6.1

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Ratios, Proportions, and the Geometric Mean

6.1 Warm-Up

ANSWER 288

1. Simplify .3x2

9x

2. Solve 2n = 18 • 32.

ANSWERx3

6.1 Warm-Up

ANSWER 6

ANSWER 24

4. Solve x = √36 • 16.

3. Solve 4x = 2x + 12.

6.1 Warm-Up

ANSWER 80º

5. A triangle has angle measures 18º and 82º. Find the measure of the third angle.

6.1 Example 1

SOLUTION

64 m : 6 ma. b. 5 ft20 in.

b. To simplify a ratio with unlike units, multiply by a conversion factor.

a. Write 64 m : 6 m as . Then divide out the units and simplify.

64 m6 m

= 323

= 32 : 3

5 ft20 in. = 60

20 = 31= 5 ft

20 in.12 in.1 ft

Simplify the ratio.

64 m6 m

6.1 Guided Practice

Simplify the ratio.

1. 24 yards to 3 yards

ANSWER 8 : 1

2. 150 cm : 6 m

ANSWER 1 : 4

6.1 Example 2

SOLUTION

Painting 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.

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

6.1 Example 2

STEP 2

Solve an equation to find x.

Formula for perimeter of rectangle

Substitute for l, w, and P.Multiply and combine like terms.

Divide each side by 22.

=2l + 2w P=2(9x) + 2(2x) 484= 48422x

Evaluate the expressions for the length and width. Substitute the value of x into each expression.

STEP 3

The wall is 198 feet long and 44 feet wide, so its area is198 ft 44 ft = 8712 ft .2

= 22x

Length = 9x = 9(22) = 198

Width = 2x = 2(22) = 44

6.1 Example 3

Combine like terms.

SOLUTION

Triangle Sum Theorem

Divide each side by 6.= 30x

=6x 180

= 180°x° + 2x° + 3x°

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°.

6.1 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°

6.1 Example 4

SOLUTION

a. 510

x16=

Multiply.

Divide each side by 10.

a. 510

x16=

= 10 x5 16

= 10 x80

= x8

Write original proportion.

Cross Products Property

Solve the proportion.ALGEBRA

6.1 Example 4

SOLUTION

Solve the proportion.ALGEBRA

Subtract 2y from each side.

b. 1y + 1

23y=

= 2 (y + 1)1 3y

= 2y + 23y

=y 2

Distributive Property

b. 1y + 1 = 2

3y

Write original proportion.

Cross Products Property

6.1 Guided Practice

5. 2 x

5 8=

6. 1x – 3

43x=

7.

y – 3 7

y14=

16 5

ANSWER

ANSWER 12

ANSWER 6

Solve the proportion.

6.1 Example 5

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.

6.1 Example 5

Simplify.

Cross Products Property

There are about 20,250 trees in the 150 acre area.

=2702

n150

=270 150 2 n

n=20,250

number of treesarea in acres Write proportion.

6.1 Example 6

Simplify.

Definition of geometric mean

Substitute 24 for a and 48 for b.

Factor.

Find the geometric mean of 24 and 48.

SOLUTION

=x ab

= 24 48

= 24 24 2

= 224

The geometric mean of 24 and 48 is 24 33.9.2

6.1 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

6.1 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

6.1 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

6.1 Lesson Quiz

4. Find the geometric mean of 42 and 12.

ANSWER 6 14

1s + 8 = 3

363. Solve

ANSWER 4

6.1 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.