8.1, 2, 7: Ratios, Proportions and DilationsObjectives:Be able to find and simplify the ratio of two numbers.Be able to use proportions to solve real-life problems.Be able to draw dilations.

Computing RatiosIf a and b are two quantities that are measured in the same units, then the ratio of a to b is a/b. The ratio of a to b can also be written as a:b. Because a ratio is a quotient, its denominator cannot be zero. Ratios are usually expressed in simplified form. For instance, the ratio of 6:8 is usually simplified to 3:4. (You divided by 2)

ExamplesSimplify the ratios:

121 )

4

61 )

18

cma

cm

ftb

in

3:1

4 :1

72

18

in

in

2 )

2 )

DFa

AE

BCb

DE

5

101

2

1

3

Example3) The perimeter of rectangle ABCD is 60 centimeters. The ratio of AB: BC is 3:2. Find the length and the width of the rectangle

w

lA

B C

D

Solution:Because the ratio of AB:BC is 3:2, you can represent the length of AB as 3x and the width of BC as 2x.

Statement

2l + 2w = P

2(3x) + 2(2x) = 60

6x + 4x = 60

10x = 60

x = 6

Reason

Formula for perimeter of a rectangle

Substitute l, w and P

Multiply

Combine like terms

Divide each side by 10

So, ABCD has a length of 18

centimeters and a width of 12 cm.

Example

4) The measures of the angles in a triangle are in the extended ratio of 2 : 5: 8. Find the measures of the angles and classify the triangle.2 5 8 180x x x

15 180x

12x 24 ,60 ,96

Using Proportions An equation that equates two ratios is called a proportion. For

instance, if the ratio of a/b is equal to the ratio c/d; then the following proportion can be written:

Means Extremes

The numbers a and d are the extremes of the proportions. The numbers b and c are the means of the proportion.

a c

b d

The product of the extremes

equals the product of the means. If , then a c

ad bcb d

Example

( )(6) (4)(9)y

6 36y 6y

(4)( 4) (3)( 3)y y

4 16 3 9y y

25y

4 4 35) 6)

9 6 3 4

y

y y

Solve the proportions.

Additional Properties of Proportions

If , then (Reciprocal Property)b d

ad bca c

If , then a c a b

b d c d

If , then a c a b c d

b d b d

Geometric MeanThe geometric mean of two positive numbers a and b is the positive number x such that

ax =

xb

2x ab

x ab

Example

8) 4 and 9 9) 6 and 12

4 92 2 3 3

6

6 122 2 2 3 3

6 2

Find the geometric mean between the two numbers.

4

9

x

x

2 4 9x

6

12

x

x

2 6 12x

DilationDilation: A type of transformation (nonrigid), in which the image and preimage are similar.

Nonrigid: Image and preimage are not congruent. Therefore, length is not preserved, thus it is not an isometry.

Similar: Polygons in which their corresponding angles are congruent and the lengths of their corresponding sides are proportional.

P

R

Q

P’

Q’

R’

A dilation may be a reduction (contraction) or an enlargement (expansion).

Assignment Read Pages 457-460 and 465-467 Define: Ratio, Proportion and Geometric

Mean Pages 461-464

#12,16,20,24,28,32,36,44,45-47,52-58 even,65-66.

Pages 468-471 #10-32 even, 48-56 even.