[Math] the area of the shaded region

areageometry

Problem 2 seems to have two ways of going about it.

Way 1

Assume the whole shape is a triangle and the unshaded region is a trapezoid. Subtract the trapezoid's area from the triangle's area.

Way 2

Assume both shaded regions are triangles. Add the shaded triangles' areas.

I'm fairly certain at least Triangle 1's area is correct because finding the area of the trapezoid containing Triangle 1 and the unshaded trapezoid yields the same area as when combining the area of Triangle 1 determined in Way 2 and the area of the trapezoid determined in Way 1: A=(a+b)*h/2 = ((12+13)+15)*11/2 = 220.

Question

Why doesn't Way 2 provide the correct answer?

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Best Answer

It's the second solution which is correct, and the first one is wrong!

The error hides in the assumption the whole figure is a triangle. If it was, the big triangle would be similar to the smaller one on the right side, hence the proportion would hold $$\frac{12+13}{11+21}=\frac{15}{21}$$ However, it does not, as $$\frac{12+13}{11+21}=0.78125 > 0.7142857 \approx \frac{15}{21}$$ and the big figure is a concave quadrangle.