Find area of the fourth triangle given the area of three triangles.

Question

This is the question that I got in TCS Ninja under the Quantitative section.

How shall I do this? Help !!

Answer 1

Denote by $[...]$ the area of the polygon $...$

Lema 1

Let $ABCD$ be a rectangle a $P$ a point inside as shown. It follows that $$[APB]+[DPC]=[APD]+[CPB]$$

Proof

Draw the parallels to $AD$ and $AB$ through $P$. Then $$[APB]+[DPC]=\frac{AB·FP}{2}+\frac{DC·EP}{2}=\frac{AB·FP+AB·DC}{2}=\frac{AB·EF}{2}=\frac{[ABCD]}{2}$$

Now back to your problem $$A1+A3=2081=A2+A4=1016+A4\iff A4=2081-1016=\color{red}{1065}$$

Thanks for the correction @Jean Marie.