Chapter 6Similarity

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 ma.

Then divide out the units and simplify.

a. Write 64 m : 6 m as 64 m6 m

.

= 323

= 32 : 3

Simplify the ratio.

64 m6 m

GUIDED PRACTICE for Example 1

Simplify the ratio.

1. 24 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 ft20 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 ProblemsExample

• 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=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

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. 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 4 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

GUIDED PRACTICE for Example 4

5. 2 x

5 8=

6. 1x – 3

43x=

7.

y – 3 7

y14=

16 5

ANSWER

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.

=2702

n150

=270 150 2 n

n=20,250

number of treesarea 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 proportiona/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 ab

= 24 48

= 24 24 2

= 224

The geometric mean of 24 and 48 is 24 33.9.2

GUIDED PRACTICE for Examples 5 and 6

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

Daily Homework Quiz

For use after Lesson 6.1

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

Daily Homework Quiz

For use after Lesson 6.1

4. Find the geometric mean of 42 and 12

ANSWER 6 14

1s + 8 = 3

363. Solve

ANSWER 4

Daily Homework Quiz

For use after Lesson 6.1

5. The extended ratio of the angles of a triangle is 5: 12: 13. Find the angle measures of the triangle.

ANSWER 30 ,72 , 78 o o 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