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

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

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?

Simplify the ratios to determine an equivalent ratio. 3 ft = 1 yard Convert 3 yd to ft 1 km = 1000 m Convert 5 km to m

Simplify the ratio Convert 2 ft to in

What is the simplified ratio of width to length?

What is the simplified ratio of width to length?

What is the simplified ratio of width to length?

Use the number line to find the ratio of the distances

Finding side lengths with ratios and perimeters A 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 = 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 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 = 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

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? =30 =2(30) = 60 = 3(30) = 90 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 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 = 2(45) = 90

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

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

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

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! z = 3 x = 18 d = 5 x = 9 m = 7

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

Find the geometric mean x = 2 x = 3

Find the geometric mean x = 9