Similar Triangles

1
Similar Triangles
An Introduction to Ratios Proportions
2
When one shape is a dilation (stretch) of another
shape, the shapes are said to be similar.
x 2
2
4
2
x 2
4
3
When one shape is a dilation (stretch) of another
shape, the shapes are said to be similar.
x 3
6
2
2
x 3
6
4
When 2 shapes are similar, their angles are
congruent.
C
D
G
H

70
70
E
F
A
B
5
When 2 shapes are similar, their angles are
congruent.
C
D
G
H

110
110
E
F
A
B
6
When 2 shapes are similar, their angles are
congruent.
C
D
G
H

70
70
E
F
A
B
7
When 2 shapes are similar, their angles are
congruent.
C
D
G
H

110
110
E
F
A
B
8
When 2 shapes are similar, their sides are
proportional.
C
D
G
H

6
4
E
F
12
A
B
18
9
When 2 shapes are similar, their sides are
proportional.
C
D
G
H

6
4
E
F
12
A
B
18
This means that the ratio of the sides in one
shape equals the ratio of the sides in the other
shape.
10
A ratio is a comparison between two or more
things.
2 apples to 3 oranges
4 dogs 1 cat
2 cups of flour5 cookies
11
When 2 ratios are equal, it is called a
proportion.
3 cups of flour8 cookies
6 cups of flour16 cookies

12
When 2 ratios are equal, it is called a
proportion.
5 cups of granola1 bag of trail mix
20 cups of granola4 bags of trail mix

13
When 2 ratios are equal, it is called a
proportion.
3 parts soap5 parts water
9 parts soap15 parts water

14
In a proportion, the cross products are equal.
3 5
9 15

15
In a proportion, the cross products are equal.
3 5
9 15

3 x 15 5 x 9
16
In a proportion, the cross products are equal.
2 7
6 21

17
In a proportion, the cross products are equal.
2 7
6 21

2 x 21 7 x 6
18
In a proportion, the cross products are equal.
2 6
5 15

19
In a proportion, the cross products are equal.
2 6
5 15

2 x 15 6 x 5
20
Now lets get back to similar shapes.
21
When 2 shapes are similar, their sides are
proportional.
C
D
G
H

6
4
E
F
12
A
B
18
22
When 2 shapes are similar, their sides are
proportional.
C
D
G
H

6
4
E
F
12
A
B
18
The cross products are equal.
6 x 12 18 x 4
23
Practice Time!
24
1) Use cross products to solve for x in the
following proportion
2 cups of flour5 cookies
x cups of flour12 cookies

25
Cross Mulitply
2 cups of flour5 cookies
x cups of flour12 cookies

2 12 5xor24 5x
26
Divide
2 cups of flour5 cookies
x cups of flour12 cookies

2 12 5xor24 5x5 54 4/5 x
27
2) Use cross products to solve for x in the
following proportion
3 ml vinegar8 ml water
10 ml vinegarx ml water

28
Cross Multiply
3 ml vinegar8 ml water
10 ml vinegarx ml water

3x 8 10 or3x 80
29
Divide
3 ml vinegar8 ml water
10 ml vinegarx ml water

3x 8 10 or3x 803 3x 26 2/3
30
3) Use cross products to solve for x in the
following proportion
3 petsx households
8 pets15 households

31
Cross Multiply
3 petsx households
8 pets15 households

3 15 8x or45 8x
32
Divide
3 petsx households
8 pets15 households

3 15 8x or45 8x8 85 5/8 x
33
4) Use cross products to solve for x in the
following proportion
1 gallon of milk10 bowls of cereal
x gallons of milk24 bowls of cereal

34
Cross Multiply
1 gallon of milk10 bowls of cereal
x gallons of milk24 bowls of cereal

1 24 10x or24 10x
35
Divide
1 gallon of milk10 bowls of cereal
x gallons of milk24 bowls of cereal

1 24 10x or24 10x10 102 4/10 x
36
5) Determine the missing side.

4
x
10
5
37
Set up a proportion.

4
x
10
5
410
x 5

38
Cross Multiply

4
x
10
5
410
x 5

4 5 10x or20 10x
39
Divide

4
x
10
5
410
x 5

4 5 10x or20 10x10 102 x
40
6) Determine the missing side.

5
3
8
x
41
Set up a proportion.

5
3
8
x
58
3 x

42
Cross Multiply

5
3
8
x
58
3 x

5x 3 8 or5x 24
43
Divide

5
3
8
x
58
3 x

5x 3 8 or5x 245 5x 4 4/5
44
7) Determine the length of the stop signs shadow.
8 ft
6 ft
x ft
10 ft
45
7) This is the same as having 2 similar trianlges.

8 ft
6 ft
x ft
10 ft
46
Set up a proportion.

8 ft
6 ft
x ft
10 ft
6x
8 10

47
Cross Multiply

8 ft
6 ft
x ft
10 ft
6x
8 10

6 10 8x or60 8x
48
Divide

8 ft
6 ft
x ft
10 ft
6x
8 10

6 10 8x or60 8x8 87 4/8 or 7 ½ x
49
8) Determine the missing side.
x
2
8
9
50
Separate these overlapping triangles.
x
2
8
9
51
Separate these overlapping triangles.
x
2
8
9
52
Set up a proportion.
x
2
8
9
x9
2 8

53
Cross multiply.
x
2
8
9
x9
2 8

9 2 8x or18 8x
54
Divide.
x
2
8
9
x9
2 8

9 2 8x or18 8x8 8 2 2/8 x
55
9) Determine the missing side. Assume that these
triangles are similar.
9
x
3
5
56
Set up a proportion.
9
x
3
5
x9
3 5

57
Cross multiply.
9
x
3
5
x9
3 5

9 3 5x or27 5x
58
Divide.
9
x
3
5
x9
3 5

9 3 5x or27 5x5 55 2/5 x
59
10) Determine the missing side. Assume that
these triangles are similar.
12
20
8
x
60
Set up a proportion.
12
20
8
x
12x
8 20

61
Cross multiply.
12
20
8
x
12x
8 20

12 20 8x or240 8x
62
Divide.
12
20
8
x
12x
8 20

12 20 8x or240 8x8 830 x
63
11) Determine the missing side.
20
x
11
15
64
Separate the overlapping triangles.
20
x
11
15
65
Separate the overlapping triangles.
20
x
11
15
66
Set up a proportion.
20
x
2015
x 11
11

15
67
Cross multiply.
20
x
2015
x 11
11

15
20 11 15x or220 15x
68
Divide.
20
x
2015
x 11
11

15
20 11 15x or220 15x15 1514 10/15
x
69
The End