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.

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)

Examples Simplify the ratios:

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.

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.

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.

Example Solve the proportions.

Additional Properties of Proportions

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

Example Find the geometric mean between the two numbers.

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

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.