Holt Geometry

7-1 Ratio and Proportion7-1 Ratio and Proportion

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

7-1 Ratio and Proportion

Warm UpFind the slope of the line through each pair of points.

1. (1, 5) and (3, 9)

2. (–6, 4) and (6, –2)

Solve each equation.

3. 4x + 5x + 6x = 45

4. (x – 5)2 = 81

5. Write in simplest form.

2

x = 3

x = 14 or x = –4

Holt Geometry

7-1 Ratio and Proportion

Write and simplify ratios.

Use proportions to solve problems.

Objectives

Holt Geometry

7-1 Ratio and Proportion

ratioproportioncross products

Vocabulary

Holt Geometry

7-1 Ratio and Proportion

The Lord of the Rings movies transport viewers to the fantasy world of Middle Earth. Many scenes feature vast fortresses, sprawling cities, and bottomless mines. To film these images, the moviemakers used ratios to help them build highly detailed miniature models.

Holt Geometry

7-1 Ratio and Proportion

A ratio compares two numbers by division. The ratio

of two numbers a and b can be written as a to b, a:b,

or , where b ≠ 0. For example, the ratios 1 to 2,

1:2, and all represent the same comparison.

In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined.

Remember!

Holt Geometry

7-1 Ratio and Proportion

In Mr. Alexander’s Geometry class, there are 12 boys

and 16 girls. Write the following ratios.

Examples:

1. Boys to Girls 2. Girls to 3. Girls to Boys

12

16

3

4

12

16

4

3

16

Students

28

4

7

Holt Geometry

7-1 Ratio and Proportion

In the United States House of Representatives there are 435 seats. Of those, 70 are occupied by women. Write the ratio of men to women in

the US House of Representatives.

36570

Holt Geometry

7-1 Ratio and Proportion

Solving problems with ratios!

To simplify a ratio, we divide out the common factor. So, when we solve we are looking for that factor.

12

16Divide 4 out

3

4How do we put the 4 back in?

3 4

4 4

12

16

So to make ratios big again we multiply!

Holt Geometry

7-1 Ratio and Proportion

The ratio of the lengths of an isosceles triangle is 4:4:7, and its perimeter is 52.5 cm. What are the lengths of the sides of the triangle?

How do we find perimeter?

___ ___ ___ 52.5

4 4 7

x x x

We know we have a missing factor. What do we call it?

X = 3.5

15x = 52.54(3.5) = 14

4(3.5) = 14

7(3.5) = 24.5

Holt Geometry

7-1 Ratio and Proportion

• One common ratio is slope, which is the comparison of the change in y to the change in x. This can also be expressed as

Rise

Run

Rate of change

m

Holt Geometry

7-1 Ratio and Proportion

Example 7

Write a ratio expressing the slope of the line.

4

6

2

3

Holt Geometry

7-1 Ratio and Proportion

Example 8

Write the slopes as a ratio for points A(7, 9) and B(2, -6).

Substitute the given values.

Simplify.

-6 – 9

2 – 7

-15

-5 =

3

1

Holt Geometry

7-1 Ratio and Proportion

A proportion is an equation stating that two ratios

are equal. When you cross multiply you create equal cross products.

For example in the proportion , ad = bc. Once you have cross multiplied you need to solve for the variable using algebra.

Holt Geometry

7-1 Ratio and Proportion

Example 9: Solving Proportions

Solve the proportion.

Cross Products Property

Simplify.

Divide both sides by10.

4(65) = k(10)

260 = 10k

k = 26

4

10 65

k

Holt Geometry

7-1 Ratio and Proportion

Example 10: Solving Proportions

Solve the proportion.

Cross Products Property p(p)= 4(9)

Simplify. p2 = 36

Find the square root of both sides. p = 6

4

9

p

p

Holt Geometry

7-1 Ratio and Proportion

Example 11

Solve the proportion.

Cross Products Property

Distribute.

Subtract both sides by 3x.

3(x + 8) = 4(x + 3)

3x + 24 = 4x + 12

24 = x + 12

Subtract both sides by 12.12 = x

3 4

3 8x x

Holt Geometry

7-1 Ratio and Proportion

Example 12

Cross Products are 7a and 5b

5b

Given that , complete the following equations.

5

7

a

b

7a =b

a

5

a 7

b

7

5

7

b 5

a

Holt Geometry

7-1 Ratio and Proportion

The following table shows equivalent forms of the Cross Products Property.

Holt Geometry

7-1 Ratio and Proportion

Example 13: Problem-Solving Application

11 Understand the Problem

The answer will be the distance from the Union Station to the Dallas Public Library.

The scale of a map of downtown Dallas is 1.5 cm:300 m. If the distance between Union Station and the Dallas Public Library is 6 cm, what is the actual distance?

Holt Geometry

7-1 Ratio and Proportion

Example 13 Continued

22 Make a Plan

Let x be the distance from the Union Station to the Dallas Public Library. Write a proportion that compares the ratios of the width to the length.

Distance in cm

Distance in m

1.5 6

300 x

Holt Geometry

7-1 Ratio and Proportion

Solve33

Example 13 Continued

Cross Products Property

Simplify.

Divide both sides by 1.5.

6(300) = x(1.5)

1800 = 1.5x

x = 1200

The distance from the Union Station to the Dallas Public Library is 1200 m.

1.5 6

300 x

Holt Geometry

7-1 Ratio and Proportion

Look Back44

Example 13 Continued

Check the answer in the original problem. The

ratio of the scale distance to actual distance is

6:1200, or 1:200. The ratio of the given scale is

also 1:200. So the ratios are equal, and the

answer is correct.

6/1200 = .005

1.5/300 = .005

Holt Geometry

7-1 Ratio and Proportion

Example 14

The $250,000 budget for a local shelter is allocated proportionally to the men’s and women’s departments according to the population in the shelter by gender. If there are 1946 women and 399 men in the shelter, what amount rounded to the nearest dollar is allocated to the men’s department?

Holt Geometry

7-1 Ratio and Proportion

Example 15

After an election in a small town, the newspaper reported that 42% of the registered voters actually voted. If 12,000 people voted, how many people are registered to vote in the town?

Holt Geometry

7-1 Ratio and Proportion

Example 16

A student wanted to find the height of a statue of a pineapple in Nambour, Australia. She measured the pineapple’s shadow at 8 ft 9in and her own shadow at 2 ft. The student’s height is 5 ft 4 in. What is the height of the pineapple?

Holt Geometry

7-1 Ratio and Proportion

Example 17

The Lincoln Memorial in Washington, D.C., is approximately 57 m long and 36 m wide. If you would want to make a scale drawing of the base of the building using a scale of 1 cm: 15 m, what would be the dimensions of the scale drawing?

Holt Geometry

7-1 Ratio and Proportion

Lesson Quiz1. The ratio of the angle measures in a triangle is

1:5:6. What is the measure of each angle?

Solve each proportion.

2. 3.

4. Given that 14a = 35b, find the ratio of a to b in

simplest form.

5. An apartment building is 90 ft tall and 55 ft

wide. If a scale model of this building is 11 in.

wide, how tall is the scale model of the building?

15°, 75°, 90°

3 7 or –7

18 in.