11.7 Ratios of Areas

Objective: After studying this section you will be able to find

ratios of areas by calculating and comparing the areas and applying

properties of similar figures.

One way to determine the ratio of the areas of two figures is to calculate the quotient of the two areas.

Computing Areas

Example Find the ratio of the area of the parallelogram to the area of the triangle.

10

9

12

8

1 1

12

A b h

A bh

9 10 90 151 48 812 82

Example

In the diagram, AB = 5 and BC = 2. Find the ratio of the area of triangle ABD to triangle CBD.

D

C B A

Similar Figures

If two triangles are similar, the ratio of any pair of their corresponding altitudes, medians, or angle bisectors equals the ratio of their corresponding sides.

W Y

X4

P

Q

R

61 1

1 1

2 22 2

1212

PQR

WXY

b hA b h

A b hb h

1 1

2 2

3 3Since and

2 2

b h

b h

3 3 9

2 2 4PQR

WXY

A

A

Theorem If two figures are similar, then the ratio of their areas equals the square of the ratio of corresponding segments. (Similar Figures Theorem)

Where A1 and A2 are areas and s1 and s2 are measures of corresponding segments.

2

1 1

2 2

A s

A s

Given the similar pentagons shown, find the ratio of their areas

12

9

Example 1

Example 2

If , find the ratio of the areas of the two triangles.ABC DEF

F E

D

B

A

C12

8

If the ratios of the areas of two similar parallelograms is 49:121, find the ratio of their bases.

Example 3AM is the median of triangle ABC. Find the ratio

:ABM ACMA A

Theorem A median of a triangle divides the triangle into two triangles with equal areas.

A

B M C

P

Q R SPQR PRSA A

Summary

State in your own words how to find the area of a figure using the corresponding segments.

Homework: worksheet