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

2. 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

3. 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?

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

5. Simplify the ratio Convert 2 ft to in

6. What is the simplified ratio of width to length?

9. Use the number line to find the ratio of the distances

10. 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

11. 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 = l ² w Length = 7x Width = 3x Area = 525 525 = 7x ² 3x 525 = 21x ² √ 25 = √ x ² 5 = x Length = 7x = 7(5) = 35 Width = 3x = 3(5) = 15

12. Triangles and ratios: finding interior angles The ratio of the 3 angles in a triangle are represented by 1:2:3. The 1 st angle is a multiple of 1, the 2 nd a multiple of 2 and the 3 rd a multiple of 3. Angle 1 = 1x Angle 2 = 2x Angle 3 = 3x What do we know about the sum of the interior angles? 1x + 2x + 3x = 180 6x = 180 X = 30 =30 =2(30) = 60 = 3(30) = 90

13. 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

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

15. 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

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

17. A little trickier 3(8) = 6(x – 3) 24 = 6x – 18 42 = 6x 7 = x

18. X’s on both sides? 3(x + 8) = 6x 3x + 24 = 6x 24 = 3x 8 = x

19. Now you try! x = 18 x = 9 m = 7 z = 3 d = 5

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

21. Find the geometric mean x = 2 x = 3

22. Find the geometric mean x = 9