Transcript Geometry - BakerMath.org

Geometry
Perimeters and Areas of
Similar Figures
Is this
statement
True or
False?
The picture is
twice as big as
our smaller TV.
Screen Size: 6  9
Area: 54
July 17, 2015
Screen Size: 12  18
Area: 216
2
Goals


July 17, 2015
Compare perimeters and areas of
similar figures.
Use perimeters and areas of similar
figures to solve problems.
3
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 ?
July 17, 2015
24
4
Example continued
When the lengths of
the sides were
doubled, did the
area double?
NO!
July 17, 2015
6
24
5
So what happened?
3
The ratio of the sides:
2
6
2 3 1
 
4 6 2
?
The ratio of the areas:
6
1

24 4
?
July 17, 2015
6
4
24
6
Notice:
3
1
The ratio of the sides:
2
2
6
1
The ratio of the areas: 4
2
1
1
2  4
 
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6
4
24
7
Theorem 10.5



July 17, 2015
The Area of Similar Polygons
If two polygons are similar with
lengths of corresponding sides in
the ratio a:b, then the ratio of their
areas is a2:b2.
Another way of saying it: the ratio
of the areas is the square of the
ratio of the sides.
8
Example 1
ABCDE ~ MNOPQ
D
C
B
P
N
E
A
O
Q
4
6
Find the ratio of the
areas of the figures.
2
M
2
4
4
2
6  3  9
 
 
July 17, 2015
9
Example 2
These figures are
similar. The area of the
smaller one is 20. Find
the area of the larger
figure.
Ratio of the sides:
4
4
2

10 5
10
2
Ratio of their areas:
July 17, 2015
4
2
 5   25
 
10
Example 2 continued
Write the proportion
and solve:
20
4

x
25
4 x  20(25)
4 x  500
20
125
x
4
10
Ratio of sides 2:5
Ratio of areas 4:25
x  125
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11
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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12
2
5
8
 
25
64
64 x
64 x
Solution
125
?
5
8
320
July 17, 2015
x

320
x

320
 25(320)
 8000
8000
x
64
x  125
13
Now do you see how it works?
5 125
Do NOT write

8 320
Ratio of sides.
5(320) = 1600 and 8(125)=1000
25 125
Instead write

64 320
Ratio of SQUARE of sides.
25(320) = 8000 and 64(125) = 8000
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14
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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15
Solution
2
5
12
A = 720
July 17, 2015
x
 5
  
 12  720
25
x

144 720
144 x  25(720)
144 x  18000
x  125
16
Problem
A = 36
4
A = 100
x
If the figures are
similar, what is the
length of the side
marked x?
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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17
Problem Solution
2
A = 36
4
A = 100
x
The ratio of the sides is
the square root of the
ratios of the areas.
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36
4
 x   100
 
4
6

x 10
6 x  40
40
x
6
x  6 23
18
Here’s nothing surprising:

If two polygons are similar, the ratio of
the perimeters is equal to the ratio of
6
the sides.
3
4
2
Perimeter = 10
Perimeter = 20
10 1

20 2
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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:
8
20

12
x
8 x  12(20)
8 x  240
x  30
July 17, 2015
20
Summary



July 17, 2015
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.
21
Do you get it? Do these problems.
1. Figure CAT is similar to BAT.
CA = 10 and BA = 15.
a.
What is the ratio of the sides? 2/3
b.
What is the ratio of the perimeters of
CAT to BAT? 2/3
c.
What is the ratio of the areas of CAT to
BAT? 4/9
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22
2. Figure RATS is similar
to DOGS.
RA = 12 and DO =
30.
a.
b.
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2
5
What is the ratio of
4
the areas of RATS to
DOGS?
25
What is the ratio of
the perimeters of
RATS to DOGS?
23
3. Two figures, ABCDE and XYZWS
are similar. The perimeter of ABCDE
is 3 3 and the perimeter of XYZWS
is 18. What is the ratio of
corresponding sides? (What is the
scale factor?)
3 3
3

18
6
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24
4. JKL ~ RST. The area of JKL is 25
and the area of RST is 64. What is
the ratio of their perimeters?
25 5

64 8
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25
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:
100
9
July 17, 2015
Ratio of Perimeters:
10
3
26
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?
10 40

3
x
10 x  120
x  12
July 17, 2015
Ratio of Perimeters:
10
3
27