1 1
1
P = 3 P = 6 P =
P =
A = B A = 4B A =
A =
1
1
11
200 Chapter 5 Similarity and Transformations
STATE
STANDARDS
MA.7.G.4.1
S
Perimeters and Areas
of Similar Figures
How do changes in dimensions of similar
geometric fi gures affect the perimeters and areas of the fi gures?
ACTIVITY: Comparing Perimeters and Areas
1
1
Work with a partner. Use pattern blocks to make a fi gure whose dimensions
are 2, 3, and 4 times larger than those of the original fi gure. Find the
perimeter P and area A of each larger fi gure.
a. Sample: Square
b. Triangle
c. Rectangle d. Parallelogram
2
1
2
1
1
1
1
1
P = 6 P = 4
A = 2 A = C
P = 4 P = 8 P = 12 P = 16
A = 1 A = 4 A = 9 A = 16
5.2
English
Spanish
Section 5.2 Perimeters and Areas of Similar Figures 201
4. IN YOUR OWN WORDS How do changes in dimensions of similar
geometric fi gures affect the perimeters and areas of the fi gures?
Use what you learned about perimeters and areas of similar fi gures
to complete Exercises 8 –11 on page 204.
Figure
Original Side
Lengths
Double Side
Lengths
Triple Side
Lengths
Quadruple
Side Lengths
Perimeters
P = 4 P = 8 P = 12 P = 16
P = 3 P = 6
P = 6
P = 4
Figure
Original
Side Lengths
Double Side
Lengths
Triple Side
Lengths
Quadruple
Side Lengths
Areas
A = 1 A = 4 A = 9 A = 16
A = BA = 4B
A = 2
A = C
Work with a partner. Copy and complete the table for the areas of the
fi gures in Activity 1. Describe the pattern.
ACTIVITY: Finding Patterns for Areas
3
3
ACTIVITY: Finding Patterns for Areas
3
3
Work with a partner. Copy and complete the table for the perimeters of
the fi gures in Activity 1. Describe the pattern.
ACTIVITY: Finding Patterns for Perimeters
2
2
English
Spanish
202 Chapter 5 Similarity and Transformations
Lesson
5.2
Perimeters of Similar Figures
If two fi gures are similar, then the ratio
E
D
F
B
C
A
of their perimeters is equal to the ratio
of their corresponding side lengths.
Perimeter of △ABC
——
Perimeter of △DEF
=
AB
—
DE
=
BC
—
EF
=
AC
—
DF
Areas of Similar Figures
If two fi gures are similar, then the
E
D
F
B
C
A
ratio of their areas is equal to the
square of the ratio of their
corresponding side lengths.
Area of △ABC
——
Area of △DEF
=
(
AB
—
DE
)
2
=
(
BC
—
EF
)
2
=
(
AC
—
DF
)
2
EXAMPLE
Finding Ratios of Perimeters
1
1
Find the ratio (red to blue) of the perimeters of the similar rectangles.
Perimeter of red rectangle
———
Perimeter of blue rectangle
=
4
—
6
=
2
—
3
The ratio of the perimeters is
2
—
3
.
1. The height of Figure A is 9 feet. The height of a similar Figure B is
15 feet. What is the ratio of the perimeter of A to the perimeter of B ?
4
6
Lesson Tutorials
English
Spanish
Section 5.2 Perimeters and Areas of Similar Figures 203
Find the ratio (red to blue) of the areas
6
10
of the similar triangles.
Area of red triangle
——
Area of blue triangle
=
(
6
—
10
)
2
=
(
3
—
5
)
2
=
9
—
25
The ratio of the areas is
9
—
25
.
EXAMPLE
Finding Ratios of Areas
2
2
EXAMPLE
Real-Life Application
3
3
You place a picture on a page of a photo album. The page and the
picture are similar rectangles.
a. How many times greater is the area of the page than the area
of the picture?
b. The area of the picture is 45 square inches. What is the area
of the page?
a. Find the ratio of the area of the page to the area of the picture.
Area of page
——
Area of picture
=
(
length of page
——
length of picture
)
2
=
(
8
—
6
)
2
=
(
4
—
3
)
2
=
16
—
9
The area of the page is
16
—
9
times greater than the area
of the picture.
b. Multiply the area of the picture by
16
—
9
.
45
⋅
16
—
9
= 80
The area of the page is 80 square inches.
2. The base of Triangle P is 8 meters. The base of a similar
Triangle Q is 7 meters. What is the ratio of the area of P
to the area of Q?
3. In Example 3, the perimeter of the picture is 27 inches.
What is the perimeter of the page?
Exercises 4 –13
8 in.
6 in.
English
Spanish
Exercises
5.2
204 Chapter 5 Similarity and Transformations
1. WRITING How are the perimeters of two similar fi gures related?
2. WRITING How are the areas of two similar fi gur
es related?
3. VOCABULARY Rectangle ABCD is similar to
R
ectangle WXYZ. The area of ABCD is 30 square inches.
W
Z
B
C
D
What is the area of WXYZ ? Explain.
AD
—
WZ
=
1
—
2
AB
—
WX
=
1
—
2
9
+(-6)=3
3
+(-3)=
4
+(-9)=
9
+(-1)=
The two fi gures are similar. Find the ratios (red to blue) of the perimeters and of the areas.
4.
11
6
5.
5
8
6.
7
4
7.
9
14
8. How does doubling the side lengths of a triangle affect its perimeter?
9. How does tripling the side lengths of a triangle affect its perimeter?
10. How does doubling the side lengths of a rectangle affect its area?
11. How does quadrupling the side lengths of a rectangle affect its area?
12. BASKETBALL The Orlando Magic’s court is similar to a playground court. The
r
atio of the corresponding side lengths is 10 : 7. What is the ratio of the areas?
13. LAPTOP The ratio of the corresponding side lengths of two similar computer
scr
eens is 13 : 15. The perimeter of the smaller screen is 39 inches. What is the
perimeter of the larger screen?
Triangle ABC is similar to Triangle DEF. Tell whether the statement is true or false.
Explain your reasoning.
14.
Perimeter of △ ABC
——
Perimeter of △DEF
=
AB
—
DE
15.
Area of △ ABC
——
Area of △DEF
=
AB
—
DE
1
1
2
2
Help with Homework
English
Spanish
Section 5.2 Perimeters and Areas of Similar Figures 205
Find the percent of change. Round to the nearest tenth of a percent,
if necessary. (Section 4.2)
21. 24 feet to 30 feet 22. 90 miles to 63 miles 23. 150 liters to 86 liters
24. MULTIPLE CHOICE A runner completes an 800-meter race in 2 minutes
40 seconds. What is the runner’s speed? (Section 3.1)
A
3 sec
—
10 m
B
160 sec
—
1 m
C
5 m
—
1 sec
D
10 m
—
3 sec
16. FABRIC The cost of the fabric is $1.31.
What would you expect to pay for a
similar piece of fabric that is 18 inches
by 42 inches?
17. AMUSEMENT PARK A model of a
merry-go-round has a base area
of about 450 square inches. What
is the percent of increase of the base
area from the model to the actual
merry-go-round? Explain.
18. CRITICAL THINKING The circumference of Circle K is
π
.
Circle L
Circle K
The circumference of Circle L is 4
π
.
a. What is the ratio of their circumferences? of their
radii? of their areas?
b. What do you notice?
19. GEOMETRY Rhombus A is similar to Rhombus B.
A
B
Area = 36 cm
2
Area = 64 cm
2
What is the ratio (A to B) of the corresponding
side lengths?
20.
A triangle with an area of 10 square meters has a base of
4 meters. A similar triangle has an area of 90 square meters. What is the
height of the larger triangle?
6 in.
450 in.
2
Model
10 ft
21 in.
9 in.
MSFL7PE_0502.indd 205 10/20/09 3:23:08 PM
English
Spanish
SECTION 4.2
SECTION 3.1