Slide #1

8.1 Ratio and ProportionGeometry

Mrs. Spitz

Spring 2005

Slide #2

Objectives/Assignment Find and simplify the

ratio of two numbers. Use proportions to

solve real-life problems, such as computing the width of a painting.

Chapter 8 Definitions Ch.8 Postulates/

Theorems WS 8.1A Due end of

class WS 8.1B Homework Quizzes after 8.3 and

8.5 and 8.7

Slide #3

Computing Ratios If a and b are two quantities that are measured

in the same units, then the ratio of a to be is a/b. The ratio of a to be can also be written as a:b. Because a ratio is a quotient, its denominator cannot be zero. Ratios are usually expressed in simplified form. For instance, the ratio of 6:8 is usually simplified to 3:4. (You divided by 2)

Slide #4

Ex. 1: Simplifying Ratios Simplify the ratios:

a. 12 cm b. 6 ft c. 9 in.

4 cm 18 ft 18 in.

Slide #5

Ex. 1: Simplifying Ratios Simplify the ratios:a. 12 cm b. 6 ft

4 m 18 in

Solution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible.

Slide #6

Ex. 1: Simplifying Ratios Simplify the ratios:

a. 12 cm

4 m

12 cm 12 cm 12 3

4 m 4∙100cm 400 100

Slide #7

Ex. 1: Simplifying Ratios Simplify the ratios:

b. 6 ft

18 in

6 ft 6∙12 in 72 in. 4 4

18 in 18 in. 18 in. 1

Slide #8

Ex. 2: Using Ratios The perimeter of

rectangle ABCD is 60 centimeters. The ratio of AB: BC is 3:2. Find the length and the width of the rectangle

w

lA

B C

D

Slide #9

Ex. 2: Using Ratios SOLUTION: Because

the ratio of AB:BC is 3:2, you can represent the length of AB as 3x and the width of BC as 2x.

w

lA

B C

D

Slide #10

Solution:Statement

2l + 2w = P

2(3x) + 2(2x) = 60

6x + 4x = 60

10x = 60

x = 6

Reason

Formula for perimeter of a rectangle

Substitute l, w and P

Multiply

Combine like terms

Divide each side by 10

So, ABCD has a length of 18 centimeters and a width of 12 cm.

Slide #11

Ex. 3: Using Extended Ratios The measures of the angles

in ∆JKL are in the extended ratio 1:2:3. Find the measures of the angles.

Begin by sketching a triangle. Then use the extended ratio of 1:2:3 to label the measures of the angles as x°, 2x°, and 3x°.

J

K

L

x°

2x°

3x°

Slide #12

Solution:Statement

x°+ 2x°+ 3x° = 180°

6x = 180

x = 30

Reason

Triangle Sum Theorem

Combine like terms

Divide each side by 6

So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.

Slide #13

Ex. 4: Using Ratios The ratios of the side

lengths of ∆DEF to the corresponding side lengths of ∆ABC are 2:1. Find the unknown lengths.

8 in.

3 in.

F

D E

C

A B

Slide #14

Ex. 4: Using Ratios SOLUTION:

DE is twice AB and DE = 8, so AB = ½(8) = 4

Use the Pythagorean Theorem to determine what side BC is.

DF is twice AC and AC = 3, so DF = 2(3) = 6

EF is twice BC and BC = 5, so EF = 2(5) or 10

8 in.

3 in.

F

D E

C

A B4 in

a2 + b2 = c2

32 + 42 = c2

9 + 16 = c2

25 = c2

5 = c

Slide #15

Using Proportions An equation that

equates two ratios is called a proportion. For instance, if the ratio of a/b is equal to the ratio c/d; then the following proportion can be written:

= Means Extremes

The numbers a and d are the extremes of the proportions. The numbers b and c are the means of the proportion.

Slide #16

Properties of proportions1. CROSS PRODUCT PROPERTY. The

product of the extremes equals the product of the means.

If

= , then ad = bc

Slide #17

Properties of proportions2. RECIPROCAL PROPERTY. If two ratios

are equal, then their reciprocals are also equal.

If = , then = ba

To solve the proportion, you find the value of the variable.

Slide #18

Ex. 5: Solving Proportions

4x

57=

Write the original proportion.

Reciprocal prop.

Multiply each side by 4

Simplify.

x4

75=

4 4

x = 285

Slide #19

Ex. 5: Solving Proportions

3y + 2

2y=

Write the original proportion.

Cross Product prop.

Distributive Property

Subtract 2y from each side.

3y = 2(y+2)

y = 4

3y = 2y+4