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5 5 5
Similar Shapes and ProportionsSimilar 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.
Using the square tiles, make a rectangle 5 tiles long and
2 tiles wide.
Add tiles to increase the width of the rectangle to 6 tiles.
Add tiles to also increase the length of
the rectangle.
Reflect1. Justify Reasoning If one dimension is changed, why does the other
dimension have to change to create a similar figure?
STEP 1
STEP 2
STEP 3
There are now sets of the original tiles along
the width of the rectangle because × = 6.
The width of the rectangle is times the width of the
original rectangle.
The width of the new rectangle is times the
width of the original. To keep the lengths of the sides
proportional, the length must also be times
the length of the original. The length of the similar
rectangle is × = 15.
The length of the similar rectangle is tiles.
L E S S O N
4.1Similar Shapes and Proportions
ESSENTIAL QUESTIONHow can you use ratios to determine if two figures are similar?
EXPLORE ACTIVITY 7.5.A
Proportionality—7.5.A Generalize the critical attributes of similarity, including ratios within and between similar shapes.
115Lesson 4.1
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12 in.8 in.
7 in.
34˚
106˚
40˚E
F
D
34˚
40˚
24 in.36 in.
21 in.
106˚R Q
S
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A C
B
D F
E
Determining Whether Two Triangles Are Similar 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.
The symbol ~ means “is similar to.” In
the figure shown, sides and angles that
are the same color correspond to each
other. △ABC ~ △DEF.
? ?
??
? ?
Explain whether the triangles are similar.
Check that the
corresponding angles of
the triangles have equal
measures.
m∠E = m∠R = 106 °
m∠F = m∠S = 34 °
m∠D = m∠Q = 40 °
Check that the corresponding side lengths are proportional._
DE corresponds to _
QR . _
EF corresponds to _
RS .
_
DF corresponds to _
QS .
DE ___ QR = EF __ RS = DF ___ QS
7 __
21 = 8 __
24 = 12
__ 36
1 _
3 = 1 _
3 = 1 _
3 ✓
Since the measures of the corresponding angles are equal and the
corresponding sides are proportional, the triangles are similar.
EXAMPLE 1
STEP 1
STEP 2
Math TalkMathematical Processes
7.5.A
Are all equilateral triangles similar?
Explain.
∠E corresponds to ∠R
∠F corresponds to ∠S
∠D corresponds to ∠Q
Write ratios using the lengths of corresponding sides.
Substitute the lengths of the sides.
Simplify.
A side of a figure can be named by its endpoints with a bar above.
Similar Figures
In two similar figures:
• the measures of their corresponding angles are equal, and
• the lengths of their corresponding sides are proportional.
Unit 2116
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129̊A C
B
10 in.
7 in.4 in.
33̊ 18̊
129̊
E
F
D16 in.
28 in.
40 in.33̊
18̊
102̊
49̊29̊
22 ft
11 ft17 ft
E D
F99̊
9 ft 5 ft
11 ftAB
C
27̊ 54̊
101̊
86̊
86̊
48̊
125̊
O
N P
M Q
R
S
T
101̊
125̊
48̊
18 mm
15 mm6 mm
6 mm 10.8 mm
9 mm10 mm
10 mm
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Explain whether the triangles are similar.
2. 3.
YOUR TURN
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.
Diana makes two sizes of earrings.
Explain whether the shapes of
the earrings are similar.
Check that the
corresponding angles have equal measures. The angle
measures are 48°, 86°, 101°, and 125° in both earrings.
Check that the corresponding side lengths are proportional._
MN corresponds to _
QR _
NO corresponds to _
RS _
OP corresponds to _
ST _
PM corresponds to _
TQ
MN ___ QR = NO ___ RS = OP ___ ST = PM ___ TQ
10 __ 6
= 10 __ 6
= 18 ____
10.8 = 15 __
9
1. _
6 = 1. _
6 = 1. _
6 = 1. _
6
The measures of the corresponding angles are equal and the
corresponding sides are proportional, so the earrings are similar.
EXAMPLEXAMPLE 2
STEP 1
STEP 2
? ? ?
???
7.5.A
Write ratios using the lengths of corresponding sides.
Substitute the lengths of the sides.
Divide.
Note that each ratio in simplest form is 5 __ 3 .
117Lesson 4.1
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48 ft
18 ft
6 ft6 ft 150˚ 110˚
54˚46˚
8 ft
3 ft
6 ft 6 ft
110˚
150˚
46˚ 54˚
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12 cm
18 cm18 cm70̊
40̊
70̊J
K
L
48 cm48 cm
28 cm
73̊73̊
Q
RP
34̊
21 ft
21 ft 21 ft
21 ft
90˚ 90˚
90˚ 90˚
14 ft
14 ft
14 ft 14 ft
90˚ 90˚
90˚ 90˚
Explain whether the shapes are similar.
4. rectangle ABCD with sides of 7 and 5 and rectangle MNOP with sides of
21 and 15
5.
YOUR TURN
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)
2. 3.
4. Describe how to use ratios to determine whether two shapes are similar.
ESSENTIAL QUESTION CHECK-IN??
Unit 2118
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Painting
MonaLisa
The DanceClass
The BlueVase
Artist
Leonardoda Vinci
Edgar Degas
PaulCézanne
OriginalSize (in.)
30 x 21
33 x 30
28 x 18
Name Class Date
Independent Practice4.1
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°.
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.
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.
7.5.A
119Lesson 4.1
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Work Area
A B
C D
4 ft15 ft
12 ft
5 ft
T S
RQ
10 cm
5 cm5 cm
10 cm
5 cm5 cm
10 cm
10 cmW X
YZ4 cm4 cm
D C
BA
8 cm
8 cm
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.
11. rectangle A and the original rectangle
12. the original rectangle and rectangle D
13. rectangle C and rectangle B
14. Analyze Relationships Which of these four-sided shapes are similar?
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.
FOCUS ON HIGHER ORDER THINKING
Unit 2120
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