Chapter 7: Right Triangles and Trigonometry

Chapter 7: Right Triangles and Trigonometry
• Lessons 7-1, 7-2, and 7-3 Solve problems using
the geometric mean, the Pythagorean Theorem,
and its converse.
• Lessons 7-4 and 7-5 Use trigonometric ratios to
solve right triangle problems.
• Lessons 7-6 and 7-7 Solve triangles using the
Law of Sines and the Law of Cosines.
Trigonometry is used to find the measures of
the sides and angles of triangles. These ratios
are frequently used in real-world applications
such as architecture, aviation, and surveying.
You will learn how surveyors use trigonometry in
Lesson 7-6.
340
Chapter 7
Right Triangles and Trigonometry
Bob Daemmrich/The Image Works
Key Vocabulary
•
•
•
•
•
geometric mean (p. 342)
Pythagorean triple (p. 352)
trigonometric ratio (p. 364)
Law of Sines (p. 377)
Law of Cosines (p. 385)
Prerequisite Skills To be successful in this chapter, you’ll need to master
these skills and be able to apply them in problem-solving situations. Review
these skills before beginning Chapter 7.
For Lesson 7-1
Proportions
Solve each proportion. Round to the nearest hundredth, if necessary. (For review, see Lesson 6-1.)
c
8
2. 3
12
1. 4
5
a
f
10
e
6
3. 20
3
5
4
6
1
4. 3
For Lesson 7-2
y
z
Pythagorean Theorem
Find the measure of the hypotenuse of each right triangle having legs with the given
measures. Round to the nearest hundredth, if necessary. (For review, see Lesson 1-3.)
5. 5 and 12
6. 6 and 8
7. 15 and 15
8. 14 and 27
For Lessons 7-3 and 7-4
Radical Expressions
Simplify each expression. (For review, see pages 744 and 745.)
9. 8
10. 102 52
7
12. 2
11. 392 362
For Lessons 7-5 through 7-7
Angle Sum Theorem
Find x. (For review, see Lesson 4-2.)
13.
14.
38˚
15.
(2x + 21)˚
40˚
44˚
x˚
x˚
155˚
B
x˚
A
C
Right Triangles and Trigonometry Make this Foldable to help you organize
your notes. Begin with seven sheets of grid paper.
Fold each sheet along the
diagonal from the corner of
one end to 2.5 inches away
from the corner of the other
end.
Staple
Staple the sheets in three
places.
2.5 in.
Stack
Stack the sheets, and fold
the rectangular part in half.
Label
Label each sheet with a
lesson number, and the
rectangular part with the
chapter title.
Ch. 7 Right Triangles
Fold
7-1
Reading and Writing As you read and study the chapter, write notes, define terms, and solve problems in
your Foldable.
Chapter 7 Right Triangles and Trigonometry 341
Geometric Mean
• Find the geometric mean between two numbers.
• Solve problems involving relationships between parts of a right triangle and the
altitude to its hypotenuse.
can the geometric mean be
used to view paintings?
Vocabulary
• geometric mean
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.
GEOMETRIC MEAN The geometric mean between two numbers is the
positive square root of their product.
Geometric Mean
Study Tip
Means and
Extremes
In the equation x2 ab,
the two x’s in x2 represent
the means, and a and b
represent the extremes of
the proportion.
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 be written using fractions as
x
a
or with cross products as x2 ab or x ab
.
x
b
Example 1 Geometric Mean
Find the geometric mean between each pair of numbers.
a. 4 and 9
Let x represent the geometric mean.
4
x
x
9
Definition of geometric mean
x2 36
Cross products
x 36
Take the positive square root of each side.
x6
Simplify.
b. 6 and 15
x
6
x
15
x2 90
Cross products
x 90
Take the positive square root of each side.
x 310
Simplify.
x 9.5
Use a calculator.
342 Chapter 7 Right Triangles and Trigonometry
Robert Brenner/PhotoEdit
Definition of geometric mean
ALTITUDE OF A TRIANGLE
Consider right triangle
XYZ 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.
Z
X
W
Y
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. Explore the relationships
among the three right triangles formed.
Z
Think and Discuss
1. Find the measures of X, XZY, Y, XWZ, XZW,
YWZ, and YZW.
2. What is the relationship between the measures of
X and YZW? What is the relationship between
the measures of Y and XZW ?
Y
W
X
3. Drag point Z to another position. Describe the
relationship between the measures of X and
YZW and between the measures of Y and XZW.
4. Make a conjecture about XYZ, XZW, and ZYW.
Study Tip
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.
The results of the Geometry Software Investigation suggest the following theorem.
Theorem 7.1
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.
Y
X
Z
W
Example: XYZ XWY YWZ
You will prove this theorem in Exercise 45.
By Theorem 7.1, since XWY YWZ, the corresponding sides are proportional.
XW
YW
YW
ZW
XW
Thus, . Notice that and Z
W
are segments of the hypotenuse of the
largest triangle.
Theorem 7.2
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 this theorem in Exercise 46.
www.geometryonline.com/extra_examples/sol
Lesson 7-1 Geometric Mean 343
Example 2 Altitude and Segments of the Hypotenuse
In PQR, RS 3 and QS 14. Find PS.
Let x PS.
RS
PS
PS
QS
x
3
14
x
Study Tip
Square Roots
Since these numbers
represent measures, you
can ignore the negative
square root value.
P
Q
S
R
RS 3, QS 14, and PS x
x2 42
Cross products
x 42
Take the positive square root of each side.
x 6.5
Use a calculator.
PS is about 6.5.
Ratios in right triangles can be used to solve problems.
Example 3 Altitude and Length of the Hypotenuse
ARCHITECTURE Mr. Martinez is designing a walkway that must pass over an
elevated train. To find the height of the elevated train, 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 distance from the street to the top of the train. Round
to the nearest tenth.
Z
X be the altitude
Draw a diagram. Let Y
drawn from the right angle of WYZ.
WX
YX
YX
ZX
5.5
8.75
8.75
ZX
WX 5.5 and YX 8.75
5.5ZX 76.5625
Y
Cross products
X
5.5 ft
ZX 13.9
Divide each side by 5.5.
8.75 ft
W
Mr. Martinez estimates that the elevated train
is 5.5 13.9 or about 19.4 feet high.
The altitude to the hypotenuse of a right triangle determines another relationship
between the segments.
Theorem 7.3
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 measures of the
hypotenuse and the segment of the hypotenuse
adjacent to that leg.
Y
X
XZ
XY
ZY
XZ
Example: and XY
XW
YZ
WZ
You will prove Theorem 7.3 in Exercise 47.
344
Chapter 7 Right Triangles and Trigonometry
W
Z
Example 4 Hypotenuse and Segment of Hypotenuse
Find x and y in PQR.
Q and R
P
Q
are legs of right triangle PQR.
Use Theorem 7.3 to write a proportion
for each leg and then solve.
Study Tip
Simplifying
Radicals
Remember that 12
4 3. Since 4 2, 12
23.
For more practice
simplifying radicals,
see pages 744 and 745.
Concept Check
PR
PQ
PQ
PS
y
6
y
2
y2 12
PS 2, PQ y, PR 6
Cross products
R
4
x
S
2
PR
RQ
RQ
SR
6
x
x
4
P
Q
y
RS 4, RQ x, PR 6
x2 24
Cross products
y 12
Take the square root.
x 24
Take the square root.
y 23
Simplify.
x 26
Simplify.
y 3.5
Use a calculator.
x 4.9
Use a calculator.
1. OPEN ENDED Find two numbers whose geometric mean is 12.
2. 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 7.3.
3. FIND THE ERROR RST is a right isosceles triangle. Holly and Ian are finding
the measure of altitude S
U
.
Ian
Holly
RU SU
= UT
SU
RS SU
= SU RT
x
9.9
= 14
x
S
7 x
= x 7
x 2 = 138.6
9.9
x
x2 = 49
x = 138.6
R
7
x= 7
U
7
T
x ≈ 11.8
Who is correct? Explain your reasoning.
Guided Practice
Find the geometric mean between each pair of numbers.
4. 9 and 4
5. 36 and 49
6. 6 and 8
7. 22 and 32
Find the measure of the altitude drawn to the hypotenuse.
8. A
9.
E
2
D
6
C
B
G
H
16
12
F
Lesson 7-1 Geometric Mean 345
Find x and y.
10.
x
B
Application
11.
C
8
D
C
y
y
B
A
3
2 √3
x
D
A
2
12. DANCES Khaliah 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, Khaliah held a book up to her eyes
so that the top and bottom of the wall were in
line with the top edge and binding of the cover.
If Khaliah’s eye level is 5 feet off the ground and
she is standing 12 feet from the wall, how high
is the wall?
Practice and Apply
For
Exercises
See
Examples
13–20
21–26
27–32
1
2
3, 4
Extra Practice
Find the geometric mean between each pair of numbers.
13. 5 and 6
14. 24 and 25
15. 45
and 80
83
63
18. and 3
17. and 1
5
5
16. 28
and 1372
22
52
19. and 5
6
13
5
20. and 6
7
Find the measure of the altitude drawn to the hypotenuse.
21. B
22. F
23. J
5
See page 766.
7
8 M
16
D
H
12
9
A
E
C
24.
25.
Q
L
G
X
26. Z
V 2W
13
10
P
21
S 7
U
R
T
28.
x
y
x
z
x
8
z
31.
x
y
z
10
4
15
y
5
y
30.
N
29.
6
z
3
2.5
Y
Find x, y, and z.
27.
8
K
12
32.
z
y
36
6x
12
x
z
8
y
x
346 Chapter 7 Right Triangles and Trigonometry
The geometric mean and one extreme are given. Find the other extreme.
33. 17
is the geometric mean between a and b. Find b if a 7.
34. 12 is the geometric mean between x and y. Find x if y 3.
Determine whether each statement is always, sometimes, or never true.
35. The geometric mean for consecutive positive integers is the average of the
two numbers.
36. The geometric mean for two perfect squares is a positive integer.
37. The geometric mean for two positive integers is another integer.
38. The measure of the altitude of a triangle is the geometric mean between the
measures of the segments of the side it intersects.
39. BIOLOGY The shape of the shell of a
chambered nautilus can be modeled by
a geometric mean. Consider the sequence
OC
of segments O
A
, O
B
, , O
D
, O
E
, O
F
,
OG
, O
H
, O
I, and O
J. 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.)
You can use geometric
mean and the Quadratic
Formula to discover the
golden mean. Visit
www.geometry
online.com/webquest
to continue work on
your WebQuest project.
G
C
J
O
B
F
H
D
A
E
I
40. RESEARCH Refer to the information at the left. Use the Internet or other
resource to write a brief description of the golden ratio.
41. CONSTRUCTION In the United States,
most building codes limit the steepness of
4
the slope of a roof to , as shown at the
3
right. A builder wants to put a support
brace from point C perpendicular to AP
.
Find the length of the brace.
P
5 yd
4 yd
A
SOCCER For Exercises 42 and 43,
refer to the graphic.
42. Find the geometric mean
between the number of
players from Indiana and
North Carolina.
43. Are there two schools whose
geometric mean is the same as
the geometric mean between
UCLA and Clemson? If so, which
schools?
3 yd
C
B
USA TODAY Snapshots®
Bruins bring skills to MLS
Universities producing the most players
in Major League Soccer this season:
15
UCLA
10
Indiana
9
Virginia
44. CRITICAL THINKING Find
the exact value of DE, given
AD 12 and BD 4.
C
North Carolina
7
Washington
7
Clemson
6
Source: MLS
A
D
www.geometryonline.com/self_check_quiz/sol
E
B
By Ellen J. Horrow and Adrienne Lewis, USA TODAY
Lesson 7-1 Geometric Mean 347
Kaz Chiba/PhotoDisc
Write the specified type proof for each theorem.
45. two-column proof
46. paragraph proof of
47. two-column proof
of Theorem 7.1
of Theorem 7.2
of Theorem 7.3
PROOF
48. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson.
How can the geometric mean be used to view paintings?
Include the following in your answer:
• an explanation of what happens when you are too far or too close to a
painting, and
• an explanation of how the curator of a museum would determine where to
place roping in front of paintings on display.
SOL/EOC Practice
Standardized
Test Practice
49. Find x and y.
A 4 and 6
C 3.6 and 6.4
B
D
8 cm
2.5 and 7.5
3 and 7
6 cm
y cm
10 cm
50. ALGEBRA Solve 5x2 405 1125.
A 15
B 12
C 43
D 4
x cm
Maintain Your Skills
Mixed Review
Find the first three iterations of each expression. (Lesson 6-6)
51. x 3, where x initially equals 12
52. 3x 2, where x initially equals 4
2
54. 2(x 3), where x initially equals 1
53. x 2, where x initially equals 3
55. 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 6-5)
Use the Exterior Angle Inequality Theorem to list all angles
that satisfy the stated condition. (Lesson 5-2)
56. all angles with a measure less than m8
57. all angles with a measure greater than m1
58. all angles with a measure less than m7
59. all angles with a measure greater than m6
7
8
3
1
2
5
6
Write an equation in slope-intercept form for the line that satisfies the given
conditions. (Lesson 3-4)
60. m 2, y-intercept 4
61. x-intercept is 2, y-intercept 8
62. passes through (2, 6) and (1, 0)
63. m 4, passes through (2, 3)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Use the Pythagorean Theorem to find the length of
the hypotenuse of each right triangle.
(To review using the Pythagorean Theorem, see Lesson 1-4.)
64.
65.
3 cm
5 ft
12 ft
66.
5 in.
4 cm
348 Chapter 7 Right Triangles and Trigonometry
3 in.
4
A Preview of Lesson 7-2
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.
Use square pieces of patty paper and algebra. Then you too can discover this
relationship among the measures of the sides of a right triangle.
Activity
Use paper folding to develop the Pythagorean Theorem.
a
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
Copy these
measures on the other
sides in the order shown
b ab
at the right. Fold the paper
to divide the square into
four sections. Label the
a a2
area of each section.
a
On another
sheet of patty paper,
mark the same lengths
a and b on the sides in
the different pattern
shown at the right.
b
Use your
straightedge and pencil
to connect the marks as
shown at the right. Let c
represent the length of
each hypotenuse.
b2
b
a
ab
b
a
b
1
2 ab
1
2 ab
c
c
c2
2
c2
c
c
Step 5 Label the area
of each section, which is
1
ab for each triangle and
b
c
c
c
for the square.
1
2 ab
c
1
2 ab
Step 6 Place the squares side by side and
color the corresponding regions that have the
1
1
same area. For example, ab ab ab.
2
a
2
1
2 ab
b
ab
1
2 ab
b2
b
c2
=
a
b
a
a2
ab
1
2 ab
1
2 ab
The parts that are not shaded tell us that
a2 b2 c2.
Model
1. Use a ruler to find actual measures for a, b, and c. Do these measures
measures confirm that a2 b2 c2?
2. Repeat the activity with different a and b values. What do you notice?
Analyze the model
3. Explain why the drawing at the right is an illustration of the
Pythagorean Theorem.
Investigating Slope-Intercept Form 349
Geometry Activity The Pythagorean Theorem 349
The Pythagorean Theorem
and Its Converse
Virginia SOL Standard G.7 The student will
solve practical problems involving right triangles by
using the Pythagorean Theorem …. Solutions will be
expressed in radical form or as decimal approximations.
• Use the Pythagorean Theorem.
• Use the converse of the Pythagorean Theorem.
are right triangles used to
build suspension bridges?
Vocabulary
• Pythagorean triple
The Talmadge Memorial Bridge over the
Savannah River 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.
Study Tip
Look Back
THE PYTHAGOREAN THEOREM In Lesson 1-3, you used the Pythagorean
To review finding the
hypotenuse of a right
triangle, see Lesson 1-3.
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.
Theorem 7.4
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.
Symbols:
a2
b2
B
c
A
a
C
b
c2
The geometric mean can be used to prove the Pythagorean Theorem.
Proof
Pythagorean Theorem
Given:
ABC with right angle at C
Prove:
a2 b2 c2
C
a
Proof:
Draw right triangle ABC so C is the right angle.
. Let AB c,
Then draw the altitude from C to AB
AC b, BC a, AD x, DB y, and CD h.
B
b
h
y D
Two geometric means now exist.
c
a
a
y
and
c
b
b
x
a2 cy
and
b2 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).
350 Chapter 7 Right Triangles and Trigonometry
x
c
A
Example 1 Find the Length of the Hypotenuse
LONGITUDE AND LATITUDE NASA Dryden
is located at about 117 degrees longitude and
34 degrees latitude. NASA Ames 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.
Maps
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
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
34 c
5.8 c
Pythagorean Theorem
a 5, b 3
Simplify.
Add.
Take the square root of each side.
Use a calculator.
The degree distance between NASA Dryden and NASA Ames is about 5.8 degrees.
Example 2 Find the Length of a Leg
Find x.
(XY)2 (YZ)2 (XZ)2
7 2 x2 142
49 x2 196
x2 147
x 147
x 73
x 12.1
X
Pythagorean Theorem
XY 7, XZ 14
Simplify.
Subtract 49 from each side.
14 in.
7 in.
Y
x in.
Z
Take the square root of each side.
Simplify.
Use a calculator.
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.
Theorem 7.5
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
b
a
C
You will prove this theorem in Exercise 38.
www.geometryonline.com/extra_examples/sol
Lesson 7-2 The Pythagorean Theorem and Its Converse 351
Example 3 Verify a Triangle is a Right Triangle
Study Tip
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.
COORDINATE GEOMETRY Verify that PQR
is a right triangle.
Use the Distance Formula to determine the lengths
of the sides.
(–3 3)2 (6 2
)2
PQ Q (–3, 6)
y
R (5, 5)
P (3, 2)
x1 3, y1 2, x2 –3, y2 6
2 (–6)
42
Subtract.
52
Simplify.
O
x
2 QR [5 (–3)]
(5 6
)2 x1 –3, y1 6, x2 5, y2 5
82 (–1)
2
Subtract.
65
Simplify.
PR (5 3
)2 (5
2)2
x1 = 3, y1 = 2, x2 = 5, y2 = 5
Subtract.
13
Simplify.
22
32
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
Converse of the Pythagorean Theorem
13
65
2
52 13 65
65 65
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.
A Pythagorean triple is three whole numbers that satisfy the equation a2 b2 c2,
where c is the greatest number. One common Pythagorean triple is 3-4-5, in which
the sides of a right triangle are in the ratio 3 : 4 : 5. If the measures of the sides of any
right triangle are whole numbers, the measures form a Pythagorean triple.
Example 4 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 15 and 8.
a2 b2
82 152
64 225
289
c2
Pythagorean Theorem
2
16
a 8, b 15, c 16
256 Simplify.
256 Add.
Since 289 256, segments with these measures cannot form a right triangle.
Therefore, they do not form a Pythagorean triple.
352
Chapter 7 Right Triangles and Trigonometry
b. 20, 48, and 52
a2 b2 c2
202 482 522
400 2304 2704
2704 2704
If you cannot quickly
identify the greatest
number, use a calculator
to find decimal values
for each number and
compare.
a 20, b 48, c 52
Simplify.
Add.
These segments form the sides of a right triangle since they satisfy the
Pythagorean Theorem. The measures are whole numbers and form a
Pythagorean triple.
Study Tip
Comparing
Numbers
Pythagorean Theorem
3 , 6 , and 3
c. 5
5
5
a2 b2 c2
Pythagorean Theorem
53 56 35
2
2
2
5
3
6
9
25
25
25
5
5
Simplify.
9
9
25
25
9
3
6
3
a , b , c Add.
9
Since , segments with these measures form a right triangle. However,
25
25
the three numbers are not whole numbers. Therefore, they do not form a
Pythagorean triple.
Concept Check
1. FIND THE ERROR Maria and Colin are determining whether 5-12-13 is a
Pythagorean triple.
Colin
Maria
a2 + b2 = c2
?
132 + 52 = 122
?
169 + 25 = 144
193 ≠ 144
no
a2 + b2 = c2
?
5 2 + 12 2 = 13 2
?
25 + 144 = 169
169 = 169
yes
Who is correct? Explain your reasoning.
2. Explain why a Pythagorean triple can represent the measures of the sides of a
right triangle.
3. OPEN ENDED Draw a pair of similar right triangles. List the corresponding
sides, the corresponding angles, and the scale factor. Are the measures of the
sides of each triangle a Pythagorean triple?
Guided Practice
Find x.
4.
6
5.
10
x
4
7
x
5
7
6.
x
37.5
20
Lesson 7-2 The Pythagorean Theorem and Its Converse 353
7. 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.
9. 40
, 20, 21
8. 15, 36, 39
Application
10. 44
, 8, 108
11. COMPUTERS Computer monitors are
usually measured along the diagonal of the
screen. A 19-inch monitor has a diagonal that
measures 19 inches. If the height of the screen
is 11.5 inches, how wide is the screen?
19 in.
11.5 in.
Practice and Apply
For
Exercises
See
Examples
14, 15
12, 13, 16, 17
18–21
22–29
1
2
3
4
Extra Practice
Find x.
12.
8
13.
8
14.
8
x
4
28
x
14
15.
20
16.
40
17.
33
See page 767.
x
x
x
25
25
15
32
x
COORDINATE GEOMETRY Determine whether QRS is a right triangle for the
given vertices. Explain.
18. Q(1, 0), R(1, 6), S(9, 0)
19. Q(3, 2), R(0, 6), S(6, 6)
20. Q(–4, 6), R(2, 11), S(4, –1)
21. 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.
22. 8, 15, 17
23. 7, 24, 25
24. 20, 21, 31
25. 37, 12, 34
1 1 74
26. , , 5 7
35
5
3 , 2 , 3
27. 2
3
36
3 4
28. , , 1
5 5
6 8 10
29. , , 7 7
7
For Exercises 30–35, use the table of Pythagorean triples.
a
b
c
30. Copy and complete the table.
5
12
13
31. A primitive Pythagorean triple is a Pythagorean triple
10
24
?
with no common factors except 1. Name any primitive
15
?
39
Pythagorean triples contained in the table.
?
48
52
32. Describe the pattern that relates these sets of
Pythagorean triples.
33. These Pythagorean triples are called a family. Why do you think this is?
34. Are the triangles described by a family of Pythagorean triples similar? Explain.
35. For each Pythagorean triple, find two triples in the same family.
a. 8, 15, 17
b. 9, 40, 41
c. 7, 24, 25
354 Chapter 7 Right Triangles and Trigonometry
Aaron Haupt
GEOGRAPHY For Exercises 36 and 37, use
the following information.
Denver is located at about 105 degrees
longitude and 40 degrees latitude. San
Francisco is located at about 122 degrees
longitude and 38 degrees latitude. Las Vegas
is located at about 115 degrees longitude and
36 degrees latitude. Using the lines of
longitude and latitude, find each degree
distance.
125°
120°
115°
110°
105°
100°
40°
San Francisco
Denver
Las Vegas
35°
30°
36. San Francisco to Denver
37. Las Vegas to Denver
38. PROOF
Write a paragraph proof of Theorem 7.5.
39. PROOF Use the Pythagorean Theorem and the
figure at the right to prove the Distance Formula.
y
A(x1, y1)
d
C (x1, y2)
B (x2, y2)
x
O
40. 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?
41. SAILING The mast of a sailboat is
supported by wires called shrouds.
What is the total length of wire
needed to form these shrouds?
16 ft
x
2 ft
12 ft
42. 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.
12 ft
3 ft
26 ft
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.
Online Research
For information about
a career in the military,
visit:
www.geometryonline.
com/careers
x
shrouds
9 ft
43. 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?
H
44. CRITICAL THINKING The figure at the right is a
rectangular prism with AB 8, BC 6, and BF 8,
and M is the midpoint of BD. Find BD and HM. How
are EM, FM, and GM related to HM?
F
D
A
www.geometryonline.com/self_check_quiz/sol
G
E
M
C
B
Lesson 7-2 The Pythagorean Theorem and Its Converse 355
Answer the question that was posed at the beginning of
the lesson.
How are right triangles used to build suspension bridges?
Include the following in your answer:
• the locations of the right triangles, and
• an explanation of which parts of the right triangle are formed by the cables.
45. WRITING IN MATH
SOL/EOC Practice
Standardized
Test Practice
Graphing
Calculator
46. In the figure, if AE 10, what is the value of h?
A 6
B 8
C 10
D 12
2
2
47. ALGEBRA If x 36 (9 x) , then find x.
A 6
B no solution
C 2.5
D 10
y
E (8, h)
A
O
C
x
PROGRAMMING For Exercises 48 and 49, use the following information.
The TI-83 Plus program uses a
procedure for finding Pythagorean triples
PROGRAM: PYTHTRIP
that was developed by Euclid around
:For (X, 2, 6)
:Disp B,A,C
320 B.C. Run the program to generate a
:For
(Y,
1,
5)
:Else
list of Pythagorean triples.
:If X > Y
:Disp A,B,C
48. List all the members of the 3-4-5
:Then
:End
family that are generated by the
2
2
:int (X Y + 0.5)➞A
:End
program.
:2XY➞B
:Pause
49. A geometry student made the
2
2
:int (X +Y +0.5)➞C
:Disp ” ”
conjecture that if three whole
:If
A
>
B
:End
numbers are a Pythagorean triple,
:Then
:End
then their product is divisible by 60.
Does this conjecture hold true for
:Stop
each triple that is produced by
the program?
Maintain Your Skills
Mixed Review
Find the geometric mean between each pair of numbers. (Lesson 7-1)
50. 3 and 12
51. 9 and 12
52. 11 and 7
53. 6 and 9
54. 2 and 7
55. 2 and 5
Find the value of each expression. Then use that value as the next x in the
expression. Repeat the process and describe your observations. (Lesson 6-6)
56. 2x
, where x initially equals 5
57. 3x, where x initially equals 1
58. x , where x initially equals 4
1
59. , where x initially equals 4
1
2
x
60. Determine whether the sides of a triangle could have the lengths 12, 13, and 25.
Explain. (Lesson 5-4)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Simplify each expression by rationalizing the denominator.
(To review simplifying radical expressions, see pages 744 and 745.)
61. 7
3
12
66. 3
356 Chapter 7 Right Triangles and Trigonometry
18
62. 2
26
67. 3
14
63. 2
311
64. 3
24
65. 2
15
68. 3
69. 2
8
25
70. 10
Special Right Triangles
Virginia SOL Standard G.7 The student will solve practical problems involving right triangles by using …
properties of special right triangles, …. Solutions will be expressed in radical form or as decimal approximations.
• Use properties of 45°-45°-90° triangles.
• Use properties of 30°-60°-90° triangles.
is triangle tiling used in wallpaper design?
Triangle tiling is the process of laying copies of a single triangle next to each
other to fill an area. One type of triangle tiling is wallpaper tiling. There are
exactly 17 types of triangle tiles that can be used for wallpaper tiling.
1
2
10
3
11
12
4
5
13
6
14
7
15
8
16
9
17
Tile 4 is made up of two 45°-45°-90° triangles that form a
square. This tile is rotated to make the wallpaper design
shown at the right.
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.
x
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.
x
d2
x2
x2
d
Pythagorean Theorem
d2 2x2
Add.
d 2x2
Take the positive square root of each side.
d 2 Factor.
x2
d x2
Simplify.
This algebraic proof verifies that the length of the hypotenuse of any 45°-45°-90°
triangle is 2 times the length of its leg. The ratio of the sides is 1 : 1 : 2.
Theorem 7.6
In a 45°-45°-90° triangle, the length of the hypotenuse
is 2
times the length of a leg.
n
45˚
n √2
n
45˚
Lesson 7-3 Special Right Triangles 357
Example 1 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. The length of the hypotenuse of one
of the triangles is 42. There are four 45°-45°-90° triangles along the diagonal of
the square. So, the length of the diagonal of the square is 442 or 162 inches.
Example 2 Find the Measure of the Legs
Study Tip
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.
Find x.
The length of the hypotenuse of a 45°-45°-90° triangle
is 2 times the length of a leg of the triangle.
C
xm
AB (AC)2
6 x2
AB 6, AC x
6
x
2
2
6
x
2
2
62
x
2
xm
45˚
B
45˚
A
6m
Divide each side by 2
.
Rationalize the denominator.
Multiply.
32 x
Divide.
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 K
M
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.
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 3x
2
J
30˚ 30˚
2x
2x
a
L
60˚
x
60˚
M
x
K
Pythagorean Theorem
JM a, LM x, JL 2x
Simplify.
Subtract x2 from each side.
Take the positive square root of each side.
a 3 x2 Factor.
a x3
Simplify.
So, in a 30°-60°-90° triangle, the measures of the sides are x, x3, and 2x. The ratio
of the sides is 1 : 3 : 2.
358
Chapter 7 Right Triangles and Trigonometry
The relationship of the side measures leads to Theorem 7.7.
Theorem 7.7
Study Tip
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
3 times the length of the shorter leg.
the longer leg is 30˚
n √3
2n
The shorter leg is opposite
the 30° angle, and the
longer leg is opposite
the 60° angle.
60˚
n
Example 3 30°-60°-90° Triangles
Find AC.
C is the longer leg, A
BC
A
B
is the shorter leg, and is the hypotenuse.
1
2
B
60˚
14 in.
AB (BC)
1
2
(14) or 7
BC 14
A
C
AC 3(AB)
3(7) or 73
AB 7
Example 4 Special Triangles in a Coordinate Plane
COORDINATE GEOMETRY Triangle PCD is a 30°-60°-90° triangle with right
angle C. C
D
is the longer leg with endpoints C(3, 2) and D(9, 2). Locate point P
in Quadrant I.
y
D lies on a horizontal gridline of
Graph C and D. C
the coordinate plane. Since PC
will be perpendicular to
CD
CD
, it lies on a vertical gridline. Find the length of .
CD 9 3 6
CD
is the longer leg. P
C
is the shorter leg.
So, CD 3(PC). Use CD to find PC.
C (3, 2)
x
O
CD 3(PC)
P
6 3(PC) CD 6
6
PC
3
6
3
PC
3 3
Divide each side by 3
.
D (9, 2)
60˚
2x
x
30˚
Rationalize the denominator.
63
PC
3
Multiply.
23 PC
Simplify.
C
x √3 6
D
Point P has the same x-coordinate as C. P is located 23 units above C.
So, the coordinates of P are 3, 2 23 or about (3, 5.46).
www.geometryonline.com/extra_examples/sol
Lesson 7-3 Special Right Triangles 359
Concept Check
Guided Practice
1. OPEN ENDED Draw a 45°-45°-90° triangle. Be sure to label the angles and the
sides and to explain how you made the drawing.
2. Explain how to draw a 30°-60°-90° triangle with the shorter leg 2
centimeters long.
3. Write an equation to find the length of a rectangle that has a diagonal twice as long
as its width.
Find x and y.
4.
5.
3
x
45˚
6.
y
10
x
30˚
45˚
y
8
x
y
Find the missing measures.
7. If c 8, find a and b.
8. If b 18, find a and c.
B
a
c
60˚
30˚
b
C
A
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.
9. A(8, 0), B(8, 3)
10. A(6, 6), B(2, 6)
Application
11. SOFTBALL Find the distance from home plate
to second base if the bases are 90 feet apart.
2nd Base
90 ft
3rd
Base
90 ft
45˚ 45˚
d
1st
Base
45˚ 45˚
90 ft
90 ft
Home Plate
Practice and Apply
For
Exercises
See
Examples
12, 13, 17,
22, 25
14–16, 18–21,
23, 24
27–31
1, 2
Find x and y.
12.
x˚ y
13.
9.6
14.
x
x
y˚
60˚
18
17
y
3
4
Extra Practice
15.
16.
17.
x
60˚
See page 767.
y
12
11 30˚
60˚
60˚
y
5
45˚
y
x
x
For Exercises 18 and 19, use the figure at the right.
18. If a 103, find CE and y.
19. If x 73, find a, CE, y, and b.
B
c
y
a
C
360 Chapter 7 Right Triangles and Trigonometry
x
E
60˚
30˚
b
A
20. The length of an altitude of an equilateral triangle is 12 feet. Find the length of a
side of the triangle.
21. The perimeter of an equilateral triangle is 45 centimeters. Find the length of an
altitude of the triangle.
22. The length of a diagonal of a square is 222 millimeters. Find the perimeter of
the square.
23. The altitude of an equilateral triangle is 7.4 meters long. Find the perimeter of
the triangle.
24. The diagonals of a rectangle are 12 inches long and intersect at an angle of 60°.
Find the perimeter of the rectangle.
25. 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.
26. Find x, y, z, and the perimeter of ABCD.
z
D
8
A
C
x
30˚
45˚
6
y
B
27. 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).
28. PGH is a 45°-45°-90° triangle with mP 90. Find the coordinates of P in
Quadrant I for G(4, 1) and H(4, 5).
29. 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).
30. PCD is a 30°-60°-90° triangle with mC 30 and hypotenuse CD
. Find the
coordinates of P for C(2, 5) and D(10, 5) if P lies above CD
.
31. If P
SR
Q
, use the figure to find a, b, c,
and d.
y
P (a, b) Q (8√3, d )
6√3
60˚
S ( 0, 0)
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
45˚
T (c, 0) U (e, 0)
R (f, 0)
x
TRIANGLE TILING For Exercises 32–35, 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.
32. How many 30°-60°-90° triangles are
used to create the basic circular
pattern?
33. Which angle of the 30°-60°-90° triangle
is being rotated to make the basic shape?
34. Explain why there are no gaps in the basic pattern.
35. Use grid paper to cut out 30°-60°-90° triangles. Color the same pattern
on each triangle. Create one basic figure that would be part of a
wallpaper tiling.
Lesson 7-3 Special Right Triangles 361
John Gollings, courtesy Federation Square
36. Find x, y, and z.
37. If BD 83 and mDHB 60,
find BH.
z
8
H
G
E
y
45˚
F
D
60˚
x
C
A
38. Each triangle in the figure is
a 30°-60°-90° triangle. Find x.
B
39. In regular hexagon UVWXYZ, each
side is 12 centimeters long. Find WY.
X
W
4
Y
V
30˚ 30˚
30˚
30˚
x
12
U
Z
40. BASEBALL The diagram at the right shows some
dimensions of Comiskey Park in Chicago, Illinois.
D is a segment from home plate to dead center
B
field, and A
E
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?
D
C
A
E
45˚
x
347 ft
45˚
347 ft
B
41. CRITICAL THINKING Given figure ABCD,
DC
with AB
, mB 60, mD 45, BC 8,
and AB 24, find the perimeter.
A
B
60˚
45˚
D
C
42. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson.
How is triangle tiling used in wallpaper design?
Include the following in your answer:
• which of the numbered designs contain 30°-60°-90° triangles and which
contain 45°-45°-90° triangles, and
• a reason why rotations of the basic design left no holes in the completed
design.
SOL/EOC Practice
Standardized
Test Practice
43. In the right triangle, what is AB if BC 6?
A 12 units
B
62 units
C
43 units
D
23 units
A
4x˚
2x˚
C
B
a2
44. SHORT RESPONSE For real numbers a and b, where b 0, if a ★ b 2 , then
b
(3 ★ 4)(5 ★ 3) ? .
362 Chapter 7 Right Triangles and Trigonometry
Maintain Your Skills
Mixed Review
Determine whether each set of measures can be the sides of a right triangle.
Then state whether they form a Pythagorean triple. (Lesson 7-2)
45. 3, 4, 5
46. 9, 40, 41
47. 20, 21, 31
48. 20, 48, 52
49. 7, 24, 25
50. 12, 34, 37
Find x, y, and z. (Lesson 7-1)
51.
52.
y
53.
10
x
12
4
z
z
z
8
y
x
15
x
5
Write an inequality or equation relating
each pair of angles. (Lesson 5-5)
54. mALK, mALN
y
A
8.5
4.7
4
K
N
L
6
6
55. mALK, mNLO
4
8.5
56. mOLK, mNLO
6.5
O
57. mKLO, mALN
58. Determine whether JKL with vertices J(3, 2), K(1, 5), and L(4, 4)
is congruent to RST with vertices R(6, 6), S(4, 3), and T(1, 4).
Explain. (Lesson 4-4)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Solve each equation.
(To review solving equations, see pages 737 and 738.)
59. 5 x
3
x
60. 0.14
7
63. 0.25
64. 9 10
k
9
m
0.8
n
13
g
61. 0.5 62. 0.2 24
65. 0.4
35
66. 0.07
x
P ractice Quiz 1
y
Lessons 7-1 through 7-3
Find the measure of the altitude drawn to the hypotenuse. (Lesson 7-1)
1.
2.
B
X
5
9
A
21
7
C
Y
Z
3. Determine whether ABC with vertices A(2, 1), B(4, 0), and C(5, 7) is a right triangle. Explain.
(Lesson 7-2)
Find x and y. (Lesson 7-3)
4.
y
3
5.
y
6
45˚
x
www.geometryonline.com/self_check_quiz/sol
30˚
x
Lesson 7-3 Special Right Triangles 363
Trigonometry
Virginia SOL Standard G.7 The student will solve practical problems involving right triangles by using … right
triangle trigonometry. Solutions will be expressed in radical form or as decimal approximations.
• Find trigonometric ratios using right triangles.
• Solve problems using trigonometric ratios.
can surveyors determine
angle measures?
Vocabulary
•
•
•
•
•
trigonometry
trigonometric ratio
sine
cosine
tangent
The old surveyor’s telescope shown
at right is called a theodolite
(thee AH duh lite). It is an optical
instrument used to measure angles
in surveying, navigation, and
meteorology. It consists of a telescope
fitted with a level and mounted on
a tripod so that it is free to rotate
about its vertical and horizontal
axes. After measuring 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
Study Tip
Reading Math
SOH-CAH-TOA is a
mnemonic device for
learning the ratios for
sine, cosine, and tangent
using the first letter of
each word in the ratios.
opp
sin A hyp
adj
hyp
cos A opp
adj
tan A Symbols
sine of A measure of leg opposite A
measure of hypotenuse
BC
sin A AB
measure of leg opposite B
measure of hypotenuse
AC
sin B AB
sine of B cosine of A measure of leg adjacent to A
measure of hypotenuse
AC
cos A AB
measure of leg adjacent to B
measure of hypotenuse
BC
cos B AB
cosine of B tangent of A measure of leg opposite A
measure of leg adjacent to A
BC
tan A AC
measure of leg opposite B
tangent of B measure of leg adjacent to B
AC
tan B BC
364 Chapter 7 Right Triangles and Trigonometry
Arthur Thevenart/CORBIS
Models
B
A
hypotenuse
leg
opposite
A
leg opposite B
C
B
hypotenuse
A
leg adjacent to A
leg
adjacent to
B
C
B leg
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.
Study Tip
Equivalent Ratios
Notice that the ratio
leg opposite A
is the
hypotenuse
leg adjacent C
same as .
hypotenuse
a
Thus, sin A cos C .
b
Likewise, cos A c
sin B .
b
Trigonometric Ratios
A
• 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.
• Fold the triangle so that there are two segments
perpendicular to B
A
. Label points D, E, F, and G as
, A
B
, B
C
, AF, AG
,
shown. Use a ruler to measure AC
FG
, AD
, AE
, and DE
to the nearest millimeter.
A
C
A
ED
G
F
B
Analyze
C
B
B
C
1. What is true of AED, AGF, and ABC ?
2. Copy the table. Write the ratio of the side lengths for each trigonometric
ratio. Then calculate a 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 with
the trigonometric ratios.
4. What is true about mA in each triangle?
As the Geometry Activity 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 1 Find Sine, Cosine, and Tangent Ratios
Find sin R, cos R, tan R, sin S, cos S, and tan S.
Express each ratio as a fraction and as a decimal.
T
S
opposite leg
hypotenuse
sin R adjacent leg
hypotenuse
cos R ST
RS
4
5
or 0.6
or 0.8
opposite leg
hypotenuse
sin S ST
RT
or 1.3
adjacent leg
hypotenuse
4
3
opposite leg
adjacent leg
tan S ST
RS
4
5
or 0.75
3
5
or 0.8
www.geometryonline.com/extra_examples/sol
opposite leg
adjacent leg
3
5
cos S R
tan R or 0.6
5
RT
RS
RT
RS
3
4
RT
ST
3
4
Lesson 7-4 Trigonometry 365
Study Tip
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.
Example 2 Use a Calculator to Evaluate Expressions
Use a calculator to find each value to the nearest ten thousandth.
a. cos 39°
KEYSTROKES: COS 39 ENTER
cos 39° 0.7771
b. sin 67°
SIN 67 ENTER
KEYSTROKES:
sin 67° 0.9205
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 3 Use Trigonometric Ratios to Find a Length
Study Tip
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 was 56°. Find the height of the cliff to the nearest
yard.
Let x be the height of the cliff in yards.
x
97
The tangent of angle A is
the same as the slope of
the line through the origin
that forms an angle of
measure A with the
positive x-axis.
leg opposite
tan 56° tan leg adjacent
Tangent
97 tan 56° x
x yd
Multiply each side by 97.
Use a calculator to find x.
56˚
KEYSTROKES: 97 TAN 56 ENTER
97 yd
143.8084139
The cliff is about 144 yards high.
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
Read as
sin A x
A sin1 (x)
A equals the inverse sine of x.
cos A y
tan A z
A
cos1
(y)
A equals the inverse cosine of y.
A
tan1
(z)
A equals the inverse tangent of z.
Example 4 Use Trigonometric Ratios to Find an Angle Measure
COORDINATE GEOMETRY Find mA 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
366
C(5, 4)
A(1, 2)
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 mA.
Chapter 7 Right Triangles and Trigonometry
O
B(6, 2)
x
Study Tip
Calculators
The second functions of
the SIN , COS , and
TAN keys are usually
the inverses.
Solve
AB (6 1)2
(2
2)2
BC (5 6
)2 (4
2)2
25 0 or 5
1 4 or 5
AC (5 1
)2 (4
2)2
16
4
20 or 25
Use the cosine ratio.
AC
AB
cos A leg adjacent
cos 25
cos A AC 25
and AB 5
hypotenuse
5
25
A cos1 5
Solve for A.
Use a calculator to find mA.
KEYSTROKES:
2nd
[COS1] 2 2nd [0] 5 )
5 ENTER
mA 26.56505118
The measure of A is about 26.6.
Examine Use the sine ratio to check the answer.
BC
AB
leg opposite
hypotenuse
sin A sin 5
sin A BC 5
and AB 5
5
KEYSTROKES:
2nd [SIN1] 2nd [0
] 5
5 ENTER
)
mA 26.56505118
The answer is correct.
Concept Check
1. Explain why trigonometric ratios do not depend on the size of the right triangle.
2. OPEN ENDED Draw a right triangle and label the measures of one acute angle
and the measure of the side opposite that angle. Then solve for the remaining
measures.
3. Compare and contrast the sine, cosine, and tangent ratios.
y
x
x
4. Explain the difference between tan A y and tan1 A.
Guided Practice
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.
5. a 14, b 48, and c 50
6. a 8, b 15, and c 17
B
a
C
c
b
A
Use a calculator to find each value. Round to the nearest ten-thousandth.
7. sin 57°
8. cos 60°
9. cos 33°
10. tan 30°
11. tan 45°
12. sin 85°
Lesson 7-4 Trigonometry 367
Find the measure of each angle to the nearest tenth of a degree.
14. sin B 0.6307
13. tan A1.4176
COORDINATE GEOMETRY Find the measure of the angle to the
nearest tenth in each right triangle ABC.
15. A in ABC, for A(6, 0), B( 4, 2), and C(0, 6)
16. B in ABC, for A(3, 3), B(7, 5), and C(7, 3)
Application
17. 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.
Practice and Apply
R
For
Exercises
See
Examples
18–21,
28–36
22–27
43–48
37–42,
52–54
1
2
3
4
Extra Practice
See page 767.
Q
Use PQR with right angle R to find sin P, cos P, tan P,
sin Q, cos Q, and tan Q. Express each ratio as a fraction,
and as a decimal to the nearest hundredth.
P
18. p 12, q 35, and r 37
19. p 6, q 23, and r 32
3
33
20. p , q , and r 3
21. p 23, q 15
, and r 33
2
2
Use a calculator to find each value. Round to the nearest ten-thousandth.
23. tan 42.8°
24. cos 77°
22. sin 6°
25. sin 85.9°
26. tan 12.7°
27. cos 22.5°
Use the figure to find each trigonometric
ratio. Express answers as a fraction and
as a decimal rounded to the nearest
ten-thousandth.
28. sin A
29. tan B
31. sin x°
32. cos x°
34. cos B
35. sin y°
C
5√26
A
x˚
5 √26
D B
25
1
30. cos A
33. tan A
36. tan x°
Find the measure of each angle to the nearest tenth of a degree.
38. cos C 0.2493
39. tan E 9.4618
37. sin B 0.7245
40. sin A 0.4567
41. cos D 0.1212
42. tan F 0.4279
Find x. Round to the nearest tenth.
44.
43.
45.
x
x
17
24˚
62˚
12
60
19
x˚
46.
34
x
31˚
368 Chapter 7 Right Triangles and Trigonometry
David R. Frazier/Photo Researchers
47.
17˚
x
6.6
y˚
48.
18
x˚
15
49. 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?
3˚
60 mi
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?
75˚
COORDINATE GEOMETRY Find the measure of each angle to the nearest tenth
in each right triangle.
52. J in JCL for J(2, 2), C(2, 2), and L(7, 2)
53. C in BCD for B(1, 5), C(6, 5), and D(1, 2)
54. X in XYZ for X(5, 0), Y(7, 0), and Z(0, 35
)
55. Find the perimeter of ABC if mA 35, mC 90, and AB 20 inches.
Find x and y. Round to the nearest tenth.
56.
57.
x˚
x
55˚
12
24
36
y˚
58.
C
y
x
24
47˚
32˚
A
B
D
y
ASTRONOMY For Exercises 59 and 60, use the following information.
One way to find the distance between the sun and a relatively close star is to
determine the angles of sight for the star exactly six months apart. Half the measure
formed by these two angles of sight is called the stellar parallax. Distances in space
are sometimes measured in astronomical units. An astronomical unit is equal to the
average distance between Earth and the sun.
Alpha
Centauri
0.00021°
(stellar parallax)
Earth
1
Sun
1 Earth (6
months
later)
0.00021°
(stellar parallax)
Astronomy
The stellar parallax is one
of several methods of
triangulation used to
determine the distance of
stars from the sun. Another
method is trigonometric
parallax, which measures
the displacement of a
nearby star relative to
a more distant star.
Source: www.infoplease.com
59. Find the distance between Alpha Centauri and the sun.
60. Make a conjecture as to why this method is used only for close stars.
61. CRITICAL THINKING Use the figure at the right to
find sin x°.
A
10
62. WRITING IN MATH
D
Answer the question that was
posed at the beginning of the lesson.
How do surveyors determine angle measures?
Include the following in your answer:
• where theodolites are used, and
• the kind of information one obtains from a theodolite.
www.geometryonline.com/self_check_quiz/sol
8
x˚
B
8
10
C
Exercise 61
Lesson 7-4 Trigonometry 369
StockTrek/CORBIS
Standardized
Test Practice
SOL/EOC Practice
63. Find cos C.
A
C
B
4
5
D
3
4
5
C
3
5
4
A
If x2 152 242 15(24), find x.
B 21
C 12
64. ALGEBRA
A 20.8
Extending
the Lesson
B
3
5
9
D
Each of the basic trigonometric ratios has a
reciprocal ratio. The reciprocals of the sine,
cosine, and tangent are called the cosecant,
secant, and the cotangent, respectively.
A
c
b
B
a
C
Reciprocal
Trigonometric Ratio
Abbreviation
Definition
1
sin A
cosecant of A
csc A
measure of the hypotenuse
c
measure of the leg opposite A
a
1
cos A
secant of A
sec A
measure of the hypotenuse
c
measure of the leg adjacent A
b
1
tan A
cotangent of A
cot A
measure of the leg adjacent A
b
measure of the leg opposite A
a
Use ABC to find csc A, sec A, cot A, csc B, sec B, and cot B. Express each ratio as
a fraction or as a radical in simplest form.
65. a 3, b 4, and c 5
66. a 12, b 5, and c 13
68. a 22, b 22, and c 4
67. a 4, b 43, and c 8
Maintain Your Skills
Mixed Review
Find each measure.
B
(Lesson 7-3)
69. If a 4, find b and c.
a
70. If b 3, find a and c.
60˚
c
30˚
C
71. If c 5, find a and b.
b
A
Determine whether each set of measures can be the sides of a right triangle.
Then state whether they form a Pythagorean triple. (Lesson 7-2)
72. 4, 5, 6
73. 5, 12, 13
74. 9, 12, 15
75. 8, 12, 16
76. 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 6-1)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Find each angle measure if h k. (To review
angles formed by parallel lines and a transversal, see Lesson 3-2.)
77. m15
78. m7
79. m3
80. m12
81. m11
1
30˚
82. m4
4
7
370 Chapter 7 Right Triangles and Trigonometry
s
6
5
3
2
t
8
9
10
11
12
117˚
15 14
h
k
Angles of Elevation
and Depression
Virginia SOL Standard G.7 The student will
solve practical problems involving right triangles
by using … right triangle trigonometry. Solutions
will be expressed in radical form or as decimal
approximations.
• Solve problems involving angles of elevation.
• Solve problems involving angles of depression.
Vocabulary
• angle of elevation
• angle of depression
do airline pilots use angles
of elevation and 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.
ANGLES OF ELEVATION An
A
•
angle of elevation is the angle between
the line of sight and the horizontal
when an observer looks upward.
B
line of sight
D angle of
elevation
C
Example 1 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 mDAC.
CD
AC
450
tan x° 2025
opposite
adjacent
tan x° tan CD 450, AC 2025
x tan1 Solve for x.
450
2025
x 12.5
Use a calculator.
The angle of elevation for the takeoff should be more than 12.5°.
Lesson 7-5 Angles of Elevation and Depression 371
Robert Holmes/CORBIS
ANGLES OF DEPRESSION An
angle of depression
B
angle of depression is the angle between
the line of sight when an observer looks
downward, and the horizontal.
A
line of sight
C
D
Standardized Example 2 Angle of Depression
Test Practice
Short-Response Test Item
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°. Find the
length of the ramp to the nearest tenth foot.
10˚
3.5 ft
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
Method 1
Method 2
10˚
A
D
A
3.5 ft
80˚
D
3.5 ft
C
10˚
B
The ground and the horizontal level
with the back of the van are parallel.
Therefore, mDABmABC since
they are alternate interior angles.
3.5
AB
sin 10° C
The horizontal line from the back of
the van and the segment from the
ground to the back of the van are
perpendicular. So, DAB and BAC
are complementary angles. Therefore,
mBAC 90 10 or 80.
AB sin 10° 3.5
3.5
sin 10°
AB B
10˚
3.5
AB
cos 80° AB cos 80° 3.5
AB 20.2
The ramp is about 20.2 feet long.
3.5
cos 80°
AB AB 20.2
Angles of elevation or depression to two different objects can be used to find the
distance between those objects.
Study Tip
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.
372
Example 3 Indirect Measurement
Olivia is 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.
CDA and CDB are right triangles, and
and CD110 85 or 195. The distance
between the boats is AB or BD AD. Use
the right triangles to find these two lengths.
Chapter 7 Right Triangles and Trigonometry
C
E
33˚
85 ft
57˚
110 ft
D
A
B
Because C
E
and D
B
are horizontal lines, they are parallel. Thus, ECB CBD
and ECA CAD because they are alternate interior angles. This means that
mCBD 33 and mCAD 57.
195
DB
tan 33° DB tan 33° 195
195
tan 33°
opposite
adjacent
tan ; mCBD 33
Multiply each side by DB.
DB Divide each side by tan 33°.
DB 300.27
Use a calculator.
195
DA
tan 57° DA tan 57° 195
opposite
adjacent
tan ; mCAD 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.
Concept Check
1. OPEN ENDED Find a real-life example of
an angle of depression. Draw a diagram and
identify the angle of depression.
P
F
2. Explain why an angle of elevation is given
that name.
3. Name the angles of depression and elevation
in the figure.
Guided Practice
T
B
4. 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.)
5. 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.
6. SALVAGE A salvage ship uses sonar to
determine that the angle of depression to a
wreck on the ocean floor is 13.25°. The depth
chart shows that the ocean floor 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?
Standardized
Test Practice
7. SHORT RESPONSE From the top of
a 150-foot high tower, an air traffic
controller observes an airplane on
the runway. To the nearest foot, how
far from the base of the tower is the
airplane?
www.geometryonline.com/extra_examples/sol
13.25˚
40 m
12˚
150 ft
Lesson 7-5 Angles of Elevation and Depression 373
Practice and Apply
8. 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?
For
Exercises
See
Examples
12, 14–18
9–11
8, 13, 19,
20
1
2
3
Extra Practice
See page 768.
9. 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.
12˚
36 yd
10. 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?
11. SLEDDING A sledding run is 300 yards long with a vertical drop of 27.6 yards.
Find the angle of depression of the run.
12. RAILROADS The Monongahela Incline overlooks the city of Pittsburgh,
Pennsylvania. Refer to the information at the left to determine the incline of the
railway.
13. 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
CIVIL ENGINEERING For Exercises 14 and 15, use the following information.
The percent grade of a highway is the ratio of the vertical rise or fall over a given
horizontal distance. The ratio is expressed as a percent to the nearest whole number.
Suppose a highway has a vertical rise of 140 feet for every 2000 feet of horizontal
distance.
14. Calculate the percent grade of the highway.
15. Find the angle of elevation that the highway makes with the horizontal.
Railroads
The Monongahela Incline
is 635 feet long with a
vertical rise of 369.39
feet. It was built at a cost
of $50,000 and opened
on May 28, 1870. It is still
used by commuters to
and from Pittsburgh.
Source: www.portauthority.org
16. 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 17 and 18, 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.
17. 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?
18. 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?
19. BIRDWATCHING 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?
374 Chapter 7 Right Triangles and Trigonometry
R. Krubner/H. Armstrong Roberts
20. 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
MEDICINE For Exercises 21–23, 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°.
21. How far off the horizontal is the raised end of the treadmill at the beginning of
the exam?
22. 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?
23. 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?
Travel
Ayers Rock is the largest
monolith, a type of rock
formation, in the world.
It is approximately
3.6 kilometers long and
2 kilometers wide. The rock
is believed to be the tip of
a mountain, two thirds
of which is underground.
Source: www.atn.com.au
24. 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?
25. 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.
orbit
29°
16°
3 mi
n
f
Online Research
Data Update Use the Internet to determine the angle
of depression formed by someone aboard the international space station
looking down to your community. Visit www.geometryonline.com/data_update
to learn more.
26. CRITICAL THINKING 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.
27. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson.
How do airline pilots use angles of elevation and depression?
Include the following in your answer:
• when pilots use angles of elevation or depression, and
• the difference between angles of elevation and depression.
www.geometryonline.com/self_check_quiz/sol
Lesson 7-5 Angles of Elevation and Depression 375
John Mead/Science Photo Library/Photo Researchers
Standardized
Test Practice
SOL/EOC Practice
28. The top of a signal tower is 120 meters
above sea level. The angle of depression
from the top of tower to a passing ship
is 25°. How many meters from the foot
of the tower is the ship?
A 283.9 m
B 257.3 m
C
132.4 m
29. ALGEBRA
A
56 m
D
y
28
25˚
120 m
x
16
1
2
7
4
If , then find x when y .
2
7
4
7
B
C
1
2
3 D
Maintain Your Skills
Mixed Review
Find the measure of each angle to the nearest tenth of a degree. (Lesson 7-4)
30. cos A 0.6717
31. sin B 0.5127
32. tan C 2.1758
33. cos D 0.3421
34. sin E 0.1455
35. tan F 0.3541
Find x and y. (Lesson 7-3)
36.
37.
x
38.
y
60˚
14
12
20
y
30˚
45˚ y
x
x
39. HOBBIES 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)
40. Copy and complete the flow proof. (Lesson 4-6)
Given: 5 6
GS
FR
F
5
G
3
4
1
Prove: 4 3
X
6
2
R
Proof:
S
5 6
Given
FR GS
FXR GXS
Given
?
a.
Vert. s are .
Getting Ready for
the Next Lesson
b.
?
?
c.
d.
e.
?
f.
?
?
PREREQUISITE SKILL Solve each proportion.
(To review solving proportions, see Lesson 6-1.)
x
35
41. 5
3
42. 12
24
43. x
x
24
44. 12
48
45. x
5
46. 28
7
47. x
3
48. 6
13
42
x
376 Chapter 7 Right Triangles and Trigonometry
x
18
45
8
17
15
x
36
40
15
26
The Law of Sines
• Use the Law of Sines to solve triangles.
• Solve problems by using the Law of Sines.
are triangles used in
radio astronomy?
Vocabulary
Study Tip
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.
The Very Large Array (VLA), one of the
world’s premier astronomical radio
observatories, consists of 27 radio antennas
in a Y-shaped configuration on the Plains of
San Agustin in New Mexico. Astronomers
use the VLA to make pictures from the radio
waves emitted by astronomical objects.
Construction of the antennas is supported
by a variety of triangles, 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.
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
C
b
a
sin A
sin B
sin C
.
a
b
c
A
Proof
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.
sin A h
b
sin B h
a
b sin A h
a sin B h
Definition of sine
C
b
A
h
D
a
B
Cross products
• Law of Sines
• solving a triangle
b sin A a sin B
sin A
sin B
a
b
Substitution
Divide each side by ab.
The proof can be completed by using a similar technique with the other altitudes
sin A
a
sin C
c
sin B
b
sin C
c
to show that and .
Lesson 7-6 The Law of Sines 377
Example 1 Use the Law of Sines
a. Find b. Round to the nearest tenth.
B
Use the Law of Sines to write a proportion.
Study Tip
Rounding
If you round before the
final answer, your results
may differ from results in
which rounding was not
done until the final answer.
sin A
sin B
a
b
sin 37°
sin 68°
3
b
b sin 37° 3 sin 68°
3 sin 68°
sin 37°
68˚
Law of Sines
37˚
A
mA 37, a 3, mB 68
3
C
b
Cross products
b Divide each side by sin 37°.
b 4.6
Use a calculator.
b. Find mZ to the nearest degree in XYZ if y 17, z 14, and mY 92.
Write a proportion relating Y, Z, y, and z.
sin Y
sin Z
y
z
sin 92°
sin Z
17
14
Law of Sines
mY 92, y 17, z 14
14 sin 92° 17 sin Z Cross products
14 sin 92°
sin Z
17
14
s
in
92°
sin1 Z
17
55° Z
Divide each side by 17.
Solve for Z.
Use a calculator.
So, mZ 55.
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.
Study Tip
Look Back
To review the Angle Sum
Theorem, see Lesson 4-2.
Example 2 Solve Triangles
a. Solve ABC if mA 33, mB 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 mC.
C
a
b
A
c
mA mB mC 180 Angle Sum Theorem
33 47 mC 180 mA 33, mB 47
80 mC 180 Add.
mC 100 Subtract 80 from each side.
sin B
b
Since we know mB and b, use proportions involving .
To find a:
To find c:
sin B
sin A
b
a
sin 47°
sin 33°
14
a
sin B
sin C
b
c
sin 47°
sin 100 °
14
c
a sin 47° 14 sin 33°
14 sin 33°
sin 47°
a a 10.4
Law of Sines
Substitute.
Cross products
Divide each side by sin 47°.
Use a calculator.
Therefore, mC 100, a 10.4, and c 18.9.
378
Chapter 7 Right Triangles and Trigonometry
c sin 47° 14 sin 100°
14 sin 100°
sin 47°
c c 18.9
B
b. Solve ABC if mC 98, b 14, and c 20. Round angle measures to the
nearest degree and side measures to the nearest tenth.
Study Tip
An Equivalent
Proportion
The Law of Sines may
also be written as
a
b
c
.
sin A
sin B
sin C
You may wish to use this
form when finding the
length of a side.
We know the measures of two sides and an angle opposite one of the sides.
sin B
sin C
b
c
sin B
sin 98°
14
20
Law of Sines
mC 98, b 14, and c 20
20 sin B 14 sin 98°
14 sin 98°
20
14 sin 98°
B sin–1 20
Cross products
sin B Divide each side by 20.
B 44°
Solve for B.
Use a calculator.
mA mB mC 180
Angle Sum Theorem
mA 44 98 180 mB 44 and mC 98
mA 142 180
mA 38
sin A
sin C
a
c
sin 38°
sin 98°
a
20
Add.
Subtract 142 from each side.
Law of Sines
mA 38, mC 98, and c 20
20 sin 38° a sin 98°
Cross products
20 sin 38°
a
sin 98°
Divide each side by sin 98°.
12.4 a
Use a calculator.
Therefore, A 38°, B 44°, and a 12.4.
USE THE LAW OF SINES TO SOLVE PROBLEMS The Law of Sines is
very useful in solving direct and indirect measurement applications.
Example 3 Indirect Measurement
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 of 30 feet long on the ground. Find the length
of the telephone pole to the nearest tenth of a foot.
Draw a diagram.
S
P
GD
Draw SD
. Then find the mGDP and mGPD.
7°
mGDP 90 7 or 83
pole
mGPD 62 83 180 or mGPD 35
Since you know the measures of two angles of the
triangle, mGDP and mGPD, and the length of
a side opposite one of the angles (G
D
is opposite
GPD) you can use the Law of Sines to find
the length of the pole.
www.geometryonline.com/extra_examples/sol
G
62°
30-foot shadow
D
(continued on the next page)
Lesson 7-6 The Law of Sines 379
PD
GD
sin DGP
sin GPD
PD
30
sin 62°
sin 35°
PD sin 35° 30 sin 62°
30 sin 62°
sin 35°
Law of Sines
mDGP 62, mGPD 35, and GD 30
Cross products
PD Divide each side by sin 35°.
PD 46.2
Use a calculator.
The telephone pole is about 46.2 feet long.
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)
Concept Check
1. FIND THE ERROR Makayla and Felipe are trying to find d in DEF.
Makayla
d
sin 59° 12
F
Felipe
12 73˚
sin 59°
sin 48°
d
12
D
Who is correct? Explain your reasoning.
d
48˚
f
E
2. 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.
3. Compare the two cases for the Law of Sines.
Guided Practice
Find each measure using the given measures of XYZ. Round angle measures
to the nearest degree and side measures to the nearest tenth.
4. Find y if x 3, mX 37, and mY 68.
5. Find x if y 12.1, mX 57, and mZ 72.
6. Find mY if y 7, z 11, and mZ 37.
7. Find mZ if y 17, z 14, and mY 92.
Solve each PQR described below. Round angle measures to the nearest degree
and side measures to the nearest tenth.
8. mR 66, mQ 59, p 72
9. p 32, r 11, mP 105
10. mP 33, mR 58, q 22
11. p 28, q 22, mP 120
12. mP 50, mQ 65, p 12
13. q 17.2, r 9.8, mQ 110.7
14. Find the perimeter of parallelogram
ABCD to the nearest tenth.
A
B
6
D
380 Chapter 7 Right Triangles and Trigonometry
32°
88°
C
Application
15. 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 from A to B as 62° and
measures the angle from C to B as 55°. Find the
distance from A to B.
C
A
B
Practice and Apply
For
Exercises
See
Examples
16–21
22–29
30–38
1
2
3
Extra Practice
See page 768.
Find each measure using the given measures of KLM. Round angle measures
to the nearest degree and side measures to the nearest tenth.
16. If mL 45, mK 63, and 22, find k.
17. If k 3.2, mL 52, and mK 70, find .
18. If m 10.5, k 18.2, and mK 73, find mM.
19. If k 10, m 4.8, and mK 96, find mM.
20. If mL 88, mK 31, and m 5.4, find .
21. If mM 59, 8.3, and m 14.8, find mL.
Solve each WXY described below. Round measures to the nearest tenth.
22. mY 71, y 7.4, mX 41
23. x 10.3, y 23.7, mY 96
24. mX 25, mW 52, y 15.6
25. mY 112, x 20, y 56
26. mW 38, mY 115, w 8.5
27. mW 36, mY 62, w 3.1
28. w 30, y 9.5, mW 107
29. x 16, w 21, mW 88
30. An isosceles triangle has a base of 46 centimeters and a vertex angle of 44°.
Find the perimeter.
31. Find the perimeter of quadrilateral ABCD to the
nearest tenth.
A
12
B
28˚
40˚
D
C
32. 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.
33. AVIATION Two radar stations that are 20 miles apart located a plane at the
same time. The first station indicated that the position of the plane made an
angle of 43° with the line between the stations. The second station indicated
that it made an angle of 48° with the same line. How far is each station from
the plane?
34. SURVEYING Maria Lopez is a surveyor who must determine the distance
across a section of the Rio Grande Gorge in New Mexico. Standing 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.
Lesson 7-6 The Law of Sines 381
35. 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?
160 ft
85˚
160 ft
MIRRORS For Exercises 36 and 37, use the following information.
Kayla, Jenna, and Paige live in a town nicknamed “Perpendicular City” because
the planners and builders took great care to have all the streets oriented north-south
or east-west. The three of them play a game where they signal each other using
mirrors. Kayla and Jenna signal each other from a distance of 1433 meters. Jenna
turns 27° to signal Paige. Kayla rotates 40° to signal Paige.
36. To the nearest tenth of a meter, how far apart are Kayla and Paige?
37. To the nearest tenth of a meter, how far apart are Jenna and Paige?
Aviation
From January 2001 to May
2001, Polly Vacher flew over
29,000 miles to become the
first woman to fly the
smallest single-engined
aircraft around the world
via Australia and the Pacific.
Source: www.worldwings.org
AVIATION For Exercises 38 and 39, use the
following information.
Keisha Jefferson is flying a small plane due west.
To avoid the jet stream, she must change her course.
She turns the plane 27° to the south and flies
60 miles. Then she makes a turn of 124° heads back
to her original course.
27˚
124˚
38. How far must she fly after the second turn to return to the original course?
39. How many miles did she add to the flight by changing course?
40. CRITICAL THINKING Does the Law of Sines apply to the acute angles of a
right triangle? Explain your answer.
41. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson.
How are triangles used in radio astronomy?
Include the following in your answer:
• a description of what the VLA is and the purpose of the VLA, and
• the purpose of the triangles in the VLA.
SOL/EOC Practice
Standardized
Test Practice
42. SHORT RESPONSE In XYZ, if x 12 , mX 48, and mY 112, solve the
triangle to the nearest tenth.
43. ALGEBRA The table below shows the customer ratings of five restaurants in
the Metro City Guide to Restaurants. The rating scale is from 1, the worst, to 10,
the best. Which of the five restaurants has the best average rating?
Restaurant
A
C
Food
Decor
Service
Value
Del Blanco’s
7
9
4
7
Aquavent
8
9
4
6
Le Circus
10
8
3
5
Sushi Mambo
7
7
5
6
Metropolis Grill
9
8
7
7
Metropolis Grill
Aquavent
382 Chapter 7 Right Triangles and Trigonometry
B
D
Le Circus
Del Blanco’s
Maintain Your Skills
Mixed Review
ARCHITECTURE For Exercises 44 and 45, use the
following information.
x
1 ft
Mr. Martinez is an architect who designs houses so that
the windows receive minimum sun in the summer and
window
maximum sun in the winter. For Columbus, Ohio, the
angle of
6 ft
angle of elevation of the sun at noon on the longest day
elevation
high
of sun
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 7-5)
44. How long should the architect make the overhang so that the window gets no
direct sunlight at noon on the longest day?
45. Using the overhang from Exercise 44, how much of the window will get direct
sunlight at noon on the shortest day?
J
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 7-4)
46. j 8, k 17, 15
47. j 20, k 29, 21
49. j 72, k 14, 72
48. j 12, k 24, 123
If K
H
is parallel to JI, find the measure of
each angle. (Lesson 4-2)
50. 1
51. 2
52. 3
Getting Ready for
the Next Lesson
L
K
54˚ 36˚
120˚
1
3 J
2
I
H
c2 a2 b2
2ab
PREREQUISITE SKILL Evaluate for the given values of a, b, and c.
(To review evaluating expressions, see page 736.)
53. a 7, b 8, c 10
56. a 16, b 4, c 13
54. a 4, b 9, c 6
57. a 3, b 10, c 9
P ractice Quiz 2
Find x to the nearest tenth. (Lesson 7-4)
1.
2.
16
K
Lessons 7-4 through 7-6
x
3.
10
x˚
55. a 5, b 8, c 10
58. a 5, b 7, c 11
9.7
4. 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.
17˚
32 53˚
5. Solve DEF.
x
D
12
E
(Lesson 7-6)
82˚
8
F
(Lesson 7-5)
www.geometryonline.com/self_check_quiz/sol
Lesson 7-6 The Law of Sines 383
A Follow-Up of Lesson 7-6
The Ambiguous Case of the Law of Sines
In Lesson 7-6, 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.
Model
and AC
. Construct a
Construct AB
circle whose center is B so that it
at two points. Then,
intersects AC
construct any radius BD
.
Using the rotate tool, move D to the
other intersection point of circle B
.
and AC
Step 5
Find the measures of B
D, A
B, and A.
Note the measures of ABD,
BDA, and AD
in ABD.
Use the rotate tool to move D so that
it lies on one of the intersection points
. In ABD, find the
of circle B and AC
D.
measures of ABD, BDA, and A
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
1. Which measures are the same in both triangles?
2. Repeat the activity using different measures for A, BD
, and A
B
. How do the results
compare to the earlier results?
Make a Conjecture
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 B
D
, A
B
, and A, to have no solution? Test your
conjecture and explain.
384 Investigating Slope-Intercept Form
384 Chapter 7 Right Triangles and Trigonometry
The Law of Cosines
• Use the Law of Cosines to solve triangles.
• Solve problems by using the Law of Cosines.
are triangles used in
building design?
Vocabulary
• Law of Cosines
The Chicago Metropolitan Correctional Center is
a 27-story triangular federal detention center. The
cells are arranged around a lounge-like common
area. The architect found that a triangular floor
plan allowed for the maximum number of cells to
be most efficiently centered around the lounge.
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.
Law of Cosines
Study Tip
Side and Angle
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.
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.
C
b
A
a2 b2 c 2 2bc cos A
a
B
c
b 2 a2 c 2 2ac cos B
c 2 a2 b 2 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 1 Two Sides and the Included Angle
Find a if c 8, b 10, and mA 60.
Use the Law of Cosines since the measures of two sides and
the included are known.
a2 b2 c2 2bc cos A
a2
102
82
2(10)(8) cos 60°
a2 164 160 cos 60°
A
60˚
10
8
Law of Cosines
b 10, c 8, and mA 60
C
a
B
Simplify.
a 164
160 cos
60°
Take the square root of each side.
a 9.2
Use a calculator.
Lesson 7-7 The Law of Cosines 385
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 2 Three Sides
Q
23
18
Find mR.
r2 q2 s2 2qs cos R
R
S
37
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
13 32
Divide each side by –1332.
1332 1164
R cos1 Solve for R.
R 29.1°
Use a calculator.
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.
Example 3 Select a Strategy
Study Tip
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.
Solve KLM. Round angle measure to the
nearest degree and side measure 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
Sines. We know the measures of two sides and the
included angle. This is SAS, so use the Law of
Cosines.
k2 2 m2 2m cos K
k2
182
142
L
14
K
k
51˚
18
M
Law of Cosines
2(18)(14) cos 51°
18, m 14, and mK 51
2 2(18)(1
k 182 14
cos
51°
4)
Take the square root of each side.
k 14.2
Use a calculator.
Next, we can find mL or mM. If we decide to find mL, 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
k
sin L
sin 51°
18
14.2
Law of Sines
18, k 14.2, and mK 51
14.2 sin L 18 sin 51°
18 sin 51°
14.2
18 sin 51°
L sin–1 14.2
Cross products
sin L L 80°
386 Chapter 7 Right Triangles and Trigonometry
Divide each side by 14.2.
Take the inverse sine of each side.
Use a calculator.
Use the Angle Sum Theorem to find mM.
mK mL mM 180 Angle Sum Theorem
51 80 mM 180 mK 51 and mL 80
mM 49
Subtract 131 from each side.
Therefore, k 14.2, mK 80, and mM 49.
Example 4 Use Law of Cosines to Solve Problems
REAL ESTATE Ms. Jenkins is buying some
property that is shaped like quadrilateral
ABCD. Find the perimeter of the property.
Use the Pythagorean Theorem to find BD
in ABD.
(AB)2 (AD)2 (BD)2
1802
AB 180, AD 240
90,000 (BD)2
Simplify.
300 BD
B
Pythagorean Theorem
(BD)2
2402
C
200 ft
60˚
180 ft
A
240 ft
D
Take the square root of each side.
Next, use the Law of Cosines to find CD in BCD.
(CD)2 (BC)2 (BD)2 2(BC)(BD) cos CBD
(CD)2
2002
3002
2(200)(300) cos 60°
(CD)2 130,000 120,000 cos 60°
Law of Cosines
BC 200, BD 300, mCBD 60
Simplify.
CD 130,00
0 120,000
60°
cos
Take the square root of each side.
CD 264.6
Use a calculator.
The perimeter of the property is 180 200 264.6 240 or about 884.6 feet.
Concept Check
1. 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.
2. Explain when you should use the Law of Sines or the Law of Cosines to solve
a triangle.
3. 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.
Guided Practice
In BCD, given the following measures, find the measure of the missing side.
5. b 107, c 94, mD 105
4. c 2, d 5, mB 45
In RST, given the lengths of the sides, find the measure of the stated angle
to the nearest degree.
7. r 2.2, s 1.3, t 1.6; mR
6. r 33, s 65, t 56; mS
Solve each triangle using the given information. Round angle measure to the
nearest degree and side measure to the nearest tenth.
8. XYZ: x 5, y 10, z 13
9. KLM: k 20, m 24, mL 47
www.geometryonline.com/extra_examples/sol
Lesson 7-7 The Law of Cosines 387
Application
10. CRAFTS Jamie, age 25, is creating a logo for herself and
two cousins, ages 10 and 5. She is using a quarter (25 cents), a
dime (10 cents), and a nickel (5 cents) to represent their ages.
She will hold the coins together by soldering a triangular piece
of silver wire so that the three vertices of the triangle lie at the
centers of the three circular coins. The diameter of the quarter
is 24 millimeters, the diameter of the nickel is 22 millimeters,
and the diameter of a dime is 10 millimeters. Find the
measures of the three angles in the triangle.
Q
D
N
Practice and Apply
For
Exercises
See
Examples
11–14
15–18, 38
22–37,
39–41
1
2
3
Extra Practice
See page 768.
In TUV, given the following measures, find the measure of the missing side.
12. t 11, u 17, mV 78
11. t 9.1, v 8.3, mU 32
13. u 11, v 17, mT 105
14. v 11, u 17, mT 59
In EFG, given the lengths of the sides, find the measure of the stated angle
to the nearest degree.
16. e 14, f 19, g 32; mE
15. e 9.1, f 8.3, g 16.7; mF
17. e 325, f 198, g 208; mF
18. e 21.9, f 18.9, g 10; mG
Solve each triangle using the given information. Round angle measures to the
nearest degree and side measures to the nearest tenth.
Q
G
19.
20.
21. C
8
40˚
F
10
11
g
H
P
p
11
38˚
10
18
M
B
D
15
22. ABC: mA 42, mC 77, c 6 23. ABC: a 10.3, b 9.5, mC 37
24. ABC: a 15, b 19, c 28
25. ABC: mA 53, mC 28, c 14.9
26. KITES Beth is building a kite like the one at
the right. If AB
is 5 feet long, BC
is 8 feet long,
2
D
is 7 feet long, find the measure of the
and B
3
angle between the short sides and the angle
between the long sides to the nearest degree.
A
B
D
C
Solve each LMN described below. Round measures to the nearest tenth.
27. m 44, 54, mL 23
28. m 18, 24, n 30
29. m 19, n 28, mL 49
30. mM 46, mL 55, n 16
31. m 256, 423, n 288
32. mM 55, 6.3, n 6.7
33. mM 27, 5, n 10
34. n 17, m 20, 14
35. 14, n 21, mM 60
36. 14, m 15, n 16
37. mL 51, 40, n 35
38. 10, m 11, n 12
388 Chapter 7 Right Triangles and Trigonometry
39. In quadrilateral ABCD, AC 188, BD 214,
and
mBPC 70, and P is the midpoint of AC
D
. Find the perimeter of quadrilateral ABCD.
B
A
B
P
D
40. In quadrilateral PQRS, PQ 721, QR 547,
RS 593, PS 756, and mP 58. Find QS,
mPQS, and mR.
C
Q
P
41. BUILDINGS Refer to the information at the left.
Find the measures of the angles of the triangular
building to the nearest tenth.
42. 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?
R
S
Carlos
40 ft
50 ft
24 ft
30 ft
22 ft
Adam
Buildings
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
43. PROOF Justify each statement for the
derivation of the Law of Cosines.
A
D
is an altitude of ABC.
Given: A
b
c
h
Prove: c2 a2 b2 2ab cos C
B
Proof:
ax
x
C
a
Statement
a. c2 (a x)2 h2
b. c2 a2 2ax x2 h2
c. x2 h2 b2
d. c2 a2 2ax b2
x
e. cos C Reasons
a. ?
b. ?
c. ?
d. ?
e. ?
f. b cos C x
g. c2 a2 2a(b cos C) b2
h. c2 a2 b2 2ab cos C
f.
g.
h.
b
?
?
?
44. CRITICAL THINKING Graph A(–6, –8), B(10, –4), C(6, 8), and D(5, 11) on a
coordinate plane. Find the measure of interior angle ABC and the measure
of exterior angle DCA.
45. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson.
How are triangles used in building construction?
Include the following in your answer:
• why the building was triangular instead of rectangular, and
• why the Law of Sines could not be used to solve the triangle.
www.geometryonline.com/self_check_quiz/sol
Lesson 7-7 The Law of Cosines 389
Standardized
Test Practice
SOL/EOC Practice
46. For DEF, find d to the nearest tenth if e 12, f 15, and mD 75.
A 18.9
B 16.6
C 15.4
D 9.8
47. ALGEBRA Ms. LaHue earns a monthly base salary of $1280 plus a commission
of 12.5% of her total monthly sales. At the end of one month, Ms. LaHue earned
$4455. What were her total sales for the month?
A $3175
B $10,240
C $25,400
D $35,640
Maintain Your Skills
Mixed Review
Find each measure using the given measures from XYZ. Round angle measure
to the nearest degree and side measure to the nearest tenth. (Lesson 7-6)
48. If y 4.7, mX 22, and mY 49, find x.
49. If y 10, x 14, and mX 50, find mY.
50. 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 7-5)
For Exercises 51–54, determine whether A
B
D. (Lesson 6-4)
C
51. AC 8.4, BD 6.3, DE 4.5, and CE 6
52. AC 7, BD 10.5, BE 22.5, and AE 15
53. AB 8, AE 9, CD 4, and CE 4
54. AB 5.4, BE 18, CD 3, and DE 10
B
A
C
D
E
Use the figure at the right to write a paragraph proof. (Lesson 6-3)
H
55. Given: JFM EFB
LFM GFB
E
Prove: JFL EFG
D
E
56. Given: JM
B
A
LM
G
B
G
J
L
F
M
B
Prove: JL E
G
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)
57. orthocenter
58. centroid
59. circumcenter
Who is Behind This Geometry Idea Anyway?
It’s time to complete your project. Use the information and
data you have gathered about your research topic, two
mathematicians, and a geometry problem to prepare a portfolio
or Web page. Be sure to include illustrations and/or tables in
the presentation.
www.geometryonline.com/webquest
390 Chapter 7 Right Triangles and Trigonometry
C
A Follow-Up of Lesson 7-7
Trigonometric Identities
In algebra, the equation 2(x 2) 2x 4 is called an identity because the equation
is true for all values of x. There are equations involving trigonometric ratios that are
true for all values of the angle measure. These are called trigonometric identities .
y
In the figure, P(x, y) is in Quadrant I. The Greek letter theta
(pronounced THAY tuh) , represents the measure of the angle
. Triangle POR is a right triangle.
formed by the x-axis and OP
Let r represent the length of the hypotenuse. Then the following
are true.
y
sin r
r
csc y
1
sin y
1
y 1 r
1
Notice that y
r
y
y
tan x
x
cot y
x
cos r
r
sec x
P (x, y)
r
O
y
r
r
y
r
y
1 1 or r
x
R
x
r
csc .
y
1
So, csc . This is known as one of the reciprocal identities .
sin Activity
Verify that cos2 sin2 1.
The expression cos2 means (cos )2. To verify an identity, work on only one side
of the equation and use what you know to show how that side is equivalent to the
other side.
cos2 sin2 1
y
xr2 r2 1
y2
x2
1
2
r
r2
x2 y2
1
r2
r2
2 1
r
11 Original equation
Substitute.
Simplify.
Combine fractions with like denominators.
Pythagorean Theorem: x2 y2 r2
Simplify.
Since 1 1, cos2 sin2 1.
Analyze
1. The identity cos2 sin2 1 is known as a Pythagorean identity . Why do
you think the word Pythagorean is used to name this?
1
cos 1
tan 2. Find two more reciprocal identities involving and .
Verify each identity.
sin 3. tan cos cos sin 4. cot 5. tan2 1 sec2 6. cot2 1 csc2 Geometry Activity Trigonometric Identities 391
Vocabulary and Concept Check
ambiguous case (p. 384)
angle of depression (p. 372)
angle of elevation (p. 371)
cosine (p. 364)
geometric mean (p. 342)
Law of Cosines (p. 385)
Law of Sines (p. 377)
Pythagorean identity (p. 391)
Pythagorean triple (p. 352)
reciprocal identities (p. 391)
sine (p. 364)
solving a triangle (p. 378)
tangent (p. 364)
trigonometric identity (p. 391)
trigonometric ratio (p. 364)
trigonometry (p. 364)
A complete list of postulates and theorems can be found on pages R1–R8.
Exercises State whether each statement is true or false. If false, replace the
underlined word or words to make a true sentence.
1. 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.
2. The tangent of A is the measure of the leg adjacent to A divided by the measure of the
leg opposite A.
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 elevation is the angle between the line of sight and the horizontal when an
observer looks upward.
5. The geometric mean between two numbers is the positive square root of their product.
6. In a 30°-60°-90° triangle, two of the sides will have the same length.
7. Looking at a city while flying in a plane is an example that uses angle of elevation.
7-1 Geometric Mean
See pages
342–348.
Concept Summary
• The geometric mean of two numbers is the square root of their product.
• You can use the geometric mean to find the altitude of a right triangle.
Examples 1
Find the geometric mean between 10 and 30.
x
10
30
x
x2 300
x 300
or 103
Definition of geometric mean
Cross products
Take the square root of each side.
2 Find NG in TGR.
T
The measure of the altitude is the geometric mean
between the measures of the two hypotenuse segments.
TN
GN
GN
RN
2
GN
GN
4
8 (GN)2
8 or 22 GN
392 Chapter 7 Right Triangles and Trigonometry
Definition of geometric mean
2
N
4
G
R
TN 2, RN 4
Cross products
Take the square root of each side.
www.geometryonline.com/vocabulary_review
Chapter 7 Study Guide and Review
Exercises
Find the geometric mean between each pair of numbers.
See Example 1 on page 342.
8. 4 and 16
9. 4 and 81
10. 20 and 35
12. In PQR, PS 8, and QS 14.
Find RS. See Example 2 on page 344.
11. 18 and 44
P
S
Q
R
7-2 The Pythagorean Theorem and Its Converse
See pages
350–356.
Example
Concept Summary
• The Pythagorean Theorem can be used to find the measures of the sides of
a right triangle.
• If the measures of the sides of a triangle form a Pythagorean triple, then
the triangle is a right triangle.
Find k.
a2 (LK)2 (JL)2
a2 82 132
a2 64 169
a2 105
a 105
a 10.2
Exercises
13.
15
J
Pythagorean Theorem
LK 8 and JL 13
13
a
Simplify.
Subtract 64 from each side.
K
Take the square root of each side.
L
8
Use a calculator.
Find x. See Example 2 on page 351.
13
14.
x
17
15.
5
17
x
13
x
21
20
7-3 Special Right Triangles
See pages
357–363.
Concept Summary
• The measure of the hypotenuse of a 45°-45°-90° triangle is 2 times the length
of the legs of the triangle. The measures of the sides are x, x, and x2.
• In a 30°-60°-90° triangle, the measures of the sides are x, x3, and 2x.
Examples 1
Find x.
The measure of the shorter leg XZ
of XYZ is half the
X
60˚
1
2
measure of the hypotenuse XY
. Therefore, XZ (26)
or 13. The measure of the longer leg is 3 times the
measure of the shorter leg. So, x 133.
Z
26
x
Y
Chapter 7 Study Guide and Review 393
Chapter 7 Study Guide and Review
2 Find x.
Q
The measure of the hypotenuse of a 45°-45°-90° triangle
4
is 2 times the length of a leg of the triangle.
x2 4
45˚
P
4
x 2
4
2
x or 22
2 2
Exercises
x
R
Find x and y. See Examples 1 and 3 on pages 358 and 359.
16.
17.
y
45˚
x
9
18.
x
13
30˚
x
y
45˚
60˚
y
x
6
y
60˚
For Exercises 19 and 20, use the figure at the right.
See Example 3 on page 359.
x
19. If y 18, find z and a.
20. If x 14, find a, z, b, and y.
b
z
30˚
a
7-4 Trigonometry
See pages
364–370.
Example
Concept Summary
• Trigonometric ratios can be used to find measures in right triangles.
Find sin A, cos A, and tan A. Express each
ratio as a fraction and as a decimal.
B
13
A
opposite leg
hypotenuse
BC
AB
5
or about 0.38
13
sin A adjacent leg
hypotenuse
AC
AB
12
or about 0.92
13
cos A 5
C
12
opposite leg
adjacent leg
BC
AC
5
or about 0.42
12
tan A Exercises 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. See Example 1 on page 365.
21. a 9, b 12, c 15
22. a 7, b 24, c 25
G
F
Find the measure of each angle to the nearest tenth of a degree.
See Example 4 on pages 366 and 367.
23. sin P 0.4522
394 Chapter 7 Right Triangles and Trigonometry
24. cos Q 0.1673
25. tan R 0.9324
c
a
b
H
Chapter 7 Study Guide and Review
7-5 Angles of Elevation and Depression
See pages
371–376.
Example
Concept Summary
• Trigonometry can be used to solve problems related to angles of elevation
and depression.
A store has a ramp near its front entrance. The ramp measures 12 feet, and has a
height of 3 feet. What is the angle of elevation?
Make a drawing.
B
12
Let x represent mBAC.
BC
AB
3
sin x° 12
opposite
hypotenuse
sin x° sin x A
3
x˚
C
BC 3 and AB 12
x sin-1 Find the inverse.
3
12
x 14.5
Use a calculator.
The angle of elevation for the ramp is about 14.5°.
Exercises
Determine the angles of elevation or depression in each situation.
See Examples 1 and 2 on pages 371 and 372.
26. An airplane must clear a 60-foot pole at the end of a runway 500 yards long.
27. An escalator descends 100 feet for each horizontal distance of 240 feet.
28. A hot-air balloon descends 50 feet for every 1000 feet traveled horizontally.
29. DAYLIGHT At a certain time of the day, the angle of elevation of the sun is 44°.
Find the length of a shadow cast by a building that is 30 yards high.
30. RAILROADS A railroad track rises 30 feet for every 400 feet of track. What is the
measure of the angle of elevation of the track?
7-6 The Law of Sines
See pages
377–383.
Example
Concept Summary
• To find the measures of a triangle by using the Law of Sines, you must
either know the measures of two angles and any side (AAS or ASA), or
two sides and an angle opposite one of these sides (SSA) of the triangle.
• To solve a triangle means to find the measures of all sides and angles.
Solve XYZ if mX 32, mY 61, and y 15. Round angle measures to the
nearest degree and side measures to the nearest tenth.
Find the measure of Z.
Y
mX mY mZ 180 Angle Sum Theorem
61˚
z
x
32 61 mZ 180 mX 32 and mY 61
93 mZ 180 Add.
mZ 87
Subtract 93 from each side.
X
32˚
y
Z
(continued on the next page)
Chapter 7 Study Guide and Review 395
• Extra Practice, see pages 766-768.
• Mixed Problem Solving, see page 788.
Since we know mY and y, use proportions involving sin Y and y.
To find x:
To find z:
sin Y
sin X
y
x
sin 61°
sin 32°
15
x
Substitute.
x sin 61° 15 sin 32°
Cross products
15 sin 32°
x sin 61°
x 9.1
sin Y
sin Z
y
z
sin 61°
sin 87°
15
z
Law of Sines
z sin 61° 15 sin 87°
15 sin 87°
sin 61°
z Divide.
z 17.1
Use a calculator.
Exercises Find each measure using the given measures of FGH. Round
angle measures to the nearest degree and side measures to the nearest tenth.
See Example 1 on page 378.
31. Find f if g 16, mG 48, and mF 82.
32. Find mH if h 10.5, g 13, and mG 65.
Solve each ABC described below. Round angle measures to the nearest
degree and side measures to the nearest tenth. See Example 2 on pages 378 and 379.
33. a 15, b 11, mA 64
34. c 12, mC 67, mA 55
35. mA 29, a 4.8, b 8.7
36. mA 29, mB 64, b 18.5
7-7 The Law of Cosines
See pages
385–390.
Example
Concept Summary
• The Law of Cosines can be used to solve triangles when you know the
measures of two sides and the included angle (SAS) or the measures of
the three sides (SSS).
Find a if b 23, c 19, and mA 54.
Since the measures of two sides and the included angle are
known, use the Law of Cosines.
Law of Cosines
a2 b c2 2bc cos A
a2 232 192 2(23)(19) cos 54°
b 23, c 19, and mA 54
a 232 192 2(23)(1
cos
54°
9)
Take the square root of each side.
a 19.4
Use a calculator.
B
19
A
a
54˚
23
Exercises In XYZ, given the following measures, find the measure of the
missing side. See Example 1 on page 385.
37. x 7.6, y 5.4, mZ 51
38. x 21, mY 73, z 16
Solve each triangle using the given information. Round angle measure to the
nearest degree and side measure to the nearest tenth.
See Example 3 on pages 386 and 387.
39. c 18, b 13, mA 64
396 Chapter 7 Right Triangles and Trigonometry
40. b 5.2, mC 53, c 6.7
C
Vocabulary and Concepts
1. State the Law of Cosines for ABC used to find mC.
2. Determine whether the geometric mean of two perfect squares will always be rational. Explain.
3. Give an example of side measures of a 30°-60°-90° triangle.
Skills and Applications
Find the geometric mean between each pair of numbers.
5. 6 and 24
4. 7 and 63
6. 10 and 50
Find the missing measures.
7.
9.
6
8.
7
x
9
x
x
5
12
10.
11.
y
12.
y
19
x
9
13
16
x
y˚
60˚
12
45˚
14. tan A
x
B
Use the figure to find each trigonometric ratio. Express answers
as a fraction.
13. cos B
8
21
15. sin A
A
Find each measure using the given measures from FGH.
Round to the nearest tenth.
16. Find g if mF 59, f 13, and mG 71.
16
15
C
17. Find mH if mF 52, f 10, and h 12.5.
18. Find f if g 15, h 13, and mF 48.
19. Find h if f 13.7, g 16.8, and mH 71.
Solve each triangle. Round each angle measure to the nearest degree and
each side measure to the nearest tenth.
20. a 15, b 17, mC 45
21. a 12.2, b 10.9, mB 48
22. a 19, b 23.2, c 21
23. 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?
24. 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?
25. STANDARDIZED TEST PRACTICE Find tan X.
SOL/EOC
Practice
A
5
12
B
12
13
C
17
12
www.geometryonline.com/chapter_test/sol
X
D
12
5
13
5
Chapter 7 Practice Test 397
SOL/EOC Practice
Part 1 Multiple Choice
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
4. In ABC, CD
is an altitude and mACB 90°.
If AD 12 and BD 3, find AC to the nearest
tenth. (Lesson 7-1)
C
1. If 4 and 3 are supplementary, which reason
could you use as the first step in proving that
1 and 2 are supplementary? (Lesson 2-7)
a
b
1
2
4
3
A
A
12
6.5
9.0
B
D 3
B
13.4
C
D
15.0
5. What is the length of R
T
? (Lesson 7-3)
R
A
Definition of similar angles
B
Definition of perpendicular lines
C
Definition of a vertical angle
D
Division Property
2. In ABC, BD
is a median. If AD 3x 5 and
CD 5x 1, find AC. (Lesson 5-1)
C
D
5 cm
135˚
T
A
5 cm
B
52 cm
C
53 cm
D
10 cm
6. An earthquake damaged a communication
tower. As a result, the top of the tower broke
off at a point 60 feet above the base. If the
fallen portion of the tower made a 36° angle
with the ground, what was the approximate
height of the original tower? (Lesson 7-4)
B
A
S
A
3
B
11
C
14
D
28
60 ft
36°
3. If pentagons ABCDE and PQRST are similar,
find SR. (Lesson 6-2)
A
11
E
D
2
3
C
B
95 ft
C
102 ft
D
162 ft
6
T
8
35 ft
P
B
14
A
Q
S
R
4
A
14
B
C
3
D
4
11
5
1
6
398 Chapter 7 Right Triangles and Trigonometry
7. Miraku drew a map showing Regina’s house,
Steve’s apartment, and Trina’s workplace.
The three locations formed RST, where
mR 34, r 14, and s 21. Which could
be mS? (Lesson 7-6)
A
15
B
22
C
57
D
84
Preparing for Standardized Tests
For test-taking strategies and more
practice, see pages 795– 810.
Part 2 Short Response/Grid In
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
8. Find mABC
if mCDA = 61.
A
69˚
(Lesson 1-6)
B x˚
Test-Taking Tip
Questions 6, 7, and 12
If a standardized test question involves trigonometric
functions, draw a diagram that represents the problem
and use a calculator (if allowed) or the table of
trigonometric relationships provided with the test to
help you find the answer.
55˚
C
D
For Questions 9 and 10, refer to the graph.
80
°F
12. Dee is parasailing at the ocean. The angle of
depression from her line of sight to the boat
is 41°. If the cable attaching Dee to the boat
is 500 feet long, how many feet is Dee above
the water? (Lesson 7-5)
60
(10, 50)
40
(0, 32)
41˚
20
500 ft
O
10
20
30 °C
9. At the International Science Fair, a Canadian
student recorded temperatures in degrees
Celsius. A student from the United States
recorded the same temperatures in degrees
Fahrenheit. They used their data to plot a
graph of Celsius versus Fahrenheit. What is
the slope of their graph? (Lesson 3-3)
10. Students used the equation of the line for
the temperature graph of Celsius versus
Fahrenheit to convert degrees Celsius to
degrees Fahrenheit. If the line goes through
points (0, 32) and (10, 50), what equation can
the students use to convert degrees Celsius
to degrees Fahrenheit? (Lesson 3-4)
Part 3 Extended Response
Record your answers on a sheet of paper.
Show your work.
13. Toby, Rani, and Sasha are practicing for a
double Dutch rope-jumping tournament.
Toby and Rani are standing at points T
and R and are turning the ropes. Sasha is
standing at S, equidistant from both Toby
and Rani. Sasha will jump into the middle
of the turning rope to point X. Prove that
when Sasha jumps into the rope, she will
be at the midpoint between Toby and Rani.
11. TUV and XYZ are similar. Calculate the
YZ
ratio . (Lesson 6-3)
(Lessons 4-5 and 4-6)
UV
S
T
X
10
V
6
U Z
Y
www.geometryonline.com/standardized_test/sol
T
X
R
Chapter 7 Standardized Test Practice 399
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