Ratios, Proportions, and the Geometric Mean

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 1000m 5000 5km m 5000m 1km

1 1600m 16 8m 5000m 50 25m Simplify the ratio 1 ft 12in 10in 2 ft 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 toP=2l+2w 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 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 = lw Length = 7x Width = 3x Length Area = 525 = 35 Width = 3x = 3(5) = 15 525 = 7x3x 525 = 21x 25 = x 5 = x

= 7x = 7(5) 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 =30 Angle 2 = 2x =2(30) = 60 = 3(30) = 90 Angle 3 = 3x

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 = 1(45) = 45 Angle 1 = 1x Angle 2 = 1x = 1(45) = 45 Angle 2 = 1x 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 mea ns extrem es 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) == 2x

12z 8 60 == 2x 12z 5 4 =z x A little trickier 3(8) = 6(x 3) 24 = 6x 18

42 = 6x 7=x Xs 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

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