Warm-up

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

D C length

A width B

Proportions

proportion: an equation that equates two ratios

Properties of Proportions 1. Cross Product Property: The

product of the extremes equals the product of the means.

If , then ad = bc.

2. Reciprocal Property: If two ratios are equal, then their reciprocals are also equal.

If , then .

€

a

b=c

d

€

a

b=c

d

€

a

b=c

d

€

b

a=d

c

Problem Solving with Proportions and Ratios

Chapter 8.1-8.2

Objective: Students will be able to apply proportions to similar figures in real-world problems.

Central Park

Empire State

Building

Times Square

The Statue of Liberty

For tourists visiting New York City, it can be hard to comprehend the actual size of the Statue of Liberty.

Your Task

To investigate the actual size of the Statue of Liberty’s nose

Given: The arm of the Statue of Liberty is 42 feet long.

Prove: The length of the Statue of Liberty’s nose.

Just to clarify:

The length of the Statue of Liberty’s nose means: Measure from the bridge of your nose

(where your nose start between your eyes)

End at the very tip of the nose

Arm length is from shoulder to the tip of your middle finger.

Hints…

What is the Statue of Liberty a statue of?

How long is your arm? How long is your nose?

Actual Length: 4.5 feet

Example 2:

International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. Use the given measurements to find the value of x.

A4 A3

210mm

x 420mm

x

Geometric Mean

geometric mean: two positive numbers

a and b is the positive number x such

that . If you solve the proportionfor x, you find that x = .

€

a

x=x

b

€

a • b

Find the geometric mean of 6 and 10.

€

6

x=x

10

Arithmetic Mean vs. Geometric Mean

Arithmetic Mean – If five students took an exam and their scores were 60%, 70%, 80%, 90% and 100%, the arithmetic class average would be 80%.

€

60 + 70 + 80 + 90 +100

5= 80

Arithmetic Mean vs. Geometric Mean

€

(1.1)(1.5)(1.3)( )1

3

= (1.1)(1.5)(1.3)3

= 1.28966159...

≈ 1.29

≈ 29% annual returns

Geometric Mean – Suppose you have an investment which

earns 10% the first year, 50% the second year, and 30% the third year. What is its average annual rate of return?

• Since your investment returns depended on the previous years. You need to use geometric mean to calculate the average annual rate of return.

Practice Time!1. The perimeter of a rectangle is 40 feet. The

ratio of the width to the length is 2:3. Find the length and the width.

4 2x

x+4AB

C

2. The measures of the angles in a triangle are in the extended ratio of 2:4:8. Find the measures of the angles.

3. Find x, if AC:BC:AB is 2:1:2.

x = 4, 8:12

€

x =126

7

255

7, 51

3

7, 102

6

7

x = 4

Solve each proportion

€

1. 3

8=x

5

€

2. 2x −13

28=

−4

7

€

3. 3x −1

2=

−2

x + 2

€

x =17

8

€

x =−11

2

€

x =−2

3, −1

Cool Down

Write down three things you learned about ratios and proportions.