11.3

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Perimeter and Area of Similar Figures

11.3 Warm-Up

1. Two polygons are similar and the ratio of corresponding sides is 3 : 4. What is the ratio of the perimeters?

2. Solve .12x = 18

6

ANSWER 3 : 4

ANSWER 4

11.3 Warm-Up

3. A rectangle has area 108 square feet and the length is three times the width. What are the dimensions of the rectangle?

ANSWER 18 ft by 6 ft

11.3 Example 1

Ratio (red to blue) of the perimeters

a.

Ratio (red to blue) of the areas

b.

In the diagram, ABC DEF. Find the indicated ratio.∆ ∆

a. By Theorem 6.1 on page 374, the ratio of the perimeters is 2:3.

b. By Theorem 11.7 above, the ratio of the areas is 22:32, or 4:9.

SOLUTION

The ratio of the lengths of corresponding sides is

128

= 32 , or 2:3.

11.3 Example 2

SOLUTION

The ratio of a side length of the den to the corresponding side length of the bedroom is 14:10, or 7:5. So, the ratio of the areas is 72:52, or 49:25. This ratio is also the ratio of the carpeting costs. Let x be the cost for the den.

11.3 Example 2

2549 = 225

x cost of carpet for dencost of carpet for bedroom

x 441= Solve for x.

ANSWER

It costs $441 to carpet the den. The correct answer is D.

11.3 Guided Practice

1. The perimeter of ∆ABC is 16 feet, and its area is 64 square feet. The perimeter of ∆DEF is 12 feet. Given ∆ABC ~ ∆DEF, find the ratio of the area of ∆ABC to the area of ∆DEF. Then find the area of ∆DEF.

; 36 ft2

916

ANSWER

11.3 Example 3

Then use Theorem 11.7. If the area ratio is a2:b2, then the length ratio is a:b.

Cooking A large rectangular baking pan is 15 inches long and 10 inches wide. A smaller pan is similar to the large pan. The area of the smaller pan is 96 square inches. Find the width of the smaller pan.

SOLUTION

First draw a diagram to represent the problem. Label dimensions and areas.

11.3 Example 3

Area of smaller panArea of large pan = 150

96 = 2516 Write ratio of known

areas. Then simplify.

Find square root of area ratio.

= 45

Length in smaller panLength in large pan

Any length in the smaller pan is , or 0.8, of the corresponding length in the large pan. So, the width of the smaller pan is 0.8(10 inches) 8 inches.

45

=

11.3 Example 4

The floor of a gazebo is a regular octagon. Each side of the floor is 8 feet, and the area is about 309 square feet. You build a small model gazebo in the shape of a regular octagon. The perimeter of the floor of the model gazebo is 24 inches. Find the area of the floor of the model gazebo to the nearest tenth of a square inch.

Gazebo

11.3 Example 4

SOLUTION

All regular octagons are similar, so the floor of the model is similar to the floor of the full-sized gazebo.

STEP 1 Find the ratio of the lengths of the two floors by finding the ratio of the perimeters. Use the same units for both lengths in the ratio.

Perimeter of full-sizedPerimeter of model = 8(8 ft)

24 in. = 64 ft2 ft = 32

1

So, the ratio of corresponding lengths (full-sized to model) is 32:1.

11.3 Example 4

STEP 2 Calculate the area of the model gazebo’s floor. Let x be this area.

(Length of full-sized)2

(Length of model)2Area of full-sizedArea of model=

309 ft2

x ft2

322

12=

Theorem 11.7

Substitute.

1024x 309= Cross Products Property

x ≈ 0.302 ft2 Solve for x.

11.3 Example 4

STEP 3 Convert the area to square inches.

0.302 ft2 144 in.2

1 ft.2 ≈ 43.5 in.2

The area of the floor of the model gazebo is about 43.5 square inches.

11.3 Guided Practice

2. The ratio of the areas of two regular decagons is 20:36. What is the ratio of their corresponding side lengths in simplest radical form?

ANSWER

53

√

11.3 Guided Practice

3. Rectangles I and II are similar. The perimeter of Rectangle I is 66 inches. Rectangle II is 35 feet long and 20 feet wide. Show the steps you would use to

find the ratio of the areas and then find the area of Rectangle I.

661320 = 1

201

400

is the ratio of sides, so the ratio of areas

is , 252 in.2

ANSWER

11.3 Lesson Quiz

Figure I Figure II. Find the ratio of the perimeters and the ratio of the areas. Then find the unknown area.

ANSWER

Ratio of perimeters: 4:3; ratio of areas: 16:9; 33.75 ft2

1.

A = 60 ft2

11.3 Lesson Quiz

Figure I Figure II. Find the ratio of the perimeters and the ratio of the areas. Then find the unknown area.

2.

A = 780 cm2

ANSWER

Ratio of perimeters: 2:3; ratio of areas: 4:9; 346 cm223

11.3 Lesson Quiz

3. Rectangle I Rectangle II. In Rectangle I, the length is 20 feet and the perimeter is 64 feet. In Rectangle II, the width is 8 yards. Find the ratio of the area of Rectangle I to the area of Rectangle II.

ANSWER 121 : 144