Perimeters and Areas 5.2 of Similar Figures

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English
5.2
Spanish
Perimeters and Areas
of Similar Figures
How do changes in dimensions of similar
geometric figures affect the perimeters and areas of the figures?
1
ACTIVITY: Comparing Perimeters and Areas
Work with a partner. Use pattern blocks to make a figure whose dimensions
are 2, 3, and 4 times greater than those of the original figure. Find the
perimeter P and area A of each larger figure.
a. Sample: Square
1
1
1
1
P=4
P=8
P = 12
P = 16
A=1
A=4
A= 9
A = 16
b. Triangle
1
1
1
P=3
P=6
P=
P=
A=B
A = 4B
A=
A=
c. Rectangle
d. Parallelogram
1
2
1
1
2
200
Chapter 5
1
1
1
P=6
P=4
A=2
A=C
Similarity and Transformations
Spanish
2
ACTIVITY: Finding Patterns for Perimeters
Work with a partner. Copy and complete the table for the perimeters of
the figures in Activity 1. Describe the pattern.
Perimeters
Figure
Original Side
Lengths
Double Side
Lengths
Triple Side
Lengths
Quadruple
Side Lengths
P=4
P=8
P = 12
P = 16
P=3
P=6
P=6
P=4
3
ACTIVITY: Finding Patterns for Areas
Work with a partner. Copy and complete the table for the areas of the
figures in Activity 1. Describe the pattern.
Figure
Areas
English
Original
Side Lengths
Double Side
Lengths
Triple Side
Lengths
Quadruple
Side Lengths
A=1
A=4
A=9
A = 16
A=B
A = 4B
A=2
A=C
4. IN YOUR OWN WORDS How do changes in dimensions of similar
geometric figures affect the perimeters and areas of the figures?
Use what you learned about perimeters and areas of similar figures
to complete Exercises 8 –11 on page 204.
Section 5.2
Perimeters and Areas of Similar Figures
201
English
5.2
Spanish
Lesson
Lesson Tutorials
Perimeters of Similar Figures
E
If two figures are similar, then the ratio
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
—— = — = — = —
EXAMPLE
1
B
D
F
C
A
Finding Ratios of Perimeters
Find the ratio (red to blue) of the perimeters of the similar rectangles.
4
6
Perimeter of red rectangle
Perimeter of blue rectangle
4
6
2
3
——— = — = —
2
3
The ratio of the perimeters is —.
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 ?
Areas of Similar Figures
E
If two figures are similar, then the
ratio of their areas is equal to the
square of the ratio of their
corresponding side lengths.
B
D
A
Area of △ABC
Area of △DEF
( ) ( ) ( )
Chapter 5
Similarity and Transformations
C
AB 2
BC 2
AC 2
= — = —
DE
EF
DF
—— = —
202
F
English
Spanish
EXAMPLE
2
Finding Ratios of Areas
Find the ratio (red to blue) of the areas
of the similar triangles.
6
( )
()
6 2
10
Area of red triangle
Area of blue triangle
—— = —
3 2
9
=—
5
25
10
= —
9
25
The ratio of the areas is —.
EXAMPLE
3
Real-Life Application
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
6 in.
)
length of page 2
length of picture
—— = ——
() ()
8 2
4 2 16
= — =—
6
3
9
= —
8 in.
16
9
The area of the page is — times greater than the area
of the picture.
16
9
b. Multiply the area of the picture by —.
⋅ 169
45 — = 80
The area of the page is 80 square inches.
Exercises 4 –13
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?
Section 5.2
Perimeters and Areas of Similar Figures
203
English
Spanish
Exercises
5.2
Help with Homework
1. WRITING How are the perimeters of two similar figures related?
2. WRITING How are the areas of two similar figures related?
3. VOCABULARY Rectangle ABCD is similar to
Rectangle WXYZ. The area of ABCD is 30 square inches.
What is the area of WXYZ ? Explain.
A
B
D
C
AD
WZ
W
X
Z
Y
1
2
AB
WX
—=—
1
2
—=—
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
The two figures are similar. Find the ratios (red to blue) of the perimeters and of the areas.
1
2
5.
4.
11
5
8
6
6.
7.
9
14
4
7
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. FOOSBALL The playing surfaces of two foosball tables are similar. The ratio 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
screens 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.
Perimeter of △ ABC
Perimeter of △DEF
AB
DE
14. —— = —
204
Chapter 5
Similarity and Transformations
Area of △ ABC
Area of △DEF
AB
DE
15. —— = —
English
Spanish
21 in.
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?
9 in.
6 in.
10 ft
17. AMUSEMENT PARK A model of a
merry-go-round has a base area
Model
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.
450 in.2
18. CRITICAL THINKING The circumference of Circle K is π.
The circumference of Circle L is 4π.
a. What is the ratio of their circumferences? of their
radii? of their areas?
Circle K
Circle L
b. What do you notice?
19. GEOMETRY Rhombus A is similar to Rhombus B.
What is the ratio (A to B) of the corresponding
side lengths?
20.
A
B
Area = 36 cm2
Area = 64 cm2
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?
Find the percent of change. Round to the nearest tenth of a percent,
if necessary. (Section
4.2)
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)3.1
SECTION
A
○
3 sec
10 m
—
B
○
160 sec
1m
—
Section 5.2
MSNA7PE2_0502.indd 205
C
○
5m
1 sec
—
D
○
10 m
3 sec
—
Perimeters and Areas of Similar Figures
205
11/23/10 2:14:19 PM

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