Perimeters and Areas of Similar Figures
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Transcript of Perimeters and Areas of Similar Figures
Geometry
Perimeters and Areas of Similar Figures
May 8, 2023 2
Screen Size: 6 9Screen Size: 12 18
The picture is twice as big as our smaller TV.
Area: 54Area: 216
Is this Is this statement statement True or True or False?False?
May 8, 2023 3
Goals
Compare perimeters and areas of similar figures.
Use perimeters and areas of similar figures to solve problems.
May 8, 2023 4
Problem
Draw a rectangle that measures 3 2.
The area is ?
6
Double the lengths of the sides and draw a rectangle that measures 6 4.The area is ? 24
May 8, 2023 5
Example continued
6When the lengths of the sides were doubled, did the area double?
NO!24
May 8, 2023 6
So what happened?
24
136 2
6
24
23
4
6
The ratio of the sides:
The ratio of the areas:6
2414
?
?
May 8, 2023 7
Notice:
1214
6
24
23
4
6
The ratio of the sides:
The ratio of the areas:
21 12 4
May 8, 2023 8
Theorem 10.5
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.
May 8, 2023 9
Example 1 ABCDE ~ MNOPQ
A
B
C D
E
M
N
O P
Q46
Find the ratio of the areas of the figures.
2 24 26 3
49
May 8, 2023 10
Example 2
410
25
4
10
These figures are similar. The area of the smaller one is 20. Find the area of the larger figure.Ratio of the sides:
Ratio of their areas: 22 4
25 5
May 8, 2023 11
Example 2 continued
4
10
Write the proportion and solve:
20 425
4 20(25)4 500
125
xxxx
20 x
Ratio of sides 2:5Ratio of areas 4:25
125
May 8, 2023 12
Your Turn
5
8
These figures are similar. The area of the larger one is 320. Find the area of the smaller figure.
320
?
May 8, 2023 13
Solution
5
8320
?
258 3202564 320
64 25(320)64 8000
800064
125
x
x
xx
x
x
125
May 8, 2023 14
Now do you see how it works?
5 1258 320Do NOT write
5(320) = 1600 and 8(125)=1000
Instead write 25 12564 320
25(320) = 8000 and 64(125) = 8000
Ratio of sides.
Ratio of SQUARE of sides.
May 8, 2023 15
Try it again…These figures are similar. Find the area of the smaller figure if the area of the larger one is 720.5
12A = 720
May 8, 2023 16
Solution
5
12A = 720
2512 72025144 720144 25(720)144 18000
125
x
x
xxx
May 8, 2023 17
Problem
A = 36
A = 100
4
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.
May 8, 2023 18
Problem Solution
A = 36
A = 100
4
x
The ratio of the sides is the square root of the ratios of the areas.
2
23
4 36100
4 610
6 40406
6
x
xx
x
x
May 8, 2023 19
Here’s nothing surprising:
If two polygons are similar, the ratio of the perimeters is equal to the ratio of the sides.
23
4
6
Perimeter = 10 Perimeter = 2010 120 2
May 8, 2023 20
Example
ABCD ~ DEFG. AB = 8 and DE = 12. The perimeter of ABCD is 20. What is the perimeter of DEFG? Solution: 8 20
128 12(20)8 240
30
xxxx
May 8, 2023 21
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.
May 8, 2023 22
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?
b. What is the ratio of the perimeters of CAT to BAT?
c. What is the ratio of the areas of CAT to BAT?
2/3
2/3
4/9
May 8, 2023 23
2. Figure RATS is similar to DOGS.RA = 12 and DO = 30.
a. What is the ratio of the perimeters of RATS to DOGS?
b. What is the ratio of the areas of RATS to DOGS?
25425
May 8, 2023 24
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?)
3 3
3 318
36
May 8, 2023 25
4. JKL ~ RST. The area of JKL is 25 and the area of RST is 64. What is the ratio of their perimeters?
2564
58
May 8, 2023 26
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:
1009
103
May 8, 2023 27
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:103
10 40310 120
12
xxx