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Chapter 6 Similarity Pre-Requisite Skills Page 354 all 6.1 Ratios, Proportions, and the Geometric Mean Objective: Solve problems by writing and solving proportions Ratio • Two numbers or quantities being compared a to b a:b a/b Simplifying Ratios • Just like simplifying a fraction • 72 : 9 EXAMPLE 1 Simplify ratios Simplify the ratio. a. 64 m : 6 m SOLUTION a. Write 64 m : 6 m as 64 m . 6m Then divide out the units and simplify. 64 m = 32 = 32 : 3 6m 3 GUIDED PRACTICE Simplify the ratio. 1. 24 yards to 3 yards ANSWER 8 : 1 for Example 1 Simplifying Ratios with Different Units • What to do: Use unit analysis to cancel out units *note: simplifying the numbers can be done before or after canceling units Example: Simplify 36in : 9ft Examples b. 5 ft 20 in. 2. 150 cm : 6 m ANSWER 1 : 4 Reading Ratios • The teacher to student ratio is 1 to 22. • What does this mean? Using Ratios and to Solve Problems Example • You are planning to paint a mural on a rectangular wall. You know the perimeter of the wall is 484 feet. The ratio of its length to its width is 9:2. Find the area of the wall. Guided Practice #3 page 357 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. Example • 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. EXAMPLE 3 Use extended ratios 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° . o o o o Triangle Sum Theorem x + 2x + 3x = 180 6x = 180 Combine like terms. Divide each side by 6. x = 30 ANSWER o o o o o The angle measures are 30 , 2(30 ) = 60 , and 3(30 ) = 90. GUIDED PRACTICE for Examples 2 and 3 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° Proportions • Ratios that are = to each other • Means and Extremes (page 358) EXAMPLE 4 Solve the proportion. ALGEBRA a. Solve proportions 5 = x 10 16 SOLUTION a. 5 10 x 16 Write original proportion. 5 16 = 10 x Cross Products Property 80 10 x Multiply. x Divide each side by 10. = = 8 = EXAMPLE 4 b. Solve proportions 2 1 = y+1 3y SOLUTION b. 1 y+1 = 2 3y Write original proportion. 1 3y = 2 (y + 1) Cross Products Property 3y = 2y + 2 Distributive Property y = 2 Subtract 2y from each side. GUIDED PRACTICE Solve the proportion. 5. 2 = 5 x 8 16 ANSWER 5 1 4 6. = x–3 3x ANSWER 7. 12 y–3 y 7 = 14 ANSWER 6 for Example 4 EXAMPLE 5 Solve a real-world problem 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. EXAMPLE 5 Solve a real-world problem 270 n = 2 150 270 150 = 2 20,250 = n number of trees Write proportion. area in acres n Cross Products Property Simplify. There are about 20,250 trees in the 150 acre area. GUIDED PRACTICE 8. for Examples 5 and 6 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. ANSWER 19,500 trees Geometric Mean • A number x that satisfies the proportion a/x = x/b • To find the geometric mean of two numbers, multiply them together and take the square root – Then simplify the square root (no decimal answers) EXAMPLE 6 Find a geometric mean 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. = 24 2 Simplify. The geometric mean of 24 and 48 is 24 2 33.9. GUIDED PRACTICE for Examples 5 and 6 Find the geometric mean of the two numbers. 9. 12 and 27 ANSWER 10. 18 and 54 ANSWER 11. 18 18 3 16 and 18 ANSWER 12 2 Daily Homework Quiz 1. Simplify the ratio 1200 cm : 1.8 m. ANSWER 2. For use after Lesson 6.1 2:3 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 Daily Homework Quiz 3. Solve ANSWER 4. For use after Lesson 6.1 1 = 3 s+8 36 4 Find the geometric mean of 42 and 12 ANSWER 6 14 Daily Homework Quiz 5. The extended ratio of the angles of a triangle is 5: 12: 13. Find the angle measures of the triangle. ANSWER 6. For use after Lesson 6.1 o o o 30 ,72 , 78 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. Homework • 1, 2 – 52 evens, 58 – 66 evens