HMMT-Nov 2017 guts, Problem 8

Regular octagon CHILDREN has area 1. Determine the area of quadrilateral LINE.

Solution: Suppose that the side length $C H=\sqrt{2} a$, then the area of the octagon is $((2+\sqrt{2}) a)^{2}-4 \cdot \frac{1}{2} a^{2}=$ $(4+4 \sqrt{2}) a^{2}$, and the area of LINE is $(\sqrt{2} a)((2+\sqrt{2}) a)=(2+2 \sqrt{2}) a^{2}$, which is exactly one-half of the area of the octagon. Therefore the area of LINE is $\frac{1}{2}$. $\fbox{\frac{1}{2}}$.

Answer: \frac{1}{2}