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Warm Up

Ratios, Proportions, and the Geometric Mean. Warm Up. Lesson Presentation. Lesson Quiz. 1. Simplify. 3 x 2. 9 x. x 3. ANSWER. 288. ANSWER. Warm-Up. 2. Solve 2 n = 18 • 32. 6. ANSWER. 4. Solve x = √ 36 • 16. 24. ANSWER. Warm-Up. 3. Solve 4 x = 2 x + 12. 80º. ANSWER.

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Warm Up

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  1. Ratios, Proportions, and the Geometric Mean Warm Up Lesson Presentation Lesson Quiz

  2. 1.Simplify . 3x2 9x x3 ANSWER 288 ANSWER Warm-Up 2.Solve 2n = 18 • 32.

  3. 6 ANSWER 4.Solvex = √36 • 16. 24 ANSWER Warm-Up 3.Solve 4x = 2x + 12.

  4. 80º ANSWER Warm-Up 5.A triangle has angle measures 18ºand82º.Find the measure of the third angle.

  5. 5 ft 5 ft 5 ft 20 in. 20 in. 20 in. b. a. 64 m : 6 m 64 m 64 m 6 m 6 m Write 64 m : 6 m as . Then divide out the units and simplify. 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 the ratio. SOLUTION

  6. 8 : 1 1 : 4 ANSWER ANSWER Guided Practice Simplify the ratio. 1. 24 yardsto 3 yards 2. 150 cm : 6 m

  7. 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 2 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. Painting SOLUTION

  8. 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 2 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

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

  10. 14 ft, 10 ft 20°, 60°, 100° ANSWER ANSWER Guided Practice 3. 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. 4.A triangle’s angle measures are in the extended ratio of 1 : 3 : 5. Find the measures of the angles.

  11. 5 x x 5 a. = 16 10 16 10 Solve the proportion. ALGEBRA 5 16 = = 80 10 x = 10 x 8 x = Example 4 SOLUTION a. Write original proportion. Cross Products Property Multiply. Divide each side by 10.

  12. 1 1 2 2 b. = y + 1 y + 1 3y 3y Solve the proportion. ALGEBRA = 1 3y 2 (y + 1) = 3y 2y + 2 = y 2 = Example 4 SOLUTION b. Write original proportion. Cross Products Property Distributive Property Subtract 2y from each side.

  13. y – 3 y 1 2 5 4 7. 5. 6. = = = x 8 x – 3 7 3x 14 16 ANSWER 5 12 6 ANSWER ANSWER Guided Practice Solve the proportion.

  14. SCIENCE As part of an environmental study, you need to estimate the number of trees in a 150 acre area. You count 270 trees in a 2 acre area and you notice that the trees seem to be evenly distributed. Estimate the total number of trees. Example 5 SOLUTION Write and solve a proportion involving two ratios that compare the number of trees with the area of the land.

  15. n number of trees = 150 area in acres 270 150 = 20,250 270 n = 2 2 n Example 5 Write proportion. Cross Products Property Simplify. There are about 20,250 trees in the 150 acre area.

  16. = ab x 2448 = 24 24 2 = = 24 2 The geometric mean of 24 and 48 is 2433.9. 2 Example 6 Find the geometric mean of 24 and 48. SOLUTION Definition of geometric mean Substitute 24 for a and 48 for b. Factor. Simplify.

  17. 8. WHAT IF?In Example 5, suppose you count 390 trees in a 3 acre area of the 150 acre area. Make a new estimate of the total number of trees. 19,500 trees ANSWER Guided Practice

  18. 18 12 ANSWER ANSWER 18 ANSWER 3 2 Guided Practice Find the geometric mean of the two numbers. 9.12 and 27 10.18 and 54 11.16 and 18

  19. 2 : 3 ANSWER ANSWER 192 mm, 72 mm Lesson Quiz 1.Simplify the ratio 1200 cm : 1.8 m. 2.The perimeter of a rectangle is 528 millimeters . The ratio of length of the width is 8 : 3. Find the length and the width.

  20. ANSWER 1 3 6 14 s + 8 36 3. Solve = ANSWER 4 Lesson Quiz 4. Find the geometric mean of 42 and 12.

  21. ANSWER 30°, 72°, 78° ANSWER 108 in. Lesson Quiz 5. The extended ratio of the angles of a triangle is 5: 12: 13. Find the angle measures of the triangle. 6. The area of a rectangle is 720 square inches. If the ratio of the length to the width is 5 : 4, find the perimeter of the rectangle.

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