LESSON
4.1
?
Similar Shapes and
Proportions
Proportionality—
7.5.A Generalize the
critical attributes of similarity,
including ratios within and
between similar shapes.
ESSENTIAL QUESTION
How can you use ratios to determine if two figures are similar?
EXPLORE ACTIVITY
7.5.A
Similar Shapes and Proportions
Similar shapes have the same shape but not necessarily the same size. You can
use square tiles to model similar figures.
A rectangle made of square tiles measures 5 tiles long and 2 tiles wide.
Find the length of a similar rectangle that measures 6 tiles wide.
STEP 1
Using the square tiles, make a rectangle 5 tiles long and
2 tiles wide.
STEP 2
Add tiles to increase the width of the rectangle to 6 tiles.
There are now
sets of the original
The width of the rectangle is
original rectangle.
STEP 3
2
2
tiles along
×
the width of the rectangle because
5
2
= 6.
2
times the width of the
5
Add tiles to also increase the length of
the rectangle.
5
5
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2
The width of the new rectangle is
times the
width of the original. To keep the lengths of the sides 2
proportional, the length must also be
times
the length of the original. The length of the similar
rectangle is
×
2
= 15.
The length of the similar rectangle is
tiles.
Reflect
1.
Justify Reasoning If one dimension is changed, why does the other
dimension have to change to create a similar figure?
Lesson 4.1
115
Determining Whether Two Triangles
Are Similar
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Similar shapes have the same shape, but not necessarily the same size.
Corresponding angles and corresponding sides of two or more similar
shapes are in the same relative position.
B
The symbol ~ means “is similar to.” In
the figure shown, sides and angles that
are the same color correspond to each A
other. △ABC ~ △DEF.
E
C
D
F
Similar Figures
In two similar figures:
• the measures of their corresponding angles are equal, and
• the lengths of their corresponding sides are proportional.
EXAMPLE 1
7.5.A
STEP 1
Math Talk
Mathematical Processes
Are all equilateral
triangles similar?
Explain.
STEP 2
S
34˚
F
34˚ 12 in.
36 in.
40˚ 24 in.
8 in.
E
D
7 in.
106˚ 40˚
106˚
R
21 in.
Check that the
corresponding angles of
the triangles have equal
measures.
m∠E = m∠R = 106 °
∠E corresponds to ∠R
m∠F = m∠S = 34 °
∠F corresponds to ∠S
m∠D = m∠Q = 40 °
∠D corresponds to ∠Q
Q
Check that the corresponding side lengths are proportional.
_
_
_
_
DE corresponds to QR.
_
_
EF corresponds to RS.
DF corresponds to QS.
Write ratios using the lengths
? ___
DE ? __
DF
___
= EF =
QR RS QS
of corresponding sides.
8 ? __
7 ? __
__
= 24
= 12
21
36
1 ? _
? _1 ✓
_
= 13 =
3
3
A side of a figure
can be named by
its endpoints
with a bar above.
Substitute the lengths of the sides.
Simplify.
Since the measures of the corresponding angles are equal and the
corresponding sides are proportional, the triangles are similar.
116
Unit 2
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Explain whether the triangles are similar.
YOUR TURN
Explain whether the triangles are similar.
2.
F
99˚C
9 ft
5 ft
B
A
11 ft
27˚
54˚ E
17 ft
29˚
102˚ 11 ft
49˚
D
22 ft
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33˚
18˚
10 in.
3. A
C
129
˚
4 in.
7 in.
B
E
16 in.
129˚
D
33˚
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and Intervention
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28 in.
40 in.
F
18˚
Determining Whether Two Four-Sided
Figures Are Similar
Shapes with four or more sides can also be similar if the corresponding angles
have equal measures and the corresponding side lengths are proportional.
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EXAMPL 2
EXAMPLE
7.5.A
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Diana makes two sizes of earrings.
Explain whether the shapes of
the earrings are similar.
101˚
O 101˚
S
18 mm
10 mm
10.8 mm
6 mm
48
˚
R
T 48˚
P
86˚ N
6 mm
9
mm
10 mm
15 mm
86˚ Q
M
125˚
125˚
STEP 1
Check that the
corresponding angles have equal measures. The angle
measures are 48°, 86°, 101°, and 125° in both earrings.
STEP 2
Check that the corresponding side lengths are proportional.
_
_
_
_
MN corresponds to QR
Note that
each ratio
in simplest
5.
form is __
3
OP corresponds to ST
OP ? ___
? ___
MN ? ___
___
= NO
= = PM
RS
ST
QR
TQ
? ____
10 ? __
18 ? __
__
= 10
=
= 15
6
6
10.8
9
_
_
_
_
_
_
_
Animated
Math
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NO corresponds to RS
PM corresponds to TQ
Write ratios using the lengths of
corresponding sides.
Substitute the lengths of the sides.
_
1.6 = 1.6 = 1.6 = 1.6
Divide.
The measures of the corresponding angles are equal and the
corresponding sides are proportional, so the earrings are similar.
Lesson 4.1
117
YOUR TURN
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Online Assessment
and Intervention
Explain whether the shapes are similar.
4.
rectangle ABCD with sides of 7 and 5 and rectangle MNOP with sides of
21 and 15
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5.
8 ft
46˚ 54˚
6 ft
6 ft
150˚
3 ft
110˚
48 ft
54˚
46˚
6 ft 150
110˚ 6 ft
˚
18 ft
Guided Practice
1. A rectangle made of square tiles measures 7 tiles long and 3 tiles
wide. What is the length of a similar rectangle whose width is 9 tiles?
(Explore Activity)
Explain whether the shapes are similar. (Examples 1 and 2)
?
?
40˚ K
18 cm 18 cm
70˚
70˚
J
L
12 cm
Q
34˚
48 cm
48 cm
73˚
73˚
P
R
28 cm
3.
21 ft
90˚
14 ft
90˚ 90˚
14 ft
21 ft
14 ft
90˚ 90˚
14 ft
ESSENTIAL QUESTION CHECK-IN
4. Describe how to use ratios to determine whether two shapes are similar.
118
Unit 2
90˚
21 ft
90˚
90˚
21 ft
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2.
Name
Class
Date
4.1 Independent Practice
Personal
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7.5.A
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Assessment and
Intervention
Determine if each statement is true or false. Justify your answer.
5. All squares are similar.
6. All right triangles are similar.
Art For 7–10, use the table. Assume all angle measures are
equal to 90°.
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7. Hugo has a small print of one of the paintings in the table.
It is similar in size to the original. The print measures
11 in. × 10 in. Of which painting is this a print? Explain.
Artist
Original
Size (in.)
Mona
Lisa
Leonardo
da Vinci
30 x 21
The Dance
Class
Edgar
Degas
33 x 30
The Blue
Vase
Paul
Cézanne
28 x 18
Painting
8. A local artist painted a copy of Cezanne’s painting. It measures
88 in. × 74 in. Is the copy similar to the original? Explain.
9. A company made a poster of da Vinci’s painting. The poster is 5 feet long and
3.5 feet wide. Is the poster similar to the original Mona Lisa? Explain.
10. The same company made a poster of The Blue Vase. The poster is 36 inches
long and 26 inches wide. Is the poster similar to the original The Blue Vase?
Explain.
Lesson 4.1
119
Problem Solving The figure shows a 12 ft by 15 ft garden divided into
four rectangular parts, each planted with a different vegetable. Explain
whether the rectangles in each pair are similar and why.
12 ft
A 5 ft B
11. rectangle A and the original rectangle
4 ft
C
15 ft
D
12. the original rectangle and rectangle D
13. rectangle C and rectangle B
FOCUS ON HIGHER ORDER THINKING
Work Area
14. Analyze Relationships Which of these four-sided shapes are similar?
10 cm
5 cm
T
R
5 cm
10 cm
S
A
4 cm
D
8 cm
8 cm
B
4 cm
C
W
10 cm
X
5 cm
Z
5 cm
10 cm
Y
15. Communicate Mathematical Ideas Describe the two tests two polygons
must pass to be proven similar.
16. Make a Conjecture Using what you know of similar figures, explain
whether you believe all rectangles are similar. Give an example or a
counterexample.
120
Unit 2
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Q