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

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

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

Warm-Up 5. A triangle has angle measures 18º and 82º. Find the measure of the third angle. ANSWER 80º

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

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

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

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

The angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°. Example 3 ALGEBRA The measures of the angles in CDE are in the extended ratio of 1 : 2 : 3. Find the measures of the angles. SOLUTION 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° Triangle Sum Theorem = 6x 180 Combine like terms. = 30 x Divide each side by 6. The angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.

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

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

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

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

Example 5 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. SOLUTION Write and solve a proportion involving two ratios that compare the number of trees with the area of the land.

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

Find the geometric mean of 24 and 48. Example 6 Find the geometric mean of 24 and 48. SOLUTION = x ab Definition of geometric mean = 24 48 Substitute 24 for a and 48 for b. = 24 24 2 Factor. = 2 24 Simplify. The geometric mean of 24 and 48 is 24 33.9. 2

Guided Practice 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. 8. ANSWER 19,500 trees

Guided Practice Find the geometric mean of the two numbers. 9. 12 and 27 ANSWER 18 10. 18 and 54 ANSWER 18 3 11. 16 and 18 ANSWER 12 2

Lesson Quiz 1. Simplify the ratio 1200 cm : 1.8 m. ANSWER 2 : 3 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. ANSWER 192 mm, 72 mm

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

Lesson Quiz 5. The extended ratio of the angles of a triangle is 5: 12: 13. Find the angle measures of the triangle. 30°, 72°, 78° ANSWER 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. ANSWER 108 in.