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Section 6.1 Ratios, Proportions, & the Geometric Mean Goal Solve problems by writing and solving proportions. Ratio of a to b A comparison of two numbers a to b or a:b or a b The ratio of triangle ABC to triangle DEF would be written as: Example 1: Simplify the ratio. a. 64 m: 6 m b. 4 ft 24in Example 2: In the diagram, AB:BC is 4:1 and AC = 30. Find AB and BC. A B C Extended Ratio: Ratios that go beyond the comparison of 2 quantities. Example 3: The measures of the interior angles of CDE are of the extended ratio of 2:3:4. Find the measures of all 3 angles. Checkpoint 1. A triangle’s angle measures are in the extended ratio of 1:4:5. Find the measures of the angles. Example 4: The perimeter of a room is 48 feet. The ratio of the length to the width is 7:5. Find the length and width of the room. Example 5: You are going to paint a rectangular wall. The perimeter of the wall is 484 feet. The ratio of the length to the width is 9:2. Find the area of the wall. Section 6.1 Ratios, Proportions, & the Geometric Mean A PROPERTY OF PROPORTIONS Proportions: An equation that states that two ratios are equal. a c b d b and c are called the means AND a and d are called the extremes Cross Products Property: In a proportion, the product of the extremes equals the product of a c the means. If where b 0 and d 0, then ad = bc b d Example 6: Solve proportions. 3 x a. 4 16 3 2 x 1 x b. GEOMETRIC MEAN The geometric mean of two positive numbers a and b is the positive number x that satisfies a x. x b So, x2 = ab and x ab Example 7: Find the geometric mean of 16 and 48. Checkpoint 2. Solve 8 2 y 5 3. Solve 4. Find the geometric mean of 14 and 16. x 3 2x 3 9