CHAPTER

7Ratio and Proportion

Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

7.1 Introduction to Ratios7.2 Rates and Unit Prices7.3 Proportions7.4 Applications of Proportions7.5 Geometric Applications

OBJECTIVES

7.5 Geometric Applications

Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

a Find lengths of sides of similar triangles usingproportions.

b Use proportions to find lengths in pairs of figures that differ only in size.

7.5 Geometric Applications

a Find lengths of sides of similar triangles usingproportions.

Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Look at the pair of triangles.

7.5 Geometric Applications

a Find lengths of sides of similar triangles usingproportions.

Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

They appear to have the same shape, but their sizes are different. These are examples of similar triangles. You can imagine using a magnifying glass to enlarge the smaller triangle and get the larger. This process works because the corresponding sides of each triangle have the same ratio. That is, the following proportion is true.

7.5 Geometric Applications

SIMILAR TRIANGLES

Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Similar triangles have the same shape. The lengths of their corresponding sides have the same ratio—that is, they are proportional.

EXAMPLE

7.5 Geometric Applications

a Find lengths of sides of similar triangles usingproportions.

1

Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The triangles below are similar triangles. Find the missing length x.

EXAMPLE

7.5 Geometric Applications

a Find lengths of sides of similar triangles usingproportions.

1

Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The ratio of x to 9 is the same as the ratio of 24 to 8 or 21 to 7. We get the proportions

EXAMPLE

7.5 Geometric Applications

a Find lengths of sides of similar triangles usingproportions.

1

Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

We can solve either one of these proportions. We use the first one:

The missing length x is 27. Other proportions could also be used.

7.5 Geometric Applications

a Find lengths of sides of similar triangles usingproportions.

Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Similar triangles and proportions can often be used to find lengths that would ordinarily be difficult to measure. For example, we could find the height of a flagpole without climbing it or the distance across a river withoutcrossing it.

EXAMPLE

7.5 Geometric Applications

b Use proportions to find lengths in pairs of figures that differ only in size.

2

Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

How tall is a flagpole that casts a 56-ft shadow at the same time that a 6-ft man casts a 5-ft shadow?

EXAMPLE

7.5 Geometric Applications

b Use proportions to find lengths in pairs of figures that differ only in size.

2

Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

If we use the sun’s rays to represent the third side of the triangle in our drawing of the situation, we see that we have similar triangles. Let p = the height of the flagpole. The ratio of 6 to p is the same as the ratio of 5 to 56. Thus, we have the proportion

EXAMPLE

7.5 Geometric Applications

b Use proportions to find lengths in pairs of figures that differ only in size.

2

Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The height of the flagpole is 67.2 ft.

7.5 Geometric Applications

b Use proportions to find lengths in pairs of figures that differ only in size.

Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

When one geometric figure is a magnification of another, the figures are similar. Thus, the corresponding lengths are proportional.

EXAMPLE

7.5 Geometric Applications

b Use proportions to find lengths in pairs of figures that differ only in size.

4

Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The sides in the photographs are proportional. Find the width of the larger photograph.

EXAMPLE

7.5 Geometric Applications

b Use proportions to find lengths in pairs of figures that differ only in size.

4

Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

We let w = the width of the photograph. Then we translate to a proportion.

EXAMPLE

7.5 Geometric Applications

b Use proportions to find lengths in pairs of figures that differ only in size.

4

Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

Thus, the width of the larger photograph is 7.5 cm.