Geometry

1
Geometry

• Perimeters and Areas of Similar Figures

2
Is this statement True or False?
The picture is twice as big as our smaller TV.
Screen Size 6 ? 9
Screen Size 12 ? 18
Area 54
Area 216
3
Goals

• Compare perimeters and areas of similar figures.
• Use perimeters and areas of similar figures to
  solve problems.

4
Problem
Draw a rectangle that measures 3 ? 2.
6
The area is ?
Double the lengths of the sides and draw a
rectangle that measures 6 ? 4.
The area is ?
24
5
Example continued
When the lengths of the sides were doubled, did
the area double? NO!
6
24
6
So what happened?
3
The ratio of the sides
6
2
?
6
The ratio of the areas
?
24
4
7
Notice
3
The ratio of the sides
6
2
The ratio of the areas
6
24
4
8
Theorem 10.5

• The Area of Similar Polygons
• If two polygons are similar with lengths of
  corresponding sides in the ratio ab, then the
  ratio of their areas is a2b2.
• Another way of saying it the ratio of the areas
  is the square of the ratio of the sides.

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Example 1 ABCDE MNOPQ
O
P
D
C
B
N
E
Q
4
A
6
M
Find the ratio of the areas of the figures.
10
Example 2
These figures are similar. The area of the
smaller one is 20. Find the area of the larger
figure.
4
Ratio of the sides Ratio of their areas
10
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Example 2 continued
Write the proportion and solve
20
x
125
4
10
Ratio of sides 25 Ratio of areas 425
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Your Turn
5
These figures are similar. The area of the larger
one is 320. Find the area of the smaller figure.
?
8
320
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Solution
5
125
?
8
320
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Now do you see how it works?
Do NOT write
Ratio of sides.
5(320) 1600 and 8(125)1000
Instead write
Ratio of SQUARE of sides.
25(320) 8000 and 64(125) 8000
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Try it again
These figures are similar. Find the area of the
smaller figure if the area of the larger one is
720.
5
12
A 720
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Solution
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Problem
If the figures are similar, what is the length of
the side marked x?
4
A 36
x
A 100
Remember the ratio of the areas is the square of
the ratios of the sides. Or, the ratio of the
sides is the square root of the ratios of the
areas.
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Problem Solution
4
A 36
x
A 100
The ratio of the sides is the square root of the
ratios of the areas.
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Heres nothing surprising

• If two polygons are similar, the ratio of the
  perimeters is equal to the ratio of the sides.

Perimeter 10
Perimeter 20
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Example

• ABCD DEFG.
• AB 8 and DE 12.
• The perimeter of ABCD is 20.
• What is the perimeter of DEFG?
• Solution

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Summary

• The ratio of the areas of similar figures is the
  square of the ratio of the sides.
• The ratio of the perimeters of similar figures is
  the same as the ratio of the sides.
• The ratio of the sides in two similar figures is
  the square root of the ratio of the areas.

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Do you get it? Do these problems.

• 1. Figure CAT is similar to BAT.
• CA 10 and BA 15.
• What is the ratio of the sides?
• What is the ratio of the perimeters of CAT to
  BAT?
• What is the ratio of the areas of CAT to BAT?

2/3
2/3
4/9
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• 2. Figure RATS is similar to DOGS.
• RA 12 and DO 30.
• What is the ratio of the perimeters of RATS to
  DOGS?
• What is the ratio of the areas of RATS to DOGS?

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3. Two figures, ABCDE and XYZWS are similar. The
perimeter of ABCDE is and the perimeter of
XYZWS is 18. What is the ratio of corresponding
sides? (What is the scale factor?)
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4. JKL RST. The area of JKL is 25 and the area
of RST is 64. What is the ratio of their
perimeters?
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5. HOT DOG. The area of HOT is 100 and the area
of DOG is 9. The perimeter of HOT is 40. What is
the perimeter of DOG? Ratio of Areas Ratio of
Perimeters
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4. HOT DOG. The area of HOT is 100 and the area
of DOG is 9. The perimeter of HOT is 40. What is
the perimeter of DOG? Ratio of Perimeters