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6.1 – Ratios, Proportions, and the Geometric Mean

6.1 – Ratios, Proportions, and the Geometric Mean. Geometry Ms. Rinaldi. Ratio. Ratio – a comparison of numbers A ratio can be written 3 ways: 1. a:b 2. 3. a to b Examples: 2 girls to 7 boys, length:width = 3:2. 5 ft. 5 ft. 5 ft. 20 in. 20 in. 20 in. b. a.

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6.1 – Ratios, Proportions, and the Geometric Mean

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  1. 6.1 – Ratios, Proportions, and the Geometric Mean Geometry Ms. Rinaldi

  2. Ratio • Ratio – a comparison of numbers • A ratio can be written 3 ways: 1. a:b 2. 3. a to b Examples: 2 girls to 7 boys, length:width = 3:2

  3. 5 ft 5 ft 5 ft 20 in. 20 in. 20 in. b. a. 64 m : 6 m 64 m 64 m Write 64 m : 6 mas . 6 m 6 m a. 3 32 1 3 = = 32 : 3 To simplify a ratio with unlike units, multiply by a conversion factor. b. 60 12 in. 20 1 ft = = = EXAMPLE 1 Simplify ratios Simplify the ratio. SOLUTION Then divide out the units and simplify.

  4. Simplify Ratios EXAMPLE 2 Simplify the ratio. 1. 24 yardsto 3 yards 2. 150 cm : 6 m

  5. Painting You are planning to paint a mural on a rectangular wall. You know that the perimeter of the wall is 484feet and that the ratio of its length to its width is 9 : 2. Find the area of the wall. STEP 1 Write expressions for the length and width. Because the ratio of length to width is 9 : 2, you can represent the length by 9xand the width by 2x. EXAMPLE 3 Use a ratio to find a dimension SOLUTION

  6. STEP 2 Solve an equation to findx. 2l + 2w = P 484 2(9x) + 2(2x) = 484 22x = x 22 = Evaluate the expressions for the length and width. Substitute the value of xinto each expression. STEP 3 The wall is 198feet long and 44feet wide, so its area is 198 ft 44 ft = 8712 ft. 2 EXAMPLE 3 Use a ratio to find a dimension (continued) Formula for perimeter of rectangle Substitute for l, w, and P. Multiply and combine like terms. Divide each side by 22. Length= 9x = 9(22) = 198 Width = 2x = 2(22) = 44

  7. EXAMPLE 4 Use a ratio to find a dimension The perimeter of a room is 48 feet and the ratio of its length to its width is 7:5. Find the length and width of the room.

  8. ALGEBRA The measures of the angles in CDE are in the extended ratio of 1 : 2 : 3. Find the measures of the angles. o o o o 180 x + 2x + 3x = 6x 180 = x = 30 ANSWER o o o o o The angle measures are 30 , 2(30 ) = 60 , and 3(30 ) = 90. EXAMPLE 5 Use extended ratios SOLUTION Begin by sketching the triangle. Then use the extended ratio of 1 : 2 : 3 to label the measures as x° , 2x° , and 3x° . Triangle Sum Theorem Combine liketerms. Divide each side by 6.

  9. Use Extended Ratios EXAMPLE 6 A triangle’s angle measures are in the extended ratio of 1 : 3 : 5. Find the measures of the angles.

  10. x 5 x 5 a. = 16 10 16 10 Solve the proportion. ALGEBRA a. = 5 16 = 80 10 x = 10 x 8 x = EXAMPLE 7 Solve proportions SOLUTION Write original proportion. Cross Products Property Multiply. Divide each side by 10.

  11. 1 1 2 2 b. = y + 1 y + 1 3y 3y b. = 1 3y 2 (y + 1) = 3y 2y + 2 = y 2 = EXAMPLE 8 Solve proportions SOLUTION Write original proportion. Cross Products Property Distributive Property Subtract 2y from each side.

  12. y – 3 y 1 2 5 4 a. b. c. = = = x 8 7 x – 3 3x 14 Solve proportions EXAMPLE 9 Solve the proportion.

  13. Geometric Mean

  14. Find a geometric mean EXAMPLE 10 Find the geometric mean of the two numbers. a) 12 and 27 b) 24 and 48

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