1. Ratios, Proportions, and the Geometric Mean
Chapter 6.1: Similarity

2. Ratios The ratio of a to b can be written 3 ways: a:b a to b
A ratio is a comparison of two numbers expressed by a fraction.
The ratio of a to b can be written 3 ways:
a:b
a to b

3. Equivalent Ratios Can you come up with your own?
Equivalent ratios are ratios that have the same value.
Examples:
1:2 and 3:6
5:15 and 1:3
6:36 and 1:6
2:18 and 1:9
4:16 and 1:4
7:35 and 1:5
Can you come up with your own?

4. Simplify the ratios to determine an equivalent ratio.

5. Simplify the ratio
12 in. = 1 ft

6. Simplify the ratios to determine an equivalent ratio.
3 ft = 1 yard
1 km = 1000 m

7. What is the simplified ratio of width to length?

8. What is the simplified ratio of width to length?

9. What is the simplified ratio of width to length?

10. What is the simplified ratio of width to length?
8 in
12 in

11. What is the simplified ratio of width to length?
2 in
22 in

12. What is the simplified ratio of width to length?
180 in
15 ft

13. What is the simplified ratio of width to length?
144 in
3 ft

14. Use the number line to find the ratio of the distances

15. Use the number line to find the ratio of the distances

16. Finding side lengths with ratios and perimeters
A rectangle has a perimeter of 56 and the ratio of length to width is 6:1.
Any ideas?
P = 2l + 2w
The length must be a multiple of 6, while the width must be a multiple of 1.
New Ratio ~ 6x:1x, where 6x = length and 1x = width
What next?

17. P = 2l + 2w
Length = 6x, width = 1x, perimeter = 56
P = 2(6x) + 2(1x)
56 = 12x + 2x
56 = 14x
4 = x
Are we done yet?
Length = 6x = 6(4) = 24
Width = 1x = 1(4) = 4

18. Finding side lengths with ratios and perimeters
A rectangle has a perimeter of 50 and the ratio of length to width is 3:2.
Any ideas?
P = 2l + 2w
Length = 3x
Width = 2x
Perimeter = 50
50 = 2(3x) + 2(2x)
50 = 6x + 4x
50 = 10x
5 = x
Length = 3x = 3(5) = 15
Width = 2x = 2(5) = 10

19. Finding side lengths with ratios and perimeters
A rectangle has a perimeter of 480 and the ratio of length to width is 5:1.
P = 2l + 2w
Length = 5x
Width = 1x
Perimeter = 480
480= 2(5x) + 2(1x)
480 = 10x + 2x
480= 12x
40 = x
Length = 5x = 5(40) = 200
Width = 1x = 1(40) = 40

20. Finding side lengths with ratios and perimeters
A rectangle has a perimeter of 36and the ratio of length to width is 8:1.
Any ideas?
P = 2l + 2w
Length = 8x
Width = 1x
Perimeter = 36
36 = 2(8x) + 2(1x)
36 = 16x + 2x
36 = 18x
2 = x
Length = 8x = 8(2) = 16
Width = 1x = 1(2) = 2

21. Finding side lengths with ratios and area
A rectangle has an area of 525 and the ratio of length to width is 7:3
Any ideas?
A = l²w
Length = 7x
Width = 3x
Area = 525
525 = 7x²3x
525 = 21x²
√25 = √x²
5 = x
Length = 7x = 7(5) = 35
Width = 3x = 3(5) = 15

22. Finding side lengths with ratios and area
A rectangle has an area of 490 and the ratio of length to width is 5:2
A = l²w
Length = 5x
Width = 2x
Area = 490
490 = 5x²2x
490= 10x²
√49 = √x²
7 = x
Length = 5x = 5(7) = 35
Width = 2x = 2(7) = 14

23. Finding side lengths with ratios and area
A rectangle has an area of 486 and the ratio of length to width is 6:1
A = l²w
Length = 6x
Width = 1x
Area = 486
486 = 6x²1x
486 = 6x²
√81 = √x²
9 = x
Length = 6x = 6(9) = 54
Width = 1x = 1(9) = 9

24. Triangles and ratios: finding interior angles
The ratio of the 3 angles in a triangle are represented by 1:2:3.
The 1st angle is a multiple of 1, the 2nd a multiple of 2 and the 3rd a multiple of 3.
Angle 1 = 1x
Angle 2 = 2x
Angle 3 = 3x
What do we know about the sum of the interior angles?

25. Angle 1 = 1x = 1(30) = 30 Angle 2 = 2x = 2(30) = 60
The sum of the interior angles is 180
Angle 1 = 1x
Angle 2 = 2x
Angle 3 = 3x
1x + 2x + 3x = 180
6x = 180
x = 30
Angle 1 = 1x = 1(30) = 30
Angle 2 = 2x = 2(30) = 60
Angle 3 = 3x = 3(30) = 90

26. Triangles and ratios: finding interior angles
The ratio of the angles in a triangle are represented by 1:1:1.
Angle 1 = 1x
Angle 2 = 1x
Angle 3 = 1x
1x + 1x + 1x = 180
1x = 180
x = 60
Angle 1 = 1x = 1(60) = 60
Angle 2 = 1x = 1(60) = 60
Angle 3 = 1x = 1(60) = 60

27. Triangles and ratios: finding interior angles
The ratio of the angles in a triangle are represented by 1:1:2.
Angle 1 = 1x
Angle 2 = 1x
Angle 3 = 2x
1x + 1x + 2x = 180
4x = 180
x = 45
Angle 1 = 1x = 1(45) = 45
Angle 2 = 1x = 1(45) = 45
Angle 3 = 2x = 1(45) = 90

28. Proportions, extremes, means
Proportion: a mathematical statement that states that 2 ratios are equal to each other.
means
extremes

29. Solving Proportions 1y = 3(3) y = 9
When you have 2 proportions or fractions that are set equal to each other, you can use cross multiplication.
1y = 3(3)
y = 9

30. Solving Proportions
1(8) = 2x
4(15) = 12z
8 = 2x
60 = 12z
5 = z
4 = x

31. Now you try it!
b = 24
a = 15
c = 54
x = 28
a = 24

32. A little trickier
3(8) = 6(x – 3)
24 = 6x – 18
42 = 6x
7 = x

33. X’s on both sides?
3(x + 8) = 6x
3x + 24 = 6x
24 = 3x
8 = x

34. Now you try!
z = 3
x = 18
d = 5
x = 9
m = 7

35. Geometric Mean
When given 2 positive numbers, a and b the geometric mean satisfies:

36. Find the geometric mean
x = 2
x = 3

37. Find the geometric mean
x = 10

38. Find the geometric mean
x = 9

39. Find the geometric mean