Ratios, Proportions, and the Geometric Mean
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Transcript Ratios, Proportions, and the Geometric Mean
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 1010
x 10 2