11.7 Ratios of Areas

Ratio of Areas: What is the area ratio between ABCD and XYZ? A B C D9 10 Y X Z 12 8

Steps: 1.Set up fraction 2.Write formulas 3.Plug in numbers 4.Solve and label with units

1. Ratio A A 2. A = b 1 h 1 A = 1/2b 2 h 2 3. = 910 1/ = =15 8

Find the ratio of ABD to CBD CA D B When AB = 5 and BC = 2 Notice that the height of both triangles are congruent. When you set up the problem, the 1/2 and the height disappear leaving only 5/2 as the ratio.

Similar triangles: Ratio corresponding of : 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 PR 6 X Y W 4 QP = 6 XW 4 =3 2

Q PR 6 X Y W 4 Ratio of area: A PQR = 1/2 b 1 h 1 A WXY 1/2 b 2 h 2 = b 1 h 1 b2h2b2h2 = 33 2 = 9 4

Area ratio is the sides ratio squared! T109: If 2 figures are similar, then the ratio of their areas equals the square of the ratio of the corresponding segments. (similar-figures Theorem) A 1 = S 1 2 A 2 S 2 When A 1 and A 2 are areas and S 1 and S 2 are measures of corresponding segments.

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 C M B Notice these are not similar figures!

A C M B 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 b 1 h 1 A ∆ACM 1/2 b 2 h 2 = 1/2 xy 1/2 xy = 1 Therefore the ratio is 1:1 They are equal !

T110: The median of a triangle divides the triangle into two triangles with equal area.

Find the ratios: 10cm 9cm 10cm 9cm

Find the ratio 2 6