What is the area of the shaded region?

Question

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?

Answer 1

You numerical calculation is correct, indeed for the first

• whole: $\frac12 \cdot 32 \cdot 25 =400$
• trapezium: $\frac12 \cdot (15+12) \cdot 11 = 148.5$

then $A=251.5$.

For the second

• triangle 1: $\frac12 \cdot 15 \cdot 21 = 157.5$
• triangle 1: $\frac12 \cdot 13 \cdot 11 = 71.5$

then $A=229$.

The discreapncy depends upon the fact that the given of the problems are wrong, indeed observe that the segment with length 15 is not orthogonal to the base indeed

$$\frac{15}{21}=\frac{5}{7}\approx 0.71423\neq \frac{25}{32}=0.78125$$

Answer 2

The first triangle is not a right triangle, so determining the base and height for the formula $A=\frac12(base)(height)$ is a little more delicate.

Indeed, for the base of triangle $1$, one needs to draw a right triangle with the base of triangle $1$ being the hypotenuse of the "new" triangle. Then we can calculate the base to have length:

$$ \sqrt{3^2+11^2}=\sqrt{130}. $$

The height is a little more challenging since we don't know the angles of the given triangles, but the length of the third side can easily be calculated using a similar technique, and one can then apply Heron's Formula to calculate the area of triangle $1$.

Answer 3

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.