Ratios, Proportions, and the
Geometric Mean
Chapter 6.1: Similarity
Ratios

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

a
b
Equivalent Ratios
 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?
Simplify the ratios to determine an
equivalent ratio.
Convert 3 yd to ft
3 ft = 1 yard
3 ft
3 yd 
 9 ft
1yd
10 ft
9 ft
1 km = 1000 m
Convert 5 km to m
1000 m 5000
5km 

m  5000 m
1km
1
1600 m 16 8m


5000 m 50 25m
Simplify the ratio
10in
2 ft
1 ft  12in
Convert 2 ft to in
12in
2 ft 
 24in
1 ft
10in 5in

24in 12in
What is the simplified ratio of width to
length?
4cm 1cm

12cm 3cm
What is the simplified ratio of width to
length?
6in. 3in.

10in. 5in.
What is the simplified ratio of width to
length?
1 ft
18in
1 ft  12in.
12in.
1 ft 
 12in.
1 ft
12in 2in

18in 3in
Use the number line to find the ratio of
the distances
AB 3

BC 2
AB 3

CD 2
EF 3

DE 1
BF
8

AC
5
Finding side lengths with ratios and perimeters
 A rectangle has a perimeter of 56 and the ratio of length to
width is 6:1. 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?
 Length = 6x, width = 1x, perimeter = 56
 56=2(6x)+2(1x)
 56=12x+2x
 56=14x
 4=x
 L = 24, w= 4
Finding side lengths with ratios and
area
 A rectangle has an area of 525 and the ratio of length to
width is 7:3
 A = l²w
 Length = 7x
 Width = 3x
Length = 7x = 7(5) = 35
 Area = 525
Width = 3x = 3(5) = 15
 525 = 7x²3x
 525 = 21x²
 √25 = √x²
 5=x
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
=30
=2(30) = 60
= 3(30) = 90
 What do we know about the sum of the interior angles?
1x + 2x + 3x = 180
6x = 180
X = 30
Triangles and ratios: finding interior angles
 The ratio of the angles in a triangle are represented by 1:1:2.
 Angle 1 = 1x
Angle 1 = 1x = 1(45) = 45
 Angle 2 = 1x
Angle 2 = 1x = 1(45) = 45
Angle 3 = 2x = 2(45) = 90
 Angle 3 = 2x
 1x + 1x + 2x = 180
 4x = 180
 x = 45
Proportions, extremes, means
 Proportion: a mathematical statement that states that 2 ratios
are equal to each other.
a c

b d
means
extremes
1 x

2 8
Solving Proportions
 When you have 2 proportions or fractions that are set equal
to each other, you can use cross multiplication.
 1y = 3(3)
y = 9
Solving Proportions
1(8)
4(15)==2x12z
860==2x12z
54 = x
z
A little trickier
3(8) = 6(x – 3)
24 = 6x – 18
42 = 6x
7=x
X’s on both sides?
3(x + 8) = 6x
3x + 24 = 6x
24 = 3x
8=x
Now you try!
x = 18
x=9
m=7
z=3
d=5
Geometric Mean
 When given 2 positive numbers, a and b the geometric mean
satisfies:
a x

x b
x 2  ab
x  ab
Find the geometric mean
x  ab
x  1(4)  4
x=2
x  ab
x  1(9)  9
x=3
Find the geometric mean
x  ab
x  3(27)  81
x=9
x  ab
x  40(5)  200  2 100  2 1010
x  10 2