6.1

ANSWER 288

1. Simplify .3x2

9x

2. Solve 2n = 18 • 32.

ANSWERx3

ANSWER 6

ANSWER 244. Solve x = √36 • 16.

3. Solve 4x = 2x + 12.

ANSWER 80º

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

6.1 Ratios, Proportions, and the Geometric MeanBell Thinger

6.1

6.1Example 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

Simplify the ratio.

64 m6 m

= 5 ft20 in.

12 in.1 ft

.

6.1Guided Practice

Simplify the ratio.

1. 24 yards to 3 yards

ANSWER 8 : 1

2. 150 cm : 6 m

ANSWER 1 : 4

6.1Example 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

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

= 22x

Length = 9x = 9(22) = 198

Width = 2x = 2(22) = 44

Example 2

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

6.1Example 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°.

6.1Guided Practice

3. The perimeter of a rectangular 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

6.1Example 4

SOLUTION

a. 510

x16=

Multiply.

Divide each side by 10.

a. 510

x16=

= 10 x80

= x8

Write original proportion.

Cross Products Property

Solve the proportion.ALGEBRA

= 10 x5 16. .

6.1

SOLUTION

Solve the proportion.ALGEBRA

Subtract 2y from each side.

b. 1y + 1

23y=

= 2y + 23y

=y 2

Distributive Property

b. 1y + 1 = 2

3y

Write original proportion.

Cross Products Property

Example 4

= 2 (y + 1)1 3y.

6.1Guided 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.

6.1Example 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.

6.1Example 5

Simplify.

Cross Products Property

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

=2702

n150

n=20,250

number of treesarea in acres Write proportion.

=270 150 2 n. .

6.1

6.1Example 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

= 224

The geometric mean of 24 and 48 is 24 ≈ 33.9.2

= 24 48.

= 24 24 2. .

6.1Guided 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

6.1Exit Slip

1. Simplify the ratio 1200 cm : 1.8 m.

ANSWER 20 : 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

6.1Exit Slip

4. Find the geometric mean of 42 and 12.

ANSWER 6 14

3. Solves + 8 1 = 36

3

ANSWER 4

6.1Exit Slip

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.

6.1

Homework

Pg 376-379#12, 16, 28, 50, 58