8.1: Ratios and Proportions

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8.1: Ratios and Proportions GEOMETRY: Chapter 8 8.1: Ratios and Proportions

If a and b are two numbers or quantities and b≠0, then the ratio of a to b is a/b. The ratio of a to b can also be written as a:b. For example, the ratio of a side length in triangle DEF to a side length in ABC can be written as ½ or 1:2. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 356.

Ex. 1: Simplify the ratio. 12km: 3km 36 in.; 9ft 100cm:2m 3 ft: 30 in. Answers: 4:1 b. 1:3 c. 1:2 d. 6:5

Ex. 2: The area of a rectangular garden is 108 square feet, and the ratio of the length to the width is 4:3. Find the length and width of fence needed to enclose the garden. Answer: length—12 feet, width—9feet

Ex. 3. The measure of the angles of triangle RST are in the extended ratio 2:3:4. Find the measures of the angles. Answers: 400, 600, 800 Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 358.

Proportions: An equation that states that two ratios are equal is called a proportion. The numbers b and c are the means of the proportion. The numbers a and d are the extremes of the proportion. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 358.

A Key Property of Proportions Cross Products Property: In a proportion, the product of the extremes equals the product of the means. Reciprocal Property—If two ratios are equal, then their reciprocals are also equal. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 358.

Ex.4 Solve the proportion.

Ex.4 Solve the proportion. Answer: a. 9; b. 5

Ex.5 As part of a science project, you need to estimate the number of blue spruce trees in a 50 acre forest. You count 36 trees in 3 acres and notice that the trees seem to be evenly distributed. Estimate the total number of blue spruce trees in the forest.

Ex.5 As part of a science project, you need to estimate the number of blue spruce trees in a 50 acre forest. You count 36 trees in 3 acres and notice that the trees seem to be evenly distributed. Estimate the total number of blue spruce trees in the forest. Answer: about 600 trees.

Geometric Mean The geometric mean of two positive numbers a and b is the positive number x that satisfies So, x2=ab and

Ex.6 Find the geometric mean of 36 and 54.

Ex.6 Find the geometric mean of 36 and 54. Answer:

6.1, p. 360, #1-25 odds, 33-45 odds (20 questions)