ratios of areas lesson 11.7

Ratio of Areas:What is the area ratio between ABCD and XYZ?

A B

CD 9

10

Y

XZ

12

8

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

Steps:

1.Set up fraction

2.Write formulas

3.Plug in numbers

4.Solve and label with units

1. Ratio AA

2. A = b1h1

A = 1/2b2h2

3. = 9•10 1/2 • 8 •12= 90

48=15

8

Find the ratio of ABD to CBD

C A

D

B

When AB = 5 and BC = 2

2. ABC = 1/2b1hCBD = 1/2b2h

3. = ½(5)h ½(2)h

1. Ratio ABDCBD

4. 5:2 or 5/2

Similar triangles:

Ratio of any pair of corresponding , altitudes, medians, angle bisectors, equals the ratio of their corresponding sides.

Given ∆ PQR ∆WXY

Find the ratio of the area.

First find the ratio of the sides.

Q

P R

6X

YW

4

QP = 6XW 4

=32

Q

P R

6

X

YW

4

Ratio of area: A PQR = 1/2 b1h1 AWXY

1/2 b2h2

= b1h1

b2h2

= 3•32•2

= 94

Because they are similar triangles the ratios of the sides and heights are the same.

Area ratio is the sides ratio squared!

Theorem 109: If 2 figures are similar, then the ratio of their areas equals the square of the ratio of the corresponding segments. (similar-figures Theorem)A1 = S1 2 A2 S2

When A1 and A2 are areas and S1 and S2 are measures of corresponding segments.

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

S1 = 12S2 9

= 4 3

A1 = 4 2 A2 3

= 16 9

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

49cm2121 cm2

A1 = S1 2 A2 S2

49 = S1 2 121

S2

7 = S1

11 S2

Corresponding Segments include:

Sides, altitudes, medians, diagonals, and radii.

Ex. AM is the median of ∆ABC. Find the ratio of

A ∆ ABM : A ∆ACM

A

CMB

Notice these are not similar figures!

A

CMB

1. Altitude from A is congruent for both triangles. Label it X.

2. BM = MC because AM is a median.

Let y = BM and MC.A∆ABM = 1/2 b1h1

A ∆ACM 1/2 b2h2

= 1/2 xy1/2 xy

= 1

Therefore the ratio is 1:1 They are equal !

Theorem 110: The median of a triangle divides the triangle into two triangles with equal area.