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7-1 Ratios and Proportions A ratio is a comparison of two quantities using division. The ratio of

quantities a and b can be expressed as a to b, a:b, or

, where .

Example: In 2007 the Boston RedSox baseball team won 96 games out of 162

games played. Write a ratio for the number of games won to the total number

of games played.

To find ratio, divide the number of games won by total number of games played.

Won: 96 games

Total: 162 games

Ratio:

Conclusion: The Boston Red Sox won about 59% of their games in 2007.

An equation stating that two ratios are equal is called a proportion.

Therefore,

is a proportion for any numbers a and c and any

nonzero numbers b and d. In a proportion, the cross products are equal.

So,

if and only if

Example: Monique randomly surveyed 30 students from her class and found

that 18 had a dog or a cat for a pet. If there is 870 students in Monique’s

school, predict the total number of students with a dog or a cat.

First set up the proportion:

Then, cross multiply:

So, Monique has predicted that 522 students will have a dog or a cat.

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Examples:

1. There are 182 girls in the sophomore class of 305. What is the ratio of girls to

total students?

2. The length of a rectangle is 8 inches and its width is 5 inches. What is the ratio

of length to width?

3. The ratio of the sides of a triangle is 8:15:17. Its perimeter is 480 inches. Find

the length of each side of the triangle.

4. The ratio of the angles in a triangle is 7:9:20. Find the measure of each angle

in the triangle.

Solve each proportion:

5.

7.

6.

8.

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9. Mayor Hefty conducted a random survey of 200 voters and found that 135

approve of the job she is doing. If there are 48,000 voters in Mayor Hefty’s town,

predict the total number of voters who approve of the job she is doing.

Write and solve a proportion that compares the number of registered voters and the number of

registered voters who approve of the job the mayor is doing.

10. If 3 DVD’s cost $44.85, find the cost of one DVD.

11. If a 6-foot post casts a shadow that is 8 feet long, how tall is an antenna that

casts a 60-foot shadow at the same time?

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7-2 Similar Polygons Similar polygons have the same shape but not necessarily the same size

Similar figures have:

1. _______________________________________

2. _______________________________________

The symbol means is similar to.

Are the following similar? If so, write a similarity statement and give your reasoning.

1.

2.

3.

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The ratio of the lengths of the corresponding sides of two similar polygons is called the scale factor. The scale factor depends on the

order of comparison.

Example: If , find the scale factor of .

The scale factor is

.

Example:

Given that the figures are similar, find the value of x.

1. 2.

3.

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Word Problem Examples:

1. An electronics company manufactures widescreen television sets

in several different sizes. The rectangular viewing area of each television size is similar to the viewing areas of the other sizes.

The company’s 42-inch widescreen television has a viewing area

perimeter of approximately 144.4 inches. What is the viewing area perimeter of the company’s 46-inch widescreen television?

2. An official Olympic-sized ice hockey rink measures 30 meters by 60 meters. The ice hockey rink at the local community college

measures 25.5 meters by 51 meters. Are the ice hockey rinks

similar? Explain your reasoning.

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7-3 Similar Triangles

Similar Triangle Tests: AA Similarity:

SSS Similarity:

SAS Similarity:

Determine whether the triangles are similar. State the test used

and write a mathematical sentence about the similarity.

1.

2.

3.

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Determine whether the triangles are similar. If so, write a similarity statement.

Explain your reasoning.

1. 2.

3. 4.

Identify the similar triangles in each figure. Explain why they are similar and find

the missing measure.

1. 2.

3. 4.

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Word Problems:

1.

2. A local furniture stores sells two versions of the same chair: one for adults,

and one for children. Find the value of x such that the chairs are similar.

3. Gavin and Brianna want to know how far a mountain peak is from their

houses. They measure the angles between the line of sight to the peak and to

each other’s houses and carefully make the drawing shown.

The actual distance between Gavin and Brianna’s house is 1.5 miles.

a. What is the actual distance of the mountain peak from Gavin’s house?

Round to the nearest tenth of a mile.

b. What is the actual distance of the mountain peak from Brianna’s house?

Round to the nearest tenth of a mile.

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7-4 Parallel Lines and Proportional Parts

Proportionality Theorem

With Triangles:

,

,

With Parallel Lines:

1.Find x 2.

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3. Find x. 4. Find x and y.

Example:

1. 2.

3.

4.

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Word Problems:

1. 2.

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7-5 Parts of Similar Triangles

Special Segments of Similar Triangles:

When two triangles are similar, corresponding altitudes, angle

bisectors, and medians are proportional to the corresponding

sides.

Example:

Find x.

1. 2.

3. 4.

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5. 6.

7.

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7-6 Similarity Transformations

A dilation is a transformation that enlarges or reduces the original figure

proportionally. Dilations are performed with respect to a fixed point called the center of dilation. The scale factor of a dilation describes

the extent of the dilation. The scale factor is the ratio of a length on the

image to a corresponding length on the preimage.

When the scale factor is greater than 1, the dilation is an enlargement,

and when the scale factor is less than 1, the dilation is a reduction.

Example: Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation.

1. 2.

3. 4.

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Graph the original figure and its dilated image. Then verify that the dilation is a

similarity transformation.

1.

2.

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7-7 Scale Drawings and Models A scale model or a scale drawing is an object or drawing with lengths

proportional to the object it represents. The scale of a model or drawing

is the ratio of a length on the model or drawing to the actual length of the object being modeled.

Example: 1. The scale on a map shown is 0.75 inches: 6 miles. Find the actual distance from

Pineham to Menlo Fields.

(The distance between Pineham and Menlo Fields is 1.25 inches.)

2.A doll house that is 15 inches tall is a scale model of a real house that

with a height of 20 feet.

a. What is the scale of the model?

b. How many times as tall as the actual house is the model?

3.

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4.

5.