100 Points! Please help with this geometry question! Give an explanation, it could be short or statement reasonings! Thanks!

168 cm²

Step-by-step explanation:

Refer to attached

Given Area of ΔDCF = 63 cm²To findArea of ΔABCSolutionLet the area of ABC be A, thenA = 1/2bh, where b= AC is the base and h is the height of the triangle

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The triangle DCF can be divided into DEC and DEF as per picture and the area of them is:

Area of ΔDEF

1/2DE*h2

Area of ΔDEC

1/2DE*h1

Since D and E are midpoints of AB and BC:

DE = 1/2AC = 1/2bAnd h1 = 1/2h, h2 = 1/2h1 = 1/2*1/2h = 1/4h

So the area of ΔDCF is:

1/2(1/2bh1) + 1/2(1/2bh2) =1/2(1/2b*1/2h) + 1/2(1/2b*1/4h) = 1/8bh + 1/16bh =3/16bh

Since the area of ABC is A = 1/2bh we have:

3/8*1/2bh = 3/8A and 3/8A = 63

Solving for A:

A = 63*8/3 = 168 cm²

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