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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.
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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:
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 Example 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. Reason Formula for perimeter of a rectangle Substitute l, w and P Multiply Combine like terms Divide each side by 10 Statement 2l + 2w = P 2(3x) + 2(2x) = 60 6x + 4x = 60 10x = 60 x = 6 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.
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: Using Proportions 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.
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 P P’ Q’ Q R’ A dilation may be a reduction (contraction) or an enlargement (expansion). R 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.
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. Assignment