1. 8.1, 2, 7: Ratios, Proportions and Dilations
Objectives:
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.

2. Computing Ratios
If 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)

3. Examples
Simplify the ratios:

4. So, ABCD has a length of 18 centimeters and a width of 12 cm.
Example
3) 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
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.

5. 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.

6. 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.
The product of the extremes
equals the product of the means.

7. Example
Solve the proportions.

8. Additional Properties of Proportions

9. Geometric Mean
The geometric mean of two positive numbers a and b is the positive number x such that a x = x b

10. Example
Find the geometric mean between the two numbers.

11. Dilation
Dilation: 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 P’ Q’ Q R’
A dilation may be a reduction (contraction) or an enlargement (expansion). R

12. Assignment Read Pages 457-460 and 465-467
Define: Ratio, Proportion and Geometric Mean
Pages #12,16,20,24,28,32,36,44,45-47,52-58 even,65-66.
Pages #10-32 even, even.