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Chapter 6.1: Similarity
Ratios, Proportions, and the Geometric Mean
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:ba to b b
a
Equivalent RatiosEquivalent ratios are ratios that
have the same value.
Examples:1:2 and 3:65:15 and 1:36:36 and 1:62:18 and 1:94:16 and 1:47:35 and 1:5Can you come up with your own?
Simplify the ratios to determine an equivalent ratio.
ftyd
ftyd 9
1
33
3 ft = 1 yard ft
ft
9
10
1 km = 1000 m
mmkm
mkm 5000
1
5000
1
10005
m
m
m
m
25
8
50
16
5000
1600
Convert 3 yd to ft
Convert 5 km to m
Simplify the ratio
ft
in
2
10
inft 121 Convert 2 ft to in
inft
inft 24
1
122
in
in
in
in
12
5
24
10
What is the simplified ratio of width to length?
cm
cm
cm
cm
3
1
12
4
What is the simplified ratio of width to length?
.5
.3
.10
.6
in
in
in
in
What is the simplified ratio of width to length?
in
ft
18
1
.121 inft
.121
.121 in
ft
inft
in
in
in
in
3
2
18
12
Use the number line to find the ratio of the distances
BC
AB
2
3
CD
AB
2
3
DE
EF
1
3
AC
BF
5
8
Finding side lengths with ratios and perimetersA rectangle has a perimeter of 56 and the ratio
of length to width is 6:1.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 = widthWhat next?Length = 6x, width = 1x, perimeter = 56
56=2(6x)+2(1x) 56=12x+2x
56=14x 4=x
L = 24, w= 4
P=2l+2w
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 = l²wLength = 7xWidth = 3xArea = 525
525 = 7x²3x525 = 21x²√25 = √x²5 = x
Length = 7x = 7(5) = 35Width = 3x = 3(5) = 15
Triangles and ratios: finding interior anglesThe 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 = 1xAngle 2 = 2xAngle 3 = 3x
What do we know about the sum of the interior angles?1x + 2x + 3x = 180
6x = 180X = 30
=30=2(30) = 60= 3(30) = 90
Triangles and ratios: finding interior angles
The ratio of the angles in a triangle are represented by 1:1:2.
Angle 1 = 1xAngle 2 = 1xAngle 3 = 2x
1x + 1x + 2x = 1804x = 180 x = 45
Angle 1 = 1x = 1(45) = 45
Angle 2 = 1x = 1(45) = 45Angle 3 = 2x = 2(45) = 90
Proportions, extremes, meansProportion: a mathematical statement that
states that 2 ratios are equal to each other.
d
c
b
a
82
1 x
meansextremextremeses
Solving ProportionsWhen 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) = 2x8 = 2x4 = x
4(15) = 12z60 = 12z5 = z
A little trickier
3(8) = 6(x – 3)24 = 6x – 18
42 = 6x
7 = x
X’s on both sides?
3(x + 8) = 6x3x + 24 = 6x24 = 3x8 = x
Now you try!
x = 18
x = 9
m = 7
z = 3
d = 5
Geometric MeanWhen given 2 positive numbers, a and b
the geometric mean satisfies:
abx
b
x
x
a
abx 2
Find the geometric mean
abx
4)4(1 x
x = 2
abx
9)9(1 x
x = 3
Find the geometric mean
abx
81)27(3 x
x = 9
abx
101021002200)5(40 x
210x
