Angles of elevation or depression to two different objects can be used to
find the distance between those objects.
EXAMPLE
Common
Misconception
The angle of
depression is often
not an angle of the
triangle but the
complement to an
angle of the triangle.
In DBC, the angle
of depression is
∠BCE, not ∠DCB.
Indirect Measurement
Olivia works in a lighthouse on a
cliff. She observes two sailboats due
east of the lighthouse. The angles of
depression to the two boats are 33°
and 57°. Find the distance between
the two sailboats to the nearest foot.
C
85 ft
E
33˚
57˚
110 ft
CDA and CDB are right triangles,
and CD = 110 + 85 or 195. The
D
A
distance between the boats is AB or
DB - DA. Use the right triangles to
find these two lengths.
−−
−−
Because CE and DB are horizontal lines, they are parallel. Thus,
∠ECB ∠CBD and ∠ECA ∠CAD because they are alternate
interior angles. This means that m∠CBD = 33 and m∠CAD = 57.
B
Use the measures of CBD to find DB.
195
tan 33° = _
DB
DB tan 33° = 195
opposite
adjacent
tan = _; m∠CBD = 33
Multiply each side by DB.
195
DB = _
Divide each side by tan 33°.
DB ≈ 300.27
Use a calculator.
tan 33°
Use the measures of CAD to find DA.
195
tan 57° = _
DA
DA tan 57° = 195
opposite
adjacent
tan = _; m∠CAD = 57
Multiply each side by DA.
195
DA = _
tan 57°
Divide each side by tan 57°.
DA ≈ 126.63
Use a calculator.
The distance between the boats is DB - DA.
DB - DA ≈ 300.27 - 126.63 or about 174 feet
3. BOATING Two boats are observed by a parasailer 75 meters above
a lake. The angles of depression are 12.5° and 7°. How far apart are
the boats?
Example 1
(p. 464)
1. AVIATION A pilot is flying at 10,000 feet and wants to take the plane up to
20,000 feet over the next 50 miles. What should be his angle of elevation to
the nearest tenth? (Hint: There are 5280 feet in a mile.)
466 Chapter 8 Right Triangles and Trigonometry
Example 2
(p. 465)
Example 3
(p. 466)
2. OCEAN ARCHAEOLOGY A salvage ship
uses sonar to determine the angle of
depression to a wreck on the ocean
floor that is 40 meters below the surface.
How far must a diver, lowered from the
salvage ship, walk along the ocean floor
to reach the wreck?
3. STANDARDIZED TEST EXAMPLE
From the top of a 150-foot high
tower, an air traffic controller
observes an airplane on the
runway. Which equation would be
used to find the distance from the
base of the tower to the airplane?
150
A x = 150 tan 12° B x = _
cos 12°
HOMEWORK
HELP
For
See
Exercises Examples
4–11
1
12, 13
2
14–17
3
13.25˚
ƒ
FT
X
150
C x=_
150
D x=_
tan 12°
4. GOLF A golfer is standing at the tee,
looking up to the green on a hill. If
the tee is 36 yards lower than the
green and the angle of elevation
from the tee to the hole is 12°, find
the distance from the tee to the hole.
5. TOURISM Crystal is on a bus in France
with her family. She sees the Eiffel
Tower at an angle of 27°. If the tower is
986 feet tall, how far away is the bus?
Round to the nearest tenth.
40 m
sin 12°
36 yd
12˚
P
T
27˚
B
CIVIL ENGINEERING For Exercises 6 and 7, use the following information.
The percent grade of a highway is the ratio of the vertical rise or fall over a
horizontal distance expressed to the nearest whole percent. Suppose a highway
has a vertical rise of 140 feet for every 2000 feet of horizontal distance.
6. Calculate the percent grade of the highway.
7. Find the angle of elevation that the highway makes with the horizontal.
8. SKIING A ski run has an angle of elevation of 24.4° and a vertical drop of
1100 feet. To the nearest foot, how long is the ski run?
GEYSERS For Exercises 9 and 10, use the following information.
Kirk visits Yellowstone Park and Old Faithful on a perfect day. His eyes
are 6 feet from the ground, and the geyser can reach heights ranging from
90 feet to 184 feet.
9. If Kirk stands 200 feet from the geyser and the eruption rises 175 feet in the
air, what is the angle of elevation to the top of the spray to the nearest tenth?
10. In the afternoon, Kirk returns and observes the geyser’s spray reach a
height of 123 feet when the angle of elevation is 37°. How far from the
geyser is Kirk standing to the nearest tenth of a foot?
Lesson 8-5 Angles of Elevation and Depression
467
11. RAILROADS Refer to the information at the left. Determine the incline of
the Monongahela Incline.
12. AVIATION After flying at an altitude of 500 meters, a helicopter starts to
descend when its ground distance from the landing pad is 11 kilometers.
What is the angle of depression for this part of the flight?
13. SLEDDING A sledding run is 300 yards long with a vertical drop of
27.6 yards. Find the angle of depression of the run.
Real-World Link
The Monongahela
Incline, in Pittsburgh,
Pennsylvania, is 635 feet
long with a vertical
rise of 369.39 feet.
Although opened on
May 28, 1870, it is still
used by commuters to
and from Pittsburgh.
Source: www.portauthority.org
14. AMUSEMENT PARKS From the top of
a roller coaster, 60 yards above the
ground, a rider looks down and sees
the merry-go-round and the Ferris
wheel. If the angles of depression
are 11° and 8°, respectively, how far
apart are the merry-go-round and
the Ferris wheel?
8˚
11˚
60 yd
15. BIRD WATCHING Two observers are 200 feet apart, in line with a tree
containing a bird’s nest. The angles of elevation to the bird’s nest are
30° and 60°. How far is each observer from the base of the tree?
16. METEOROLOGY The altitude of the base of a cloud
formation is called the ceiling. To find the ceiling
one night, a meteorologist directed a spotlight
vertically at the clouds. Using a theodolite placed
83 meters from the spotlight and 1.5 meters above
the ground, he found the angle of elevation to be
62.7°. How high was the ceiling?
62.7˚˚
1.5 m
83 m
17. TRAVEL Kwan-Yong uses a theodolite to measure the angle of elevation
from the ground to the top of Ayers Rock to be 15.85°. He walks half a
kilometer closer and measures the angle of elevation to be 25.6°. How high
is Ayers Rock to the nearest meter?
18. PHOTOGRAPHY A digital camera with a panoramic lens is described as
having a view with an angle of elevation of 38º. If the camera is on a 3-foot
tripod aimed directly at a 124-foot monument, how far from the monument
should you place the tripod to see the entire monument in your photograph?
MEDICINE For Exercises 19–21, use the following information.
A doctor is using a treadmill to assess the strength of a patient’s heart.
At the beginning of the exam, the 48-inch long treadmill is set at an
incline of 10°.
19. How far off the horizontal is the raised end of the treadmill at the
beginning of the exam?
20. During one stage of the exam, the end of the treadmill is 10 inches above
the horizontal. What is the incline of the treadmill to the nearest degree?
21. Suppose the exam is divided into five stages and the incline of the
treadmill is increased 2° for each stage. Does the end of the treadmill rise
the same distance between each stage?
468 Chapter 8 Right Triangles and Trigonometry
R. Krubner/H. Armstrong Roberts
EXTRA
PRACTICE
See pages 816, 835.
Self-Check Quiz at
geometryonline.com
H.O.T. Problems
22. AEROSPACE On July 20, 1969, Neil
Armstrong became the first human to
walk on the Moon. During this mission,
Apollo 11 orbited the Moon three miles
above the surface. At one point in the
orbit, the onboard guidance system
measured the angles of depression to
the far and near edges of a large crater.
The angles measured 16° and 29°,
respectively. Find the distance across
the crater.
œÀLˆÌ
£Èc
әc
Îʓˆ
˜
v
23. OPEN ENDED Find a real-life example of an angle of depression.
Draw a diagram and identify the angle of depression.
24. REASONING Explain why an angle of elevation is given that name.
25. CHALLENGE Two weather observation stations are 7 miles apart.
A weather balloon is located between the stations. From Station 1,
the angle of elevation to the weather balloon is 33°. From Station 2,
the angle of elevation to the balloon is 52°. Find the altitude of the
balloon to the nearest tenth of a mile.
Writing in Math Describe how an airline pilot would use angles
of elevation and depression. Make a diagram and label the angles of
elevation and depression. Then describe the difference between
the two.
26.
27. The top of a signal tower is 120 meters
above sea level. The angle of depression
from the top of the tower to a passing
ship is 25°. Which is closest to the
distance from the foot of the tower to
the ship?
25˚
28. REVIEW What will happen to the slope
of line p if the line is shifted so that the
y-intercept decreases and the x-intercept
remains the same?
y
8
6
4
2
⫺8⫺6⫺4⫺2 O
120 m
2 4 6 8x
⫺4
⫺6
⫺8
sin 25° ≈ 0.42
cos 25° ≈ 0.91
tan 25° ≈ 0.47
F The slope will change from negative
to positive.
G The slope will become undefined.
A 283.9 m
C 132.4 m
H The slope will decrease.
B 257.3 m
D 56.0 m
J The slope will increase.
Lesson 8-5 Angles of Elevation and Depression
469
Find the measure of each angle to the nearest tenth of a degree. (Lesson 8-4)
29. cos A = 0.6717
30. sin B = 0.5127
1
, find tan B. (Lesson 8-4)
32. If cos B = _
"
4
!
Find x and y. (Lesson 8-3)
33.
31. tan C = 2.1758
#
34.
35.
y
x
y
14
12
30˚
45˚ y
60˚
20
x
x
36. LANDSCAPING Paulo is designing two gardens shaped like similar triangles.
One garden has a perimeter of 53.5 feet, and the longest side is 25 feet. He
wants the second garden to have a perimeter of 32.1 feet. Find the length of
the longest side of this garden. (Lesson 7-5)
37. MODEL AIRPLANES A twin-engine airplane used for medium-range flights
has a length of 78 meters and a wingspan of 90 meters. If a scale model is
made with a wingspan of 36 centimeters, find its length. (Lesson 6-2)
38. Copy and complete the flow proof. (Lesson 4-6)
Given: ∠5 ∠6
−− −−
FR GS
F
5
G
3
4
1
Prove: ∠4 ∠3
2
X
R
Proof:
6
S
⬔5 ⬔6
Given
FR GS
䉭FXR 䉭GXS
Given
?
a.
Vert. ⬔s are .
b.
c.
?
d.
?
e.
?
f.
?
?
Determine the truth value of the following statement for each set of conditions.
If you have a fever, then you are sick. (Lesson 2-3)
39. You do not have a fever, and you are sick.
40. You have a fever, and you are not sick.
41. You do not have a fever, and you are not sick.
42. You have a fever, and you are sick.
PREREQUISITE SKILL Solve each proportion. (Lesson 7-1)
35
x
43. _
=_
6
42
3
5
_
44. _
x=
45
470 Chapter 8 Right Triangles and Trigonometry
12
24
45. _
=_
x
17
x
24
46. _
=_
36
15
8-6
The Law of Sines
Main Ideas
• Use the Law of Sines
to solve triangles.
• Solve problems by
using the Law of
Sines.
New Vocabulary
Law of Sines
solving a triangle
The Statue of Liberty was designed by
Frederic-Auguste Bartholdi between
1865 and 1875. Copper sheets were
hammered and fastened to an interior
skeletal framework, which was
designed by Alexandre-Gustave Eiffel.
The skeleton is 94 feet high and
composed of wrought iron bars. These
bars are arranged in triangular shapes,
many of which are not right triangles.
The Law of Sines In trigonometry, the Law of Sines can be used to find
missing parts of triangles that are not right triangles.
8.8
Obtuse Angles
There are also values
for sin A, cos A, and
tan A, when A ≥ 90°.
Values of the ratios for
these angles will be
found using the
trigonometric functions
on your calculator.
Law of Sines
Let ABC be any triangle with a, b, and c
representing the measures of the sides
opposite the angles with measures
A, B, and C, respectively. Then
sin C
sin A
sin B
_
= _ = _.
a
PROOF
b
a
A
c
b
C
B
c
Law of Sines
ABC is a triangle with an altitude
−−
from C that intersects AB at D. Let h
−−−
represent the measure of CD. Since
ADC and BDC are right triangles,
we can find sin A and sin B.
A
h
sin A = _
h
a
D
B
a sin B = h Cross products
b sin A = a sin B
sin A
sin B
_
=_
a
b
h
sin B = _
a Definition of sine
b
b sin A = h
C
b
Substitution
Divide each side by ab.
The proof can be completed by using a similar technique with the
sin A
sin C
sin B
sin C
_
_
other altitudes to show that _
=_
a = c and
c .
b
Lesson 8-6 The Law of Sines
Rohan/Stone/Getty Images
471
EXAMPLE
Use the Law of Sines
Given measures of ABC, find the indicated measure. Round angle
measures to the nearest degree and side measures to the nearest tenth.
a. If m∠A = 37, m∠B = 68, and a = 3, find b.
Use the Law of Sines to write a proportion.
sin A
sin B
_
=_
a
b
sin
37°
sin
68°
_=_
3
b
Rounding
If you round before
the final answer, your
results may differ from
results in which
rounding was not
done until the final
answer.
Law of Sines
m∠A = 37, a = 3, m∠B = 68
b sin 37° = 3 sin 68° Cross products
3 sin 68°
b=_
Divide each side by sin 37°.
sin 37°
b ≈ 4.6
Use a calculator.
b. If b = 17, c = 14, and m∠B = 92, find m∠C.
Write a proportion relating ∠B, ∠C, b, and c.
sin B
sin C
_
=_
c
b
sin
92°
sin
C
_=_
17
14
Law of Sines
m∠B = 92, b = 17, c = 14
14 sin 92° = 17 sin C Cross products
14 sin 92°
_
= sin C
17
14
sin
92°
sin-1 _
=C
17
)
(
55° ≈ C
Divide each side by 17.
Solve for C.
Use a calculator.
So, m∠C ≈ 55.
1A. If m∠B = 32, m∠C = 51, c = 12, find a.
1B. If a = 22, b = 18, m∠A = 25, find m∠B.
The Law of Sines can be used to solve a triangle. Solving a triangle means
finding the measures of all of the angles and sides of a triangle.
EXAMPLE
Look Back
To review the Angle
Sum Theorem, see
Lesson 4-2.
Solve Triangles
a. Solve ABC if m∠A = 33, m∠B = 47, and
b = 14. Round angle measures to the nearest
degree and side measures to the nearest tenth.
We know the measures of two angles of the
triangle. Use the Angle Sum Theorem to
find m∠C.
472 Chapter 8 Right Triangles and Trigonometry
C
a
b
A
c
Extra Examples at geometryonline.com
B
An Equivalent
Proportion
The Law of Sines may
also be written as
c
a
b
_
=_
= _.
sin A
sin B
m∠A + m∠B + m∠C = 180
33 + 47 + m∠C = 180
80 + m∠C = 180
m∠C = 100
Angle Sum Theorem
m∠A = 33, m∠B = 47
Add.
Subtract 80 from each side.
sin B
Since we know m∠B and b, use proportions involving _
.
b
sin C
You may wish to use
this form when finding
the length of a side.
To find a:
To find c:
sin B
sin A
_
=_
a
b
sin 47°
sin 33°
_
_
= a
14
a sin 47° = 14 sin 33°
Law of Sines
sin B
sin C
_
=_
Substitute.
sin 47°
sin 100°
_
=_
b
14
Divide each side by sin 47°.
sin 47°
a ≈ 10.4
c
c sin 47° = 14 sin 100°
Cross products
14 sin 33°
a=_
c
14 sin 100°
c=_
sin 47°
c ≈ 18.9
Use a calculator.
Therefore, m∠C = 100, a ≈ 10.4, and c ≈ 18.9.
b. Solve ABC if m∠C = 98, b = 14, and c = 20. Round angle measures to
the nearest degree and side measures to the nearest tenth.
sin B
sin C
_
=_
Law of Sines
c
b
sin B
sin 98°
_
_
=
20
14
m∠C = 98, b = 14, and c = 20
20 sin B = 14 sin 98°
Cross products
14 sin 98°
sin B = _
20
(
Divide each side by 20.
)
14 sin 98°
B = sin-1 _
Solve for B.
20
B ≈ 44°
m∠A + m∠B + m∠C = 180
m∠A + 44 + 98 = 180
m∠A + 142 = 180
m∠A = 38
sin A
sin C
_
=_
a
c
sin 38°
sin 98°
_
=_
a
20
Use a calculator.
Angle Sum Theorem
m∠B = 44 and m∠C = 98
Add.
Subtract 142 from each side.
Law of Sines
m∠A = 38, m∠C = 98, and c = 20
20 sin 38° = a sin 98°
Cross products
20 sin 38°
_
=a
Divide each side by sin 98°.
sin 98°
12.4 ≈ a
Use a calculator.
Therefore, A ≈ 38°, B ≈ 44°, and a ≈ 12.4.
Interactive Lab
geometryonline.com
Find the missing angles and sides of PQR. Round angle measures to
the nearest degree and side measures to the nearest tenth.
2A. m∠R = 66, m∠Q = 59, p = 72
2B. p = 32, r = 11, m∠P = 105
Lesson 8-6 The Law of Sines
473
Use the Law of Sines to Solve Problems The Law of Sines is very useful in
solving direct and indirect measurement applications.
Indirect Measurement
ENGINEERING When the angle of elevation to the Sun is 62°, a telephone
pole tilted at an angle of 7° from the vertical casts a shadow 30 feet
long on the ground. Find the length of the telephone pole to the
nearest tenth of a foot.
3
Draw a diagram.
0
−− −−−
Draw SD ⊥ GD. Then find m∠GDP and
m∠GPD.
POLE
—
m∠GDP = 90 - 7 or 83
m∠GPD + 62 + 83 = 180 or m∠GPD = 35
'
—
FOOTSHADOW
$
Since you know the measures of two angles of the triangle, m∠GDP and
−−−
m∠GPD, and the length of a side opposite one of the angles (GD is opposite
∠GPD) you can use the Law of Sines to find the length of the pole.
GD
PD
_
=_
Law of Sines
sin ∠DGP
sin ∠GPD
30
PD
_
=_
m∠DGP = 62, m∠GPD = 35, and GD = 30
sin 62°
sin 35°
PD sin 35° = 30 sin 62° Cross products
30 sin 62°
PD = _
Divide each side by sin 35°.
sin 35°
PD ≈ 46.2
Use a calculator.
The telephone pole is about 46.2 feet long.
3. AVIATION Two radar stations that are 35 miles apart located a plane at the
same time. The first station indicated that the position of the plane made
an angle of 37° with the line between the stations. The second station
indicated that it made an angle of 54° with the same line. How far is each
station from the plane?
Personal Tutor at geometryonline.com
Law of Sines
Case 2 of the Law of
Sines can lead to two
different triangles.
This is called the
ambiguous case of the
Law of Sines.
Law of Sines
The Law of Sines can be used to solve a triangle in the following cases.
Case 1 You know the measures of two angles and any side of a triangle.
(AAS or ASA)
Case 2 You know the measures of two sides and an angle opposite one of these
sides of the triangle. (SSA)
474 Chapter 8 Right Triangles and Trigonometry
Example 1
(p. 472)
Find each measure using the given measures of XYZ. Round angle
measures to the nearest degree and side measures to the nearest tenth.
1. If x = 3, m∠X = 37, and m∠Y = 68, find y.
2. If y = 12.1, m∠X = 57, and m∠Z = 72, find x.
3. If y = 7, z = 11, and m∠Z = 37, find m∠Y.
4. If y = 17, z = 14, and m∠Y = 92, find m∠Z.
5. SURVEYING To find the distance between
two points A and B that are on opposite
sides of a river, a surveyor measures the
distance to point C on the same side of the
river as point A. The distance from A to C
is 240 feet. He then measures the angle
across from A to B as 62° and measures the
angle across from C to B as 55°. Find the
distance from A to B.
Example 2
(p. 472)
Example 3
(p. 474)
C
A
B
Solve each PQR described below. Round angle measures to the nearest
degree and side measures to the nearest tenth.
6. m∠P = 33, m∠R = 58, q = 22
7. p = 28, q = 22, m∠P = 120
8. m∠P = 50, m∠Q = 65, p = 12
9. q = 17.2, r = 9.8, m∠Q = 110.7
10. m∠P = 49, m∠R = 57, p = 8
11. m∠P = 40, m∠Q = 60, r = 20
12. Find the perimeter of parallelogram
ABCD to the nearest tenth.
A
B
6
D
HOMEWORK
HELP
For
See
Exercises Examples
13–18
1
19–26
2
27, 28
3
32°
88°
C
Find each measure using the given measures of KLM. Round angle
measures to the nearest degree and side measures to the nearest tenth.
13. If k = 3.2, m∠L = 52, and m∠K = 70, find .
14. If m = 10.5, k = 18.2, and m∠K = 73, find m∠M.
15. If k = 10, m = 4.8, and m∠K = 96, find m∠M.
16. If m∠M = 59, = 8.3, and m = 14.8, find m∠L.
17. If m∠L = 45, m∠M = 72, and = 22, find k.
18. If m∠M = 61, m∠K = 31, and m = 5.4, find .
Solve each WXY described below. Round measures to the nearest tenth.
19. m∠Y = 71, y = 7.4, m∠X = 41
20. x = 10.3, y = 23.7, m∠Y = 96
21. m∠X = 25, m∠W = 52, y = 15.6 22. m∠Y = 112, x = 20, y = 56
23. m∠W = 38, m∠Y = 115, w = 8.5 24. m∠W = 36, m∠Y = 62, w = 3.1
25. w = 30, y = 9.5, m∠W = 107
26. x = 16, w = 21, m∠W = 88
Lesson 8-6 The Law of Sines
475
IN
27. TELEVISIONS To gain better reception on his
antique TV, Mr. Ramirez positioned the two
antennae 13 inches apart with an angle between
them of approximately 82°. If one antenna is
5 inches long, about how long is the other
antenna?
IN ƒ XIN
28. REAL ESTATE A house is built on a triangular plot
of land. Two sides of the plot are 160 feet long,
and they meet at an angle of 85°. If a fence is to
be placed along the perimeter of the property,
how much fencing material is needed?
85˚
160 ft
160 ft
29. An isosceles triangle has a base of 46 centimeters and a vertex angle
of 44°. Find the perimeter.
30. Find the perimeter of quadrilateral ABCD to the
nearest tenth.
A
B
28˚
12
40˚
C
D
31. SURVEYING Maria Lopez is a surveyor who must determine the
distance across a section of the Rio Grande Gorge in New Mexico.
On one side of the ridge, she measures the angle formed by the
edge of the ridge and the line of sight to a tree on the other side of
the ridge. She then walks along the ridge 315 feet, passing the tree
and measures the angle formed by the edge of the ridge and the
new line of sight to the same tree. If the first angle is 80° and the
second angle is 85°, find the distance across the gorge.
EXTRA
PRACTICE
See pages 816, 835.
Self-Check Quiz at
geometryonline.com
H.O.T. Problems
HIKING For Exercises 32 and 33, use the following information.
Kayla, Jenna, and Paige are hiking at a state park and they get
separated. Kayla and Jenna are 120 feet apart. Paige sends up a signal.
Jenna turns 95° in the direction of the signal and Kayla rotates 60°.
32. To the nearest foot, how far apart are Kayla and Paige?
33. To the nearest foot, how far apart are Jenna and Paige?
34. FIND THE ERROR Makayla and Felipe are trying to find d in DEF.
Who is correct? Explain your reasoning.
Makayla
d
sin 59° = _
12
F
Felipe
12 73˚
sin 59° _
_
= sin 48°
d
12
D
f
d
48˚
E
35. OPEN ENDED Draw an acute triangle and label the measures of two
angles and the length of one side. Explain how to solve the triangle.
36. CHALLENGE Does the Law of Sines apply to the acute angles of a
right triangle? Explain your answer.
37.
Writing in Math Refer to the information on the Statue of Liberty
on page 471. Describe how triangles are used in structural support.
476 Chapter 8 Right Triangles and Trigonometry
38. Soledad is looking at the top of a
150-foot tall Ferris wheel at an angle
of 75°.
39. REVIEW Which inequality best
describes the graph below?
5
4
3
2
1
sin 75° ≈ 0.97
cos 75° ≈ 0.26
FT
ƒ
⫺3⫺2⫺1 O
tan 75° ≈ 3.73
1 2 3 4 5x
⫺2
⫺3
FT
If she is 5 feet tall, how far is Soledad
from the Ferris wheel?
y
F y ≥ -x + 2
G y≤x+2
A 15.0 ft
C 75.8 ft
H y ≥ -3x + 2
B 38.9 ft
D 541.1 ft
J y ≤ 3x + 2
ARCHITECTURE For Exercises 40 and 41, use the following
information.
Mr. Martinez is an architect who designs houses so that
the windows receive minimum Sun in the summer and
maximum Sun in the winter. For Columbus, Ohio, the
angle of elevation of the Sun at noon on the longest day
is 73.5° and on the shortest day is 26.5°. Suppose a house
is designed with a south-facing window that is 6 feet tall.
The top of the window is to be installed 1 foot below the
overhang. (Lesson 8-5)
x
1 ft
window
6 ft
high
angle of
elevation
of sun
40. How long should the architect make the overhang so that the window gets
no direct sunlight at noon on the longest day?
41. Using the overhang from Exercise 40, how much of the window will get direct
sunlight at noon on the shortest day?
Use JKL to find sin J, cos J, tan J, sin L, cos L, and tan L. Express each
ratio as a fraction and as a decimal to the nearest hundredth. (Lesson 8-4)
42. j = 8, k = 17, l = 15
43. j = 20, k = 29, l = 21
44. j = 12, k = 24, l = 12 √3
45. j = 7 √2, k = 14, l = 7 √2
J
K
L
_
2
2
2
PREREQUISITE SKILL Evaluate c - a - b for the given values of a, b, and c. (Page 780)
-2ab
46. a = 7, b = 8, c = 10
47. a = 4, b = 9, c = 6
48. a = 5, b = 8, c = 10
49. a = 16, b = 4, c = 13
50. a = 3, b = 10, c = 9
51. a = 5, b = 7, c = 11
Lesson 8-6 The Law of Sines
477
Geometry Software Lab
EXTEND
8-6
The Ambiguous Case
of the Law of Sines
In Lesson 8-6, you learned that you could solve a triangle using the Law of
Sines if you know the measures of two angles and any side of the triangle
(AAS or ASA). You can also solve a triangle by the Law of Sines if you know
the measures of two sides and an angle opposite one of the sides (SSA). When
you use SSA to solve a triangle, and the given angle is acute, sometimes it is
possible to find two different triangles. You can use The Geometer’s Sketchpad
to explore this case, called the ambiguous case, of the Law of Sines.
ACTIVITY
. Construct a circle
and AC
Step 1 Construct AB
whose center is B so that it intersects AC
at two points. Then, construct any
−−
radius BD.
−− −−
Step 2 Find the measures of BD, AB, and ∠A.
Step 3 Use the rotate tool to move D so that it
lies on one of the intersection points of
. In ABD, find the
circle B and AC
−−−
measures of ∠ABD, ∠BDA, and AD.
Step 4 Using the rotate tool, move D to the
other intersection point of circle B
.
and AC
Step 5 Note the measures of ∠ABD,
−−−
∠BDA, and AD in ABD.
Ambiguous Case
BD = 3.50 cm
m∠ABD = 97.44˚
AB = 5.79 cm
m∠ADB = 53.48˚
m∠BAC = 29.26˚
AD = 7.15 cm
Ambiguous Case
BD = 3.50 cm
m∠ABD = 24.81˚ AD = 3.00 cm
AB = 5.79 cm
m∠ADB = 125.82˚
m∠BAC = 29.26˚
B
B
A
C
D
C
A
D
ANALYZE THE RESULTS
1. Which measures are the same in both triangles?
−−
−−
2. Repeat the activity using different measures for ∠A, BD, and AB. How do the
results compare to the earlier results?
3. Compare your results with those of your classmates. How do the results compare?
4. What would have to be true about circle B in order for there to be one unique
solution? Test your conjecture by repeating the activity.
−− −−
5. Is it possible, given the measures of BD, AB, and ∠A, to have no solution?
Test your conjecture and explain.
478 Chapter 8 Right Triangles and Trigonometry
8-7
Main Ideas
• Use the Law of
Cosines to solve
triangles.
• Solve problems by
using the Law of
Cosines.
The Law of Cosines
German architect Ludwig Mies van der
Rohe entered the design at the right in the
Friedrichstrasses Skyscraper Competition
in Berlin in 1921. The skyscraper was to be
built on a triangular plot of land. In order to
maximize space, the design called for three
towers in a triangular shape. However, the
skyscraper was never built.
New Vocabulary
Law of Cosines
The Law of Cosines Suppose you know the lengths of the sides of the
triangular building and want to solve the triangle. The Law of Cosines
allows us to solve a triangle when the Law of Sines cannot be used.
8.9
Law of Cosines
Let ABC be any triangle with a, b,
and c representing the measures of
sides opposite angles A, B, and C,
respectively. Then the following
equations are true.
Side and Angle
a2 = b2 + c2 - 2bc cos A
Note that the letter of
the side length on the
left-hand side of each
equation corresponds
to the angle measure
used with the cosine.
b2
=
a2
+
c2
C
b
A
a
B
c
- 2ac cos B
c2 = a2 + b2 - 2ab cos C
The Law of Cosines can be used to find missing measures in a triangle if
you know the measures of two sides and their included angle.
EXAMPLE
A
Two Sides and the Included Angle
Find a if c = 8, b = 10, and m∠A = 60.
Use the Law of Cosines since the measures of
two sides and the included are known.
a2
=
b2
+
c2
- 2bc cos A
C
a
8
B
Law of Cosines
a2 = 102 + 82 - 2(10)(8) cos 60°
b = 10, c = 8, and m∠A = 60
a2
Simplify.
= 164 - 160 cos 60°
60˚
10
a = √
164 - 160 cos 60°
Take the square root of each side.
a ≈ 9.2
Use a calculator.
1. In DEF, e = 19, f = 28, and m∠D = 49. Find d.
Extra Examples at geometryonline.com
Digital Image ©The Museum of Modern Art/Licensed by SCALA/Art Resource, NY
Lesson 8-7 The Law of Cosines
479
You can also use the Law of Cosines to find the measures of angles of a
triangle when you know the measures of the three sides.
EXAMPLE
Three Sides
Q
23
18
Find m∠R.
R
r2 = q2 + s2 - 2qs cos R
Law of Cosines
232 = 372 + 182 - 2(37)(18) cos R
r = 23, q = 37, s = 18
529 = 1693 - 1332 cos R
Simplify.
-1164 = -1332 cos R
Subtract 1693 from each side.
-1164
_
= cos R
Divide each side by -1332.
-1332
( 1332 )
S
37
1164
R = cos-1 _
Solve for R.
R ≈ 29.1°
Use a calculator.
2. In TVW, v = 18, t = 24, and w = 30. Find m∠W.
Use the Law of Cosines to Solve Problems Most problems can be solved
using more than one method. Choosing the most efficient way to solve a
problem is sometimes not obvious.
When solving right triangles, you can use sine, cosine, or tangent ratios.
When solving other triangles, you can use the Law of Sines or the Law of
Cosines. You must decide how to solve each problem depending on the
given information.
Solving a Triangle
To solve
Right triangle
Any triangle
EXAMPLE
Given
Begin by using
two legs
tangent
leg and hypotenuse
sine or cosine
angle and hypotenuse
sine or cosine
angle and a leg
sine, cosine, or tangent
two angles and any side
Law of Sines
two sides and the angle opposite
one of them
Law of Sines
two sides and the included angle
Law of Cosines
three sides
Law of Cosines
Select a Strategy
Solve KLM. Round angle measures to the nearest
degree and side measures to the nearest tenth.
We do not know whether KLM is a right triangle,
so we must use the Law of Cosines or the Law of
K
Sines. We know the measures of two sides and the
included angle. This is SAS, so use the Law of Cosines.
480 Chapter 8 Right Triangles and Trigonometry
L
k
14
51˚
18
M
Law of Cosines
If you use the Law
of Cosines to find
another measure,
your answer may
differ slightly from
one found using the
Law of Sines. This is
due to rounding.
k2 = 2 + m2 - 2m cos K
Law of Cosines
k2
+
- 2(18)(14) cos 51°
k = √
182 + 142 - 2(18)(14) cos 51°
= 18, m = 14, and m∠K = 51
k ≈ 14.2
Use a calculator.
=
182
142
Take the square root of each side.
Next, we can find m∠L or m∠M. If we decide to find m∠L, we can use
either the Law of Sines or the Law of Cosines to find this value. In this
case, we will use the Law of Sines.
sin L
sin K
_
=_
Law of Sines
k
sin L
sin 51°
_
_
≈
18
14.2
= 18, k ≈ 14.2, and m∠K = 51
14.2 sin L ≈ 18 sin 51°
Cross products
18 sin 51°
sin L ≈ _
14.2
18 sin 51°
L ≈ sin-1 _
14.2
(
Divide each side by 14.2.
)
L ≈ 80°
Take the inverse sine of each side.
Use a calculator.
Use the Angle Sum Theorem to find m∠M.
m∠K + m∠L + m∠M = 180
51 + 80 + m∠M ≈ 180
m∠M ≈ 49
Angle Sum Theorem
m∠K = 51 and m∠L ≈ 80
Subtract 131 from each side.
Therefore, k ≈ 14.2, m∠K ≈ 80, and m∠M ≈ 49.
3. Solve XYZ for x = 10, y = 11, and z = 12.
Personal Tutor at ca.geometryonline.com
Use the Law of Cosines
REAL ESTATE Ms. Jenkins is buying some property
that is shaped like quadrilateral ABCD. Find the
perimeter of the property.
C
200 ft
B
Use the Pythagorean Theorem to find BD in ABD.
(AB)2 + (AD)2 = (BD)2
1802
+
2402
=
(BD)2
90,000 = (BD)2
300 = BD
60˚
Pythagorean Theorem
180 ft
AB = 180, AD = 240
Simplify.
A
Take the square root of each side.
D
240 ft
Next, use the Law of Cosines to find CD in BCD.
(CD)2 = (BC)2 + (BD)2 - 2(BC)(BD) cos ∠CBD
Law of Cosines
(CD)2 = 2002 + 3002 - 2(200)(300) cos 60°
BC = 200, BD = 300, m∠CBD = 60
(CD)2 = 130,000 - 120,000 cos 60°
Simplify.
CD = √
130,000 - 120,000 cos 60°
Take the square root of each side.
CD ≈ 264.6
Use a calculator.
The perimeter is 180 + 200 + 264.6 + 240 or about 884.6 feet.
Lesson 8-7 The Law of Cosines
481
K
4. ARCHITECTURE An architect is designing a
playground in the shape of a quadrilateral.
Find the perimeter of the playground to the
nearest tenth.
15 m
13
32˚
J
Example 1
(p. 479)
(p. 480)
2. b = 107, c = 94, m∠D = 105
In RST, given the lengths of the sides, find the measure of the stated
angle to the nearest degree.
3. r = 33, s = 65, t = 56; m∠S
Example 3
(p. 480)
4. r = 2.2, s = 1.3, t = 1.6; m∠R
Solve each triangle using the given information. Round angle measures
to the nearest degree and side measures to the nearest tenth.
5. XYZ: x = 5, y = 10, z = 13
Example 4
(p. 481)
HOMEWORK
HELP
For
See
Exercises Examples
8–11
1
12–15
2
16–22,
3
25–32
23, 24
4
M
18 m
In BCD, given the following measures, find the measure of the
missing side.
2 , d = 5, m∠B = 45
1. c = √
Example 2
L
6. JKL: j = 20, = 24, m∠K = 47
7. BASKETBALL Josh and Brian are playing
basketball. Josh passes the ball to Brian,
who takes a shot. Josh is 12 feet from the
hoop and 10 feet from Brian. The angle
formed by the hoop, Josh, and Brian is 34°.
Find the distance Brian is from the hoop.
*OSH
"RIAN
In TUV, given the following measures, find the measure of the
missing side.
8. t = 9.1, v = 8.3, m∠U = 32
10. u = 11, v = 17, m∠T = 105
9. t = 11, u = 17, m∠V = 78
11. v = 11, u = 17, m∠T = 59
In EFG, given the lengths of the sides, find the measure of the
stated angle to the nearest degree.
12. e = 9.1, f = 8.3, g = 16.7; m∠F
14. e = 325, f = 198, g = 208; m∠F
13. e = 14, f = 19, g = 32; m∠E
15. e = 21.9, f = 18.9, g = 10; m∠G
Solve each triangle using the given information. Round angle measures
to the nearest degree and side measures to the nearest tenth.
16.
17.
G
8
40˚
F
18.
Q
C
10
11
g
H
482 Chapter 8 Right Triangles and Trigonometry
11
M
B
38˚
P
10
18
p
15
D
Solve each triangle using the given information. Round angle measures
to the nearest degree and side measures to the nearest tenth.
19. ABC: m∠A = 42, m∠C = 77, c = 6
20. ABC: a = 10.3, b = 9.5, m∠C = 37
21. ABC: a = 15, b = 19, c = 28
22. ABC: m∠A = 53, m∠C = 28, c = 14.9
23. KITES Beth is building a kite like the one at the
−−
−−
right. If AB is 5 feet long, BC is 8 feet long, and
−−
2
BD is 7_
feet long, find the measures of the angle
3
between the short sides and the angle between
the long sides to the nearest degree.
A
B
D
C
Real-World Link
The Swissôtel in Chicago,
Illinois, is built in the
shape of a triangular
prism. The lengths of the
sides of the triangle are
180 feet, 186 feet, and
174 feet.
Source: Swissôtel
24. BUILDINGS Refer to the information at the left.
Find the measures of the angles of the triangular
building to the nearest tenth.
Solve each LMN described below. Round measures to the nearest tenth.
25. m = 44, = 54, m∠L = 23
26. m∠M = 46, m∠L = 55, n = 16
27. m = 256, = 423, n = 288
28. m∠M = 55, = 6.3, n = 6.7
29. m∠M = 27, = 5, n = 10
30. n = 17, m = 20, = 14
31. = 14, m = 15, n = 16
32. m∠L = 51, = 40, n = 35
33. In quadrilateral ABCD,
34. In quadrilateral PQRS, PQ = 721,
AC = 188, BD = 214, m∠BPC = 70,
QR = 547, RS = 593, PS = 756, and
−−
and P is the midpoint of AC and
m∠P = 58. Find QS, m∠PQS, and
−−
BD. Find the perimeter of ABCD.
m∠R.
A
Q
P
B
P
D
C
R
S
35. SOCCER Carlos and Adam are playing soccer.
Carlos is standing 40 feet from one post of
the goal and 50 feet from the other post.
Adam is standing 30 feet from one post of
the goal and 22 feet from the other post. If
the goal is 24 feet wide, which player has a
greater angle to make a shot on goal?
Carlos
40 ft
50 ft
24 ft
30 ft
22 ft
Adam
36. Each side of regular hexagon ABCDEF is
18 feet long. What is the length of the
−−
diagonal BD? Explain your reasoning.
"
#
!
$
&
%
Lesson 8-7 The Law of Cosines
Pierre Burnaugh/PhotoEdit
483
EXTRA
PRACTICE
See pages 816, 835.
Self-Check Quiz at
geometryonline.com
37. PROOF Justify each statement for the derivation
of the Law of Cosines.
−−−
Given: AD is an altitude of ABC.
Prove: c2 = a2 + b2 - 2ab cos C
"
!
C
H
$
X
AX
#
A
Proof:
Statement
Reasons
a. c2 = (a - x)2 + h2
a.
?
b. c2 = a2 - 2ax + x2 + h2
b.
?
c. x2 + h2 = b2
c.
?
d. c2 = a2 - 2ax + b2
d.
?
x
e. cos C = _
e.
?
f. b cos C = x
f.
?
g. c2 = a2 - 2a(b cos C) + b2
g.
?
h. c2 = a2 + b2 - 2ab cos C
h.
?
b
H.O.T. Problems
B
38. OPEN ENDED Draw and label one acute and one obtuse triangle,
illustrating when you can use the Law of Cosines to find the
missing measures.
39. REASONING Find a counterexample for the following statement.
The Law of Cosines can be used to find the length of a missing side in any triangle.
40. CHALLENGE Graph A(-6, -8), B(10, -4), C(6, 8),
and D(5, 11) on the coordinate plane. Find the
measure of interior angle ABC and the measure
of exterior angle DCA.
14
12
10
8
6
4
2
⫺8⫺6⫺4⫺2 O
⫺4
⫺6
⫺8
A (⫺6, ⫺8)⫺10
y
D (5, 11)
C (6, 8)
x
2 4 6 8 10 12
B (10, ⫺4)
41. Which One Doesn’t Belong? Analyze the four terms and determine which
does not belong with the others.
42.
Pythagorean triple
Pythagorean Theorem
Law of Cosines
cosine
Writing in Math Refer to the information about the Friedrichstrasses
Skyscraper Competition on page 479. Describe how triangles were used
in van der Rohe’s design. Explain why the Law of Cosines could not be
used to solve the triangle.
484 Chapter 8 Right Triangles and Trigonometry
43. In the figure below, cos B = 0.8.
"
45. REVIEW The scatter plot shows the
responses of swim coaches to a survey
about the hours of swim team practice
and the number of team wins.
!
4EAM7INS
#
−−
What is the length of AB?
A 12.8
B 16.8
0RACTICEH
C 20.0
D 28.8
44. REVIEW Which of the following
shows 2x2 - 24xy - 72y2 factored
completely?
F (2x - 18y)(x + 4y)
G 2(x - 6y)(x + 6y)
H (2x - 8y)(x - 9)
J 2(x - 6y)(x + 18y)
Which statement best describes
the relationship between the two
quantities?
A As the number of practice hours
increases, the number of team wins
increases.
B As the number of practice hours
increases, the number of team wins
decreases.
C As the number of practice hours
increases, the number of team wins
at first decreases, then increases.
D There is no relationship between the
number of practice hours and the
number of team wins.
Find each measure using the given measures from XYZ. Round angle
measure to the nearest degree and side measure to the nearest tenth. (Lesson 8-6)
46. If y = 4.7, m∠X = 22, and m∠Y = 49, find x.
47. If y = 10, x = 14, and m∠X = 50, find m∠Y.
48. SURVEYING A surveyor is 100 meters from a building and finds that the angle
of elevation to the top of the building is 23°. If the surveyor’s eye level is
1.55 meters above the ground, find the height of the building. (Lesson 8-5)
−− −−
A
For Exercises 49–51, determine whether AB CD. (Lesson 7-4)
49. AC = 8.4, BD = 6.3, DE = 4.5, and CE = 6
50. AC = 7, BD = 10.5, BE = 22.5, and AE = 15
51. AB = 8, AE = 9, CD = 4, and CE = 4
C
B
D
E
COORDINATE GEOMETRY The vertices of XYZ are X(8, 0), Y(-4, 8), and Z(0, 12). Find
the coordinates of the points of concurrency of XYZ to the nearest tenth. (Lesson 5-1)
52. orthocenter
53. centroid
54. circumcenter
Lesson 8-7 The Law of Cosines
485
CH
APTER
8
Study Guide
and Review
Download Vocabulary
Review from geometryonline.com
3IGHT5RIANGLES
Key Vocabulary
Be sure the following
Key Concepts are noted
in your Foldable.
angle of depression (p. 465)
angle of elevation (p. 464)
cosine (p. 456)
geometric mean (p. 432)
Pythagorean triple (p. 443)
Key Concepts
Geometric Mean
sine (p. 456)
solving a triangle (p. 472)
tangent (p. 456)
trigonometric ratio (p. 456)
trigonometry (p. 456)
(Lesson 8-1)
• For two positive numbers a and b, the geometric
mean is the positive number x where the
proportion a : x = x : b is true. This proportion
a
x
can be written using fractions as _x = _ or with
b
.
cross products as x2 = ab or x = √ab
Pythagorean Theorem
(Lesson 8-2)
• In a right triangle, the sum of the squares of the
measures of the legs equals the square of the
hypotenuse.
Special Right Triangles
(Lesson 8-3)
• The measures of the sides of a 45°–45°–90°
triangle are x, x, and x √2.
• The measures of the sides of a 30°–60°–90°
triangle are x, x √3, and 2x.
Trigonometry
(Lesson 8-4)
• Trigonometric Ratios:
opposite leg
hypotenuse
sin A = _
Vocabulary Check
State whether each sentence is true or false.
If false, replace the underlined word or
number to make a true sentence.
1. To solve a triangle means to find the
measures of all its sides and angles.
2. The Law of Sines can be applied if you
know the measures of two sides and an
angle opposite one of these sides of the
triangle.
3. In any triangle, the sum of the squares of
the measures of the legs equals the square
of the measure of the hypotenuse.
4. An angle of depression is the angle
between the line of sight and the
horizontal when an observer looks
upward.
adjacent leg
cos A = _
hypotenuse
opposite leg
adjacent leg
tan A = _
Laws of Sines and Cosines
(Lessons 8-6 and 8-7)
Let ABC be any triangle with a, b, and c
representing the measures of the sides opposite the
angles with measures A, B, and C, respectively.
5. The geometric mean between two
numbers is the positive square root of
their product.
6. A 30°-60°-90° triangle is isosceles.
sin A _
sin C
sin B
• Law of Sines: _
=_
a =
c
7. Looking at a city while flying in a plane is
an example that uses an angle of
elevation.
• Law of Cosines: a 2 = b 2 + c 2 - 2bc cos A
b 2 = a 2 + c 2 - 2ac cos B
c 2 = a 2 + b 2 - 2ab cos C
8. The numbers 3, 4, and 5 form a
Pythagorean identity.
b
486 Chapter 8 Right Triangles and Trigonometry
Vocabulary Review at geometryonline.com
Lesson-by-Lesson Review
8-1
Geometric Mean
(pp. 432–438)
Example 1 Find the geometric mean
between 10 and 30.
Find the geometric mean between each
pair of numbers.
10. 4 and 81
9. 4 and 16
11. 20 and 35
13. In PQR, PS = 8, and QS = 14.
Find RS.
Q
P
Cross products
Simplify.
Example 2 Find NG in TGR.
R
The Pythagorean Theorem and Its Converse
The measure of the altitude
is the geometric mean
between the measures
of the two hypotenuse
segments.
GN
TN
_
=_
RN
GN
GN
2
_
_
=
GN
4
8 = (GN)2
√
8 or 2 √
2 = GN
T
2
N
4
G
R
Definition of geometric mean
TN = 2, RN = 4
Cross products
Take the square root of
each side.
(pp. 440–446)
Example 3 Use JKL to find a.
Find x.
15.
15
x2 = 300
30
x = √
300 or 10 √
3
14. INDIRECT MEASUREMENT To estimate
the height of the Space Needle in
Seattle, Washington, James held a book
up to his eyes so that the top and
bottom of the building were in line
with the bottom edge and binding of
the cover. If James’ eye level is 6 feet
from the ground and he is standing
60 feet from the tower, how tall is the
tower?
8-2
Definition of geometric mean
x
12. 18 and 44
S
10
x
_
=_
16.
x
13
17
J
5
17
x
13
a
20
17. FARMING A farmer wishes to create a
maze in his corn field. He cuts a path
625 feet across the diagonal of the
rectangular field. Did the farmer create
two right triangles? Explain.
K
a 2 + (LK) 2 = (JL) 2
2
2
a + 8 = 13
2
2
a + 64 = 169
a 2 = 105
FT
8
L
Pythagorean Theorem
LK = 8 and JL = 13
Simplify.
Subtract 64 from each side.
a = √
105
Take the square root of
each side.
a ≈ 10.2
Use a calculator.
FT
Chapter 8 Study Guide and Review
487
CH
A PT ER
8
8-3
Study Guide and Review
Special Right Triangles
Find x and y.
18.
y
45˚
(pp. 448–454)
Example 4 Find x.
−−
The shorter leg, XZ,
of XYZ is half
the measure of the
−−
hypotenuse XY.
30˚
19.
x
9
x
y
60˚
6
x
30˚
z
b
Y
Example 5 Find x.
a
The hypotenuse of a
45°-45°-90° triangle is
√
2 times the length
of a leg.
Q
4
P
x √2 = 4
45˚
x
x
R
4
x=_
√
2
4
x=_
√
2
Trigonometry
26
x
22. ORIGAMI To create a bird, Michelle first
folded a square piece of origami paper
along one of the diagonals. If the
diagonal measured 8 centimeters, find
the length of one side of the square.
8-4
60˚
1
Therefore, XZ = _
(26) Z
2
or 13. The longer leg is √3 times the
measure of the shorter leg.
So, x = 13 √3.
For Exercises 20 and 21, use the figure.
20. If y = 18, find z
y
60˚
and a.
21. If x = 14, find a,
z, b, and y.
X
•
√
2
_
or 2 √
2
√
2
(pp. 456–462)
Use FGH to find sin F, cos F, tan F,
sin G, cos G, and tan G. Express each
ratio as a fraction and as a decimal to
the nearest hundredth.
G
23. f = 9, g = 12, h = 15
h
24. f = 7, g = 24, h = 25
25. f = 9, g = 40, h = 41
Example 6 Find sin A, cos A, and tan A.
Express as a fraction and as a decimal.
B
13
A
f
12
5
C
opposite leg
adjacent leg
cos A = _
hypotenuse
hypotenuse
BC
AC
=_
=_
AB
AB
5
12
or about 0.38
=_
or
=_
13
13
sin A = _
F
g
H
26. SPACE FLIGHT A space shuttle is
directed towards the Moon but drifts
0.8° from its calculated path. If the
distance from Earth to the Moon is
240,000 miles, how far has the space
shuttle drifted from its path when it
reaches the Moon?
488 Chapter 8 Right Triangles and Trigonometry
about 0.92
opposite leg
tan A = _
adjacent leg
BC
=_
AC
_
= 5 or about 0.42
12
Mixed Problem Solving
For mixed problem-solving practice,
see page 835.
8-5
Angles of Elevation and Depression
(pp. 464–470)
Determine the angles of elevation or
depression in each situation.
27. Upon takeoff, an airplane must clear a
60-foot pole at the end of a runway 500
yards long.
28. An escalator descends 100 feet for each
horizontal distance of 240 feet.
29. A hot-air balloon ascends 50 feet for
every 1000 feet traveled horizontally.
30. EAGLES An eagle, 1350 feet in the air,
notices a rabbit on the ground. If the
horizontal distance between the eagle
and the rabbit is 700 feet, at what angle
of depression must the eagle swoop
down to catch the rabbit and fly in a
straight path?
8-6
The Law of Sines
Example 7 The ramp of a loading dock
measures 12 feet and has a height of 3
feet. What is the angle of elevation?
Make a drawing.
B
12
3
x˚
A
C
Let x represent m∠BAC.
opposite leg
hypotenuse
BC
sin x° = _
sin x = _
AB
3
sin x° = _
12
BC = 3 and AB = 12
( 12 )
3
x = sin⁻¹ _
Find the inverse.
x ≈ 14.5
Use a calculator.
The angle of elevation for the ramp is
about 14.5°.
(pp. 471–477)
Find each measure using the given
measures of FGH. Round angle
measures to the nearest degree and
side measures to the nearest tenth.
Example 8 Find x if y = 15. Round to the
nearest tenth.
Y
32. Find m∠H if h = 10.5, g = 13, and
m∠G = 65.
33. GARDENING Elena is planning a
triangular garden. She wants to build a
fence around the garden to keep out
the deer. The length of one side of the
garden is 26 feet. If the angles at the
end of this side are 78° and 44°, find the
length of fence needed to enclose the
garden.
61˚
z
31. Find f if g = 16, m∠G = 48, and
m∠F = 82.
X
32˚
y
x
Z
To find x and z, use proportions involving
sin Y and y.
sin Y
sin X
_
=_
y
x
sin 32°
sin 61°
_
_
= x
15
x sin 61° = 15 sin 32°
Law of Sines
Substitute.
Cross Products
15 sin 32°
x=_
Divide.
x ≈ 9.1
Use a calculator.
sin 61°
Chapter 8 Study Guide and Review
489
CH
A PT ER
8
8-7
Study Guide and Review
The Law of Cosines
(pp. 479–485)
In XYZ, given the following measures,
find the measures of the missing side.
Example 9 Find a.
B
34. x = 7.6, y = 5.4, m∠Z = 51
19
35. x = 21, m∠Y = 73, z = 16
54˚
a
Solve each triangle using the given
information. Round angle measures to
the nearest degree and side measures to
the nearest tenth.
a 2 = b 2 + c 2 - 2bc cos A Law of Cosines
36. c = 18, b = 13, m∠A = 64
a 2 = 23 2 + 19 2 - 2(23)(19) cos 54°
A
23
C
37. b = 5.2, m∠C = 53, c = 6.7
38. ART Adelina is creating a piece of art
that is in the shape of a parallelogram.
Its dimensions are 35 inches by 28
inches and one angle is 80°. Find the
lengths of both diagonals.
490 Chapter 8 Right Triangles and Trigonometry
a 2 = 890 - 874 cos 54°
b = 23,
c = 19, and
m∠A = 54
Simplify.
a = √
890 - 874 cos 54° Take the square root
of each side.
a ≈ 19.4
Use a calculator.
CH
A PT ER
8
Practice Test
Find the geometric mean between each pair
of numbers.
1. 7 and 63
2. 6 and 24
3. 10 and 50
Find the missing measures.
4.
6
x
5.
7
x
21. CIVIL ENGINEERING A section of freeway
has a steady incline of 10°. If the horizontal
distance from the beginning of the incline
to the end is 5 miles, how high does the
incline reach?
13
5
6.
7.
9
x
9
12
y
9.
y
x
60˚
8
45˚
16
y˚
12
x
x
17
C _
12
B _
12
D _
13
B
21
A
16
15
C
24. Solve DEF.
Solve each triangle. Round each angle
measure to the nearest degree and each side
measure to the nearest tenth.
17. a = 15, b = 17, m∠C = 45
18. a = 12.2, b = 10.9, m∠B = 48
19. a = 19, b = 23.2, c = 21
Chapter Test at geometryonline.com
12
5
23. COMMUNICATIONS To secure a 500-foot
radio tower against high winds, guy wires
are attached to the tower 5 feet from the top.
The wires form a 15° angle with the tower.
Find the distance from the centerline of the
tower to the anchor point of the wires.
Find each measure using the given measures
from FGH. Round to the nearest tenth.
13. Find g if m∠F = 59, f = 13, and m∠G = 71.
14. Find m∠H if m∠F = 52, f = 10, and
h = 12.5.
15. Find f if g = 15, h = 13, and m∠F = 48.
16. Find h if f = 13.7, g = 16.8, and m∠H = 71.
£Î
5
A _
12
8
Use the figure to find each
trigonometric ratio. Express
answers as a fraction.
10. cos B
11. tan A
12. sin A
22. MULTIPLE CHOICE Find tan X.
19
x
8.
20. TRAVEL From an airplane, Janara looked
down to see a city. If she looked down at an
angle of 9° and the airplane was half a mile
above the ground, what was the horizontal
distance to the city?
D
12
E
82˚
8
F
25. MULTIPLE CHOICE The top of the Boone
Island Lighthouse in Boone Island, Maine,
is 137 feet above sea level. The angle of
depression from the light on the top of the
tower to a passing ferry is 37°. How many
feet from the foot of the lighthouse is the
ferry?
F 181.8 ft
H 109.4 ft
G 171.5 ft
J 103.2 ft
Chapter 8 Practice Test
491
CH
A PT ER
8
Standardized Test Practice
Cumulative, Chapters 1–8
Read each question. Then fill in the correct
answer on the answer document provided
by your teacher or on a sheet of paper.
3. A detour has been set up on the interstate
due to a gas leak. The diagram below shows
the detour route. How many extra miles will
drivers have to travel due to the detour?
1. A diagram from a proof of the Pythagorean
Theorem is pictured below. Which statement
would be used in the proof of the
Pythagorean Theorem?
a
St
at
e
18 miles
Ro
gh
wa
y
ut
e
Hi
24 miles
b
te
Intersta
A
B
C
D
c
A The area of the larger square equals
(a + b) 2.
B The area of the inner square is equal to
half of the area of the larger square.
C The area of the larger square is equal to
the sum of the areas of the smaller square
and the four congruent triangles.
12 miles
30 miles
42 miles
80 miles
and 8.
4. A right triangle has legs of length √39
What is the length of the hypotenuse?
F 10
H 11
G √103
J 4 √26
D The four right triangles are similar.
2. In the figure below, if tan x = __43 , what are
cos x and sin x?
Question 4 If a standardized test question involves
trigonometric ratios, draw a diagram that represents the
problem. Use a calculator (if allowed) or the table of
trigonometric values provided to help you find the answer.
−− −−
5. Given: BD AE
4
#
x
3
F
G
H
J
3
4
cos x = __
, sin x = __
5
4
3
5
, sin x = __
cos x = __
4
4
3
4
, sin x = __
cos x = __
5
5
3
5
, sin x = __
cos x = __
5
4
492 Chapter 8 Right Triangles and Trigonometry
"
!
$
%
What theorem or postulate can be used to
prove ACE ∼ BCD?
A SSS
B SAS
C ASA
D AA
Standardized Test Practice at geometryonline.com
Preparing for
Standardized Tests
For test-taking strategies and more practice,
see pages 841–856.
−−
6. In ABC, D is the midpoint of AB, and E is
10. Rhombus ABCD is shown.
−−
the midpoint of AC.
Which pair of triangles
can be aestablished to be
congruent to prove
−−
−−
that AC bisects BD?
A
2
1
E
4
D
"
!
#
A ABD and CBD
C
B ACD and ACB
3
C AEB and BEC
B
G ABC ∼ ADE
11. What is the shortest side of quadrilateral
DEFG?
AE
AD = ___
J ____
DB
$
D AEB and CED
Which of the following is not true?
−− −−
F ∠1 ∠4
H DE BC
EC
H
G
65˚
7. If the sum of the measures of the interior
angles of a polygon is 900, how many sides
does the polygon have?
A5
B 7
55˚
C 8
D 10
12. An extension ladder leans against the side of
a house while gutters are being cleaned. The
base of the ladder is 12 feet from the house, and
the top of the ladder rests 16 feet up the side of
the house.
a. Draw a figure representing this situation.
What is the length of the ladder?
b. For safety, a ladder should have a
climbing angle of no less than 75°. Is
the climbing angle of this ladder safe?
c. If not, what distance from the house
should the ladder be placed so that it
still rests 16 feet up the side of the house
at a 75° climbing angle and to what new
length will the ladder need to be adjusted?
40 m
Main Street East
9. ALGEBRA Find (x 2 + 2x - 24) ÷ (x - 4).
J x+6
−−−
H DG
−−
J DE
Record your answer on a sheet of paper.
Show your work.
46 m
G x+8
E
Pre-AP
Ma
in S
tre
et W
est
H x-6
F
55˚
85˚
D
triangular traffic median on Main Street to
provide more green space in the downtown
area. The planner builds a model so that
the section of the median facing Main Street
East measures 20 centimeters. What is the
perimeter, in centimeters, of the model of
the traffic median?
F x-8
35˚
−−
F GF
−−
G FE
8. GRIDDABLE A city planner designs a
23 m
%
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Chapter 8 Standardized Test Practice
493
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