440 Chapter 7 SimilaritySimilar Polygons7-2Objective.docx

440 Chapter 7 Similarity
Similar Polygons7-2
Objective To identify and apply similar polygons
A movie theater screen is in the shape of
a rectangle 45 ft wide by 25 ft high.
Which of the TV screen formats at the
right do you think would show the most
complete scene from a movie shown on the
theater screen? Explain.
Similar fi gures have the same shape but not necessarily the
same size. You can
abbreviate is similar to with the symbol ,.
Essential Understanding You can use ratios and proportions to
decide whether
two polygons are similar and to fi nd unknown side lengths of
similar fi gures.
You write a similarity statement with corresponding vertices in
order, just as you write
a congruence statement. When three or more ratios are equal,
you can write an
extended proportion. Th e proportion ABGH 5
BC
HI 5

CD
IJ 5
AD
GJ is an extended proportion.
A scale factor is the ratio of corresponding linear measurements
of two similar fi gures. Th e ratio of the lengths of
corresponding
sides BC and YZ , or more simply stated, the ratio of
corresponding sides, is BCYZ 5
20
8 5
5
2. So the scale factor
of nABC to nXYZ is 52 or 5 : 2.
Key Concept Similar Polygons
Defi ne
Two polygons are
similar polygons if
corresponding angles are
congruent and if the
lengths of corresponding
sides are proportional.
Diagram
ABCD , GHIJ
Symbols
/A > /G
/B > /H

/C > /I
/D > /J
ABGH 5
BC
HI 5
CD
IJ 5
AD
GJ
CB
A D
IH
G J
C X
Y
Z
B
A
ABC XYZ
15 20

25
6 8
10
Dynamic Activity
Similar Polygons
A
C T I V I T I
E S
D
S
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC
Lesson
Vocabulary
• similar fi gures
• similar polygons
• extended

proportion
• scale factor
• scale drawing
• scale
L
V
L
V
• s
LL
VVV
• s
a
W
r
c
t
You learned about
ratios in the last
lesson. Can you use
ratios to help you
solve the problem?
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Problem 1

Got It?
Problem 2
Got It?
Lesson 7-2 Similar Polygons 441
Understanding Similarity
kMNP , kSRT
A What are the pairs of congruent angles?
/M > /S, /N > /R, and /P > /T
B What is the extended proportion for the ratios of
corresponding sides?
MNSR 5
NP
RT 5
MP
ST
1. DEFG , HJKL.
a. What are the pairs of congruent angles?
b. What is the extended proportion for the ratios of the lengths
of
Determining Similarity

Are the polygons similar? If they are, write a similarity
statement
and give the scale factor.
A JKLM and TUVW
Step 1 Identify pairs of congruent angles.
/J > /T, /K > /U, /L > /V, and /M > /W
Step 2 Compare the ratios of corresponding sides.
JK
TU 5
12
6 5
2
1
KL
UV 5
24
16 5
3
2
LMVW 5
24
14 5
12

7
JM
TW 5
6
6 5
1
1
Corresponding sides are not proportional, so the polygons are
not similar.
B kABC and kEFD
Step 1 Identify pairs of congruent angles.
/A > /D, /B > /E , and /C > /F
Step 2 Compare the ratios of corresponding sides.
ABDE 5
12
15 5
4
5
BC
EF 5
16
20 5
4

5
AC
DF 5
8
10 5
4
5
Yes; nABC , nDEF and the scale factor is 45 or 4 i 5.
2. Are the polygons similar? If they are, write a similarity
statement and give
the scale factor.
a. b.
R
ST
N
M P
M
L
J K T U
V
W

12
24
24 14
16
6 6
6
A B E F
DC
12 20
15 1016
8
K L
M Z Y
XW
N
10 20
1515 E
A B
R S

T
V U
CD
9
9
12
12
18
18
18
9
6
6
G
A
G
How can you use the
similarity statement
to write ratios of
Use the order of the

sides in the similarity
statement. MN
corresponds to SR ,
so MNSR is a ratio of
corresponding sides.
How do you identify
The included side
between a pair of
angles of one polygon
corresponds to the
included side between
the corresponding pair
of congruent angles of
another polygon.
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Problem 4
Got It?
Problem 3
Got It?

Using Similar Polygons
Algebra ABCD M EFGD. What is the value of x?
4.5 7.2
5 11.25
FGBC 5
ED
AD
Corresponding sides of similar
polygons are proportional.
x7.5 5
6
9 Substitute.
9x 5 45 Cross Products Property
x 5 5 Divide each side by 9.
Th e value of x is 5. Th e correct answer is B.
3. Use the diagram in Problem 3. What is the value of y?
Using Similarity
Design Your class is making a rectangular poster for a rally. Th
e poster’s design is
6 in. high by 10 in. wide. Th e space allowed for the poster is 4
ft high by 8 ft wide.
What are the dimensions of the largest poster that will fi t in the
space?

Step 1 Determine whether the height or width will fi ll the space
fi rst.
Height: 4 ft 5 48 in. Width: 8 ft 5 96 in.
48 in. 4 6 in. 5 8 96 in. 4 10 in. 5 9.6
Th e design can be enlarged at most 8 times.
Step 2 Th e greatest height is 48 in., so fi nd the width.
648 5
10
x Corresponding sides of similar polygons are proportional.
6x 5 480 Cross Products Property
x 5 80 Divide each side by 6.
Th e largest poster is 48 in. by 80 in. or 4 ft by 623 ft.
4. Use the same poster design in Problem 4. What are the
dimensions of the largest complete poster that will fi t in a
space
3 ft high by 4 ft wide?
D
E
A B
7.5
6
5

x
y F
G C
9
ACan you rely on the
diagram alone to set
up the proportion?
No, you need to use the
similarity statement to
identify corresponding
sides in order to write
ratios that are equal.
W
S
You can’t solve the
problem until you know
which dimension fi lls the
space fi rst.
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Problem 5
Got It?

G
Why is it helpful
to use a scale in
different units?
1 cm i 200 m in the
same units would be
1 cm i 20,000 cm. When
solving the problem, 1200
is easier to work with
than 120,000.
In a scale drawing, all lengths are proportional to their
corresponding actual lengths.
Th e scale is the ratio that compares each length in the scale
drawing to the actual
length. Th e lengths used in a scale can be in diff erent units.
For example, a scale might
be written as 1 cm to 50 km, 1 in. 5 100 mi, or 1 in. : 10 ft.
You can use proportions to fi nd the actual dimensions
represented in a scale drawing.
Using a Scale Drawing
Design Th e diagram shows a scale drawing of the Golden Gate
Bridge
in San Francisco. Th e distance between the two towers is the
main
span. What is the actual length of the main span of the bridge?
Th e length of the main span in the scale drawing is 6.4 cm. Let
s represent
the main span of the bridge. Use the scale to set up a

proportion.
1200 5
6.4
s
length in drawing (cm)
actual length (m)
s 5 1280 Cross Products Property
Th e actual length of the main span of the bridge is 1280 m.
5. a. Use the scale drawing in Problem 5. What is the actual
height of the towers
above the roadway?
b. Reasoning Th e Space Needle in Seattle is 605 ft tall. A
classmate wants
to make a scale drawing of the Space Needle on an 812 in.–by-
11 in. sheet
of paper. He decides to use the scale 1 in. 5 50 ft. Is this a
reasonable
scale? Explain.
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Lesson Check

Practice and Problem-Solving Exercises
List the pairs of congruent angles and the extended proportion
that
relates the corresponding sides for the similar polygons.
9. RSTV , DEFG 10. nCAB , nWVT 11. KLMNP , HGFDC
R
S
D
E
F
G
T
V
A B W
VT
C
K L
M
H G
F
D
C

N
P
Determine whether the polygons are similar. If so, write a
similarity
statement and give the scale factor. If not, explain.
12. B
A
C
E
F D
1515
9
9
9
15
13.
A
B
D

4
4 4
4 6 6
6 6
E
F
G
C
14. K
J L
R
Q
P
30
16
34 17
15
8
PracticeA See Problem 1.

See Problem 2.
Do you know HOW?
JDRT M WHYX . Complete each statement.
1. /D > 9 2. RTYX 5
j
WX
3. Are the polygons similar? If they are, write a similarity
statement and give the scale factor.
4. nFGH , nMNP. What is the value of x?
P N
M
G
F
H 20
15 12 x
10
Do you UNDERSTAND?
5. Vocabulary What does the scale on a scale drawing
indicate?
6. Error Analysis Th e polygons
at the right are similar. Which
similarity statement is not

correct? Explain.
A. TRUV , NPQU
B. RUVT , QUNP
7. Reasoning Is similarity refl exive? Transitive?
Symmetric? Justify your reasoning.
8. Th e triangles at the right are
similar. What are three similarity
statements for the triangles?
1624
18
24
12
R Q
L
PD
H G
E 12
8
16
N
U Q
P

R
TV
B
A R
P
S
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15. M
H T
A R U
LE
12
10
9
8
16.

S
18
14
15
24
T
H
E
L G
NI
17.
E
J
RAB L
15
18 24
25.4 20
80
43
70

6321.3
Algebra Th e polygons are similar. Find the value of each
variable.
18.
6
8
5
10
y
x
6
8
5
10
y
x
19.
12
9
6
3.5

6
y
z
x 20.
x
15
25.5
37
30
y
21. Web Page Design Th e space allowed for the mascot on a
school’s Web page is
120 pixels wide by 90 pixels high. Its digital image is 500
pixels wide by 375 pixels
high. What is the largest image of the mascot that will fi t on
the Web page?
22. Art Th e design for a mural is 16 in. wide and 9 in. high.
What are the dimensions
of the largest possible complete mural that can be painted on a
wall 24 ft wide by
14 ft high?
23. Architecture You want to make a scale drawing of New
York City’s Empire
State Building using the scale 1 in. 5 250 ft. If the building is
1250 ft tall,
how tall should you make the building in your scale drawing?

24. Cartography A cartographer is making a map of
Pennsylvania. She uses the scale
1 in. = 10 mi. Th e actual distance between Harrisburg and
Philadelphia is about
95 mi. How far apart should she place the two cities on the
map?
In the diagram below, kDFG M kHKM . Find each of the
following.
25. the scale factor of nHKM to nDFG 26. m/K
27. GDMH 28. MK 29. GD
30. Flags A company produces a standard-size U.S. fl ag that is
3 ft
by 5 ft. Th e company also produces a giant-size fl ag that is
similar to the standard-size fl ag. If the shorter side of the
giant-size fl ag is 36 ft, what is the length of its longer side?
31. a. Coordinate Geometry What are the measures of /A, /ABC
, /BCD,
/CDA, /E , /F , and /G? Explain.
b. What are the lengths of AB, BC , CD, DA, AE , EF , FG, and
AG?
c. Is ABCD similar to AEFG? Justify your answer.
See Problem 3.
See Problem 4.
See Problem 5.
ApplyB

30
27.5
15
18
59
70
D F H K
M
G
y
O
x
6
2
1 4 6
E
F
G
C

D
A
B
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32. Think About a Plan Th e Davis family is planning to
drive from San Antonio to Houston. About how far will
they have to drive?
• How can you fi nd the distance between the two cities
on the map?
• What proportion can you set up to solve the problem?
33. Reasoning Two polygons have corresponding side
lengths that are proportional. Can you conclude that
the polygons are similar? Justify your reasoning.
34. Writing Explain why two congruent fi gures must also be
similar. Include scale factor in your explanation.
35. nJLK and nRTS are similar. Th e scale factor of nJLK to
nRTS is 3 i 1. What is the scale factor of nRTS to nJLK ?
36. Open-Ended Draw and label two diff erent similar
quadrilaterals. Write a similarity
statement for each and give the scale factor.

Algebra Find the value of x. Give the scale factor of the
polygons.
37. nWLJ , nQBV 38. GKNM , VRPT
J
L
W x 6
x
B
VQ 8
5 M
G
K
R
P
T
VN
3
3
4

4 3x 2
x 4
8.4
6.3
Sports Choose a scale and make a scale drawing of each
rectangular playing surface.
39. A soccer fi eld is 110 yd by 60 yd. 40. A volleyball court is
60 ft by 30 ft.
41. A tennis court is 78 ft by 36 ft. 42. A football fi eld is 360
ft by 160 ft.
Determine whether each statement is always, sometimes, or
never true.
43. Any two regular pentagons are similar. 44. A hexagon and a
triangle are similar.
45. A square and a rhombus are similar. 46. Two similar
rectangles are congruent.
47. Architecture Th e scale drawing at the right is part of a fl
oor plan for a home.
Th e scale is 1 cm 5 10 ft. What are the actual dimensions of the
family room?
48. Th e lengths of the sides of a triangle are in the extended
ratio 2 i 3 i 4. Th e
perimeter of the triangle is 54 in.
a. Th e length of the shortest side of a similar triangle is 16 in.

What are the lengths
of the other two sides of this triangle?
b. Compare the ratio of the perimeters of the two triangles to
their scale factor.
What do you notice?
Master
bedroom
Family
room
Dining
Kitchen
ChallengeC
San Antonio Galveston
Corpus
Christi
Brownsville
Laredo
Scale
1 cm : 112 km
Del Rio
Austin
Houston

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49. In rectangle BCEG, BC i CE = 2 i 3. In rectangle LJAW, LJ
i JA = 2 i 3. Show that
BCEG , LJAW.
50. Prove the following statement: If nABC , nDEF and nDEF
, nGHK , then
nABC , nGHK.
Mixed Review
If x7 5
y
9, complete each statement using the properties of proportions.
55. 9x 5 j 56. xy 5
j
j
57. x 1 77 5
j
j
Use the diagram for Exercises 58–61.
58. Name the isosceles triangles in the fi gure.
59. CD > 9 > 9
60. AE 5 9 61. m/A 5 9

Get Ready! To prepare for Lesson 7-3, do Exercises 62–64.
How can you prove that the triangles are congruent?
62. 63. 64.
See Lesson 7-1.
See Lesson 4-5.A
C
B
D E
F
5
42
3
See Lessons 4-2 and 4-3.
Standardized Test Prep
51. PQRS , JKLM with a scale factor of 4 i 3. QR 5 8 cm.
What is the value of KL?
6 cm 8 cm 1023 cm 24 cm
52. Which of the following is NOT a property of an isosceles
trapezoid?
Th e base angles are congruent. Th e diagonals are

perpendicular.
Th e legs are congruent. Th e diagonals are congruent.
53. In the diagram at the right, what is m&1?
45 75 125 135
54. A high school community-action club plans to build a
circular play area in a city
park. Th e club members need to buy materials to enclose the
area and sand to fi ll
the area. For a 9-ft-diameter play area, what will be the
circumference and area
rounded to the nearest hundredth?
SAT/ACT
60
75 1Short
Response
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copyright:
Lesson 8-3 Trigonometry 507
8-3 Trigonometry

Objective To use the sine, cosine, and tangent ratios to
determine side lengths and
angle measures in right triangles
What is the ratio of the length of the shorter leg to the length of
the
hypotenuse for each of kADF, kAEG, and kABC? Make a
conjecture
based on your results.
Essential Understanding If you know certain combinations of
side lengths
and angle measures of a right triangle, you can use ratios to fi
nd other side lengths and
angle measures.
Any two right triangles that have a pair of congruent acute
angles are similar by
the AA Similarity Postulate. Similar right triangles have
equivalent ratios for their
corresponding sides called trigonometric ratios.
B
E
D
A
F G C62 4
4
Dynamic Activity
Trigonometric
Ratios

T
A
C
T I V I T I
E
S
D
TT
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC
Lesson
Vocabulary
• trigonometric
ratios
• sine
• cosine
• tangent
L

V
L
V
• t
LL
VVV
• t
Key Concept Trigonometric Ratios
sine of /A 5
length of leg opposite /A
length of hypotenuse
5
a
c
cosine of /A 5
length of leg adjacent to /A
length of hypotenuse
5 bc
tangent of /A 5
length of leg opposite /A
length of leg adjacent to /A
5
a
b

A
B
C
c
a
b
Here are ratios
in triangles once
again! This must
be “similar” to
something you’ve
seen before.
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5º
150 ft
Problem 1
Got It?
Problem 2
508 Chapter 8 Right Triangles and Trigonometry

You can abbreviate the ratios as
sin A 5
opposite
hypotenuse
, cos A 5
adjacent
hypotenuse
, and tan A 5
opposite
adjacent
.
Writing Trigonometric Ratios
What are the sine, cosine, and tangent ratios for lT ?
sin T 5
opposite
hypotenuse
5
8
17
cos T 5
adjacent
hypotenuse
5

15
17
tan T 5
opposite
adjacent
5
8
15
1. Use the triangle in Problem 1. What are the sine, cosine, and
tangent ratios
for /G?
In Chapter 7, you used similar triangles to measure distances
indirectly.
You can also use trigonometry for indirect measurement.
Using a Trigonometric Ratio to Find Distance
Landmarks In 1990, the Leaning Tower of
Pisa was closed to the public due to safety
concerns. Th e tower reopened in 2001
after a 10-year project to reduce its tilt from
vertical. Engineers’ eff orts were successful
and resulted in a tilt of 58, reduced from 5.58.
Suppose someone drops an object from the
tower at a height of 150 ft. How far from the base
of the tower will the object land? Round to the
nearest foot.
Th e given side is adjacent to the given angle. Th e
side you want to fi nd is opposite the given angle.

tan 58 5
x
150 Use the tangent ratio.
x 5 150(tan 58) Multiply each side by 150.
150 tan 5 enter Use a calculator.
x < 13.12329953
Th e object will land about 13 ft
from the base of the tower.
17
8
G
RT 15
G
W
G
How do the sides
relate to lT ?
GR is across from, or
opposite, /T . TR is next
to, or adjacent to, /T .
TG is the hypotenuse
because it is opposite the
908 angle.

n
Th
s
What is the fi rst
step?
Look at the triangle and
determine how the sides
of the triangle relate to
the given angle.
tml
Got It?
Problem 3
Got It?
G
When should you use
an inverse?
Use an inverse when you
know two side lengths of
a right triangle and you
want to fi nd the measure
of one of the acute

angles.
2. For parts (a)–(c), fi nd the value of w to the nearest tenth.
a. b. c.
d. A section of Filbert Street in San Francisco rises at an angle
of about 178.
If you walk 150 ft up this section, what is your vertical rise?
Round to the
nearest foot.
If you know the sine, cosine, or tangent ratio for an angle, you
can use an inverse
(sin21, cos21, or tan21) to fi nd the measure of the angle.
Using Inverses
What is mlX to the nearest degree?
A B
You know the lengths of the
hypotenuse and the side
opposite /X .
Use the sine ratio.
sin X 5
6
10 Write the ratio.
m/X 5 sin21 Q
6

10R Use the inverse.
sin–1 6 10 enter Use a calculator.
m/X < 36.86989765
< 37
3. a. Use the fi gure at the right. What is m/Y to
the nearest degree?
b. Reasoning Suppose you know the lengths
of all three sides of a right triangle. Does it
matter which trigonometric ratio you use to
fi nd the measure of any of the three angles? Explain.
w17
54
w
28
1.0
33
4.5
w
H
B
6 10

X
15
M
X
N
20
P
Y
T
100
41
You know the lengths of the
hypotenuse and the side
adjacent to /X .
Use the cosine ratio.
cos X 5
15
20
m/X 5 cos21Q
15
20R
cos–1 15 20 enter

m/X < 41.40962211
< 41
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Lesson Check
Practice and Problem-Solving Exercises
Write the ratios for sin M, cos M, and tan M.
11. 12. 13.
Find the value of x. Round to the nearest tenth.
14. 15. 16.
17. 18. 19.
20. Recreation A skateboarding ramp is 12 in. high and rises at
an angle of 178. How
long is the base of the ramp? Round to the nearest inch.
21. Public Transportation An escalator in the subway station
has a vertical rise of
195 ft 9.5 in., and rises at an angle of 10.48. How long is the
escalator? Round to
the nearest foot.

PracticeA See Problem 1.
7
25
24
L
K
M
4 V2
7M K
L
9
K L
M
4
2
2 V3
See Problem 2.
35
20

x
41
11
x
64
7 x
x
36
10 62
28
50x x
10
25
Do you know HOW?
Write each ratio.
1. sin A 2. cos A
3. tan A 4. sin B
5. cos B 6. tan B
What is the value of x? Round to the nearest tenth.
7. 8.
Do you UNDERSTAND?

9. Vocabulary Some people use SOH-CAH-TOA to
remember the trigonometric ratios for sine, cosine,
and tangent. Why do you think that word might help?
(Hint: Th ink of the fi rst letters of the ratios.)
10. Error Analysis A student states that sin A . sin X
because the lengths of the sides of nABC are greater
than the lengths of the sides of nXYZ . What is the
student’s error? Explain.
35
C
35
Y
Z X
B
A
A
C B
10
8
6
39
15

x
27
32x
tml
Find the value of x. Round to the nearest degree.
22. 23. 24.
25. 26. 27.
28. Th e lengths of the diagonals of a rhombus are 2 in. and 5
in. Find the measures of
the angles of the rhombus to the nearest degree.
29. Think About a Plan Carlos plans to build a grain bin with a
radius of 15 ft. Th e recommended slant of the roof is 258. He
wants the roof to overhang the edge of the bin by 1 ft. What
should the length x be? Give your answer in feet and inches.
• What is the position of the side of length x in relation to
the given angle?
• What information do you need to fi nd a side length of a
right triangle?
• Which trigonometric ratio could you use?

An identity is an equation that is true for all the allowed values
of the variable.
Use what you know about trigonometric ratios to show that each
equation is
an identity.
30. tan X 5 sin Xcos X 31. sin X 5 cos X ? tan X 32. cos X 5
sin X
tan X
Find the values of w and then x. Round lengths to the nearest
tenth and angle
measures to the nearest degree.
33. 34. 35.
36. Pyramids All but two of the pyramids built by the
ancient Egyptians have faces inclined at 528 angles.
Suppose an archaeologist discovers the ruins of a
pyramid. Most of the pyramid has eroded, but the
archaeologist is able to determine that the length of
a side of the square base is 82 m. How tall was the
pyramid, assuming its faces were inclined at 528?
Round your answer to the nearest meter.
See Problem 3.
5
x
14
5
x

8
x
9
13
3.0
5.8x 17
41
x
0.15
0.34
x
ApplyB
1 ft
over-
hang
x
15 ft
25
6
4x w
30

10
xw
56 34
102 102
x
w
42
52
82 m82 m
52
tml
37. a. In nABC at the right, how does sin A compare to
cos B? Is this true for the acute angles of other
right triangles?
b. Reading Math Th e word cosine is derived from the
words complement’s sine. Which angle in nABC is
the complement of /A? Of /B?
c. Explain why the derivation of the word cosine makes sense.

38. For right nABC with right /C , prove each of the following.
a. sin A , 1
b. cos A , 1
39. a. Writing Explain why tan 608 5 !3. Include a diagram
with your explanation.
b. Make a Conjecture How are the sine and cosine of a 608
angle related? Explain.
Th e sine, cosine, and tangent ratios each have a reciprocal
ratio. Th e reciprocal ratios are cosecant (csc), secant (sec),
and cotangent (cot). Use kABC and the defi nitions below to
write each ratio.
csc X 5 1sin X sec X 5
1
cos X cot X 5
1
tan X
40. csc A 41. sec A 42. cot A
43. csc B 44. sec B 45. cot B
46. Graphing Calculator Use the table feature of your graphing
calculator to study
sin X as X gets close to (but not equal to) 90. In the y= screen,
enter Y1 5 sin X .
a. Use the tblset feature so that X starts at 80 and changes by
1. Access the table .
From the table, what is sin X for X 5 89?

b. Perform a “numerical zoom-in.” Use the tblset feature, so
that X starts with 89
and changes by 0.1. What is sin X for X 5 89.9?
c. Continue to zoom-in numerically on values close to 90. What
is the greatest
value you can get for sin X on your calculator? How close is X
to 90? Does your
result contradict what you are asked to prove in Exercise 38a?
d. Use right triangles to explain the behavior of sin X found
above.
47. a. Reasoning Does tan A 1 tan B 5 tan (A 1 B) when A 1 B
, 90? Explain.
b. Does tan A 2 tan B 5 tan (A 2 B) when A 2 B . 0? Use part
(a) and indirect
reasoning to explain.
Verify that each equation is an identity by showing that each
expression on the
left simplifi es to 1.
48. (sin A)2 1 (cos A)2 5 1 49. (sin B)2 1 (cos B)2 5 1
50. 1
(cos A) 2
2 (tan A)2 5 1 51. 1
(sin A) 2
2
1
(tan A) 2

5 1
52. Show that (tan A)2 2 (sin A)2 5 (tan A)2 ? (sin A)2 is
an identity.
A
B
C
34
30
16
Proof
A
BC
15
12
9
ChallengeC
B
C
a c

b A
tml
53. Astronomy Th e Polish astronomer Nicolaus Copernicus
devised
a method for determining the sizes of the orbits of planets
farther
from the sun than Earth. His method involved noting the number
of days between the times that a planet was in the positions
labeled
A and B in the diagram. Using this time and the number of days
in
each planet’s year, he calculated c and d.
a. For Mars, c 5 55.2 and d 5 103.8. How far is Mars from the
sun
in astronomical units (AU)? One astronomical unit is defi ned
as
the average distance from Earth to the center of the sun, about
93 million miles.
b. For Jupiter, c 5 21.9 and d 5 100.8. How far is Jupiter from
the
sun in astronomical units?
d˚
c̊

Sun
Not to scale
AA
B
B
1 AU1 AU
Ea
rth’s orbit
Ou
ter p
lanet’s orbit
Mixed Review
57. Th e length of the hypotenuse of a 308-608-908 triangle is
8. What are the
lengths of the legs?
58. A diagonal of a square is 10 units. Find the length of a side
of the square.
Express your answer in simplest radical form.
Get Ready! To prepare for Lesson 8-4, do Exercises 59–62.
Use rectangle ABCD to complete each statement.
59. /1 > 9

60. /5 > 9
61. /3 > 9
62. m/1 1 m/5 5 9
See Lesson 8-2.
See Lessons 3-2 and 6-4.
C
A B
D
10
11
1 6 8
573
Standardized Test Prep
54. Grove Street has a grade of 20%. Th at means that the street
rises
20 ft for every 100 ft of horizontal distance. To the nearest
tenth,
at what angle does Grove Street rise?
11.38 78.58
11.58 78.78
55. Which of the following fi gures is NOT a parallelogram?

square trapezoid rhombus rectangle
56. In nABC, AB . BC . AC . One angle has a measure of 168.
What are all the
possible whole-number values for the measure of /A? Explain.
20 ft
100 ft
Grove
StreetSAT/ACT
Short
Response
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