6.1 – Ratios, Proportions, and the Geometric Mean

GeometryMs. Rinaldi

Ratio

• Ratio – a comparison of numbers• A ratio can be written 3 ways:

1. a:b

2.

3. a to b

Examples: 2 girls to 7 boys, length:width = 3:2

b

a

EXAMPLE 1 Simplify ratios

SOLUTION

64 m : 6 ma.

Then divide out the units and simplify.

b. 5 ft20 in.

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

a. Write 64 m : 6 m as 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

EXAMPLE 2 Simplify Ratios

Simplify the ratio.

1. 24 yards to 3 yards

2. 150 cm : 6 m

EXAMPLE 3 Use a ratio to find a dimension

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

EXAMPLE 3 Use a ratio to find a dimension (continued)

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) = 198Width = 2x = 2(22) = 44

EXAMPLE 4 Use a ratio to find a dimension

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 5 Use extended ratios

Combine like terms.

SOLUTION

Triangle Sum Theorem

Divide each side by 6.= 30x=6x 180= 180

ox + 2x + 3x o oo

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.o o o o o

ANSWER

EXAMPLE 6 Use Extended Ratios

A triangle’s angle measures are in the extended ratio of 1 : 3 : 5. Find the measures of the angles.

EXAMPLE 7 Solve proportions

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

EXAMPLE 8 Solve proportions

Subtract 2y from each side.

1y + 1

23y

b. =

= 2 (y + 1)1 3y

= 2y + 23y

=y 2

Distributive Property

SOLUTION

b. 1y + 1 = 2

3y

Write original proportion.

Cross Products Property

EXAMPLE 9 Solve proportions

a. 2 x

5 8=

b. 1x – 3

43x=

c.

y – 3 7

y14=

Solve the proportion.

Geometric Mean

EXAMPLE 10 Find a geometric mean

Find the geometric mean of the two numbers.

a) 12 and 27

b) 24 and 48