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 SOLUTION 64 m : 6 m a. Then divide out the units and simplify. a. Write 64 m : 6 m as 64 m 6 m. = 32 3 =32 : 3 Simplify the ratio. 64 m 6 m

GUIDED PRACTICE for Example 1 Simplify the ratio yards to 3 yards ANSWER 8 : 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 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 Combine like terms. SOLUTION Triangle Sum Theorem Divide each side by 6. =30x = 6x6x180 = o x + 2x + 3x ooo 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°. The angle measures are 30, 2(30 ) = 60, and 3(30 ) = 90. oo oo o ANSWER

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 proportions SOLUTION a x 16 = Multiply. Divide each side by 10. a x 16 = = 10 x5 16 = 10 x80 = x8 Write original proportion. Cross Products Property Solve the proportion. ALGEBRA

EXAMPLE 4 Solve proportions Subtract 2y from each side. 1 y y3y b. = = 2 (y + 1) 1 3y = 2y + 23y3y = y2 Distributive Property SOLUTION b. 1 y + 1 = 2 3y3y Write original proportion. Cross Products Property

GUIDED PRACTICE for Example x 5 8 = 6. 1 x – 3 4 3x3x = 7. y – 3 7 y 14 = 16 5 ANSWER 12 ANSWER 6 Solve the proportion.

EXAMPLE 5 Solve a real-world problem SOLUTION 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. 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 Simplify. Cross Products Property There are about 20,250 trees in the 150 acre area. = n 150 = n n = 20,250 number of trees area in acres Write proportion.

GUIDED PRACTICE 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. 8. 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 Simplify. Definition of geometric mean Substitute 24 for a and 48 for b. Factor. Find the geometric mean of 24 and 48. SOLUTION = x abab = = = 224 The geometric mean of 24 and 48 is

GUIDED PRACTICE for Examples 5 and 6 Find the geometric mean of the two numbers and 27 ANSWER and 54 ANSWER and 18 ANSWER 12 2

Daily Homework Quiz For use after Lesson 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

Daily Homework Quiz For use after Lesson Find the geometric mean of 42 and 12 ANSWER s + 8 = Solve ANSWER 4

Daily Homework Quiz For use after Lesson The extended ratio of the angles of a triangle is 5: 12: 13. Find the angle measures of the triangle. ANSWER 30,72, 78 oo o 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.

Homework 1, 2 – 52 evens, 58 – 66 evens