20 pts in △abc, point d∈ ab with ad: db=2: 3, point m is the midpoint of cd , and aabc=30 in2. find aacd, aamc, admb, and acmb.

First, draw your diagram. Make sure all of the specifications are there.

AD : DB = 2:3

Area of ΔABC = 30 in²

This means that the sides that make up AB (AD and DB) are 2:3. The total length of side AB = 5. The altitude of ΔABC is CD, and that is 0.5 (5 × x), and if you solve that, x = 12. So CD = 12.

Since point M is the midpoint, it divides CD into 2 equal lengths, DM and CM, both of which equal 6 (12 ÷ 2 = 6)

Now:

ΔDMB = 0.5 (6 × 3) = 0.5 (18) = 9 in²

ΔDMA = 0.5 (6 × 2) = 0.5 (12) = 6 in²

ΔACD = 5/2 = 30/x, 5x = 60, x = 12 in² (Main Problem 2: If 2 triangles have congruent altitudes, then the ratio of their areas is equal to the ratio of the sides to which those altitudes are drawn.)

ΔAMC = 12 in² - 6 in² = 6 in² (Area of ΔACD - Area of ΔDMA = Area of ΔAMC)

ΔBDC = 5/3 = 30/x, 5x = 90, x = 18 in² (Main Problem 2: If 2 triangles have congruent altitudes, then the ratio of their areas is equal to the ratio of the sides to which those altitudes are drawn.)

ΔBMC = 18 in² - 9 in² = 9 in² (Area of ΔBDC - Area of ΔDMA = Area of ΔBMC)

So the answers are:

Area of ΔACD = 12 in²

Area of ΔAMC = 6 in²

Area of ΔDMB = 9 in²

Area of ΔCMB= 9 in²

Hope this helped!

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