5. Consider a square of side length 1. Draw four lines that each connect a midpoint of a side with a corner not on that side, such that each midpoint and each corner is touched by only one line. Find the area of the region completely bounded by these lines.
Solution: In unit square $A B C D$, denote by $E, F, G, H$ the respective midpoints of sides $A B, B C, C D, D A$. Let $I$ be the intersection of $A F$ and $D E$, let $J$ be the intersection of $B G$ and $A F$, let $K$ be the intersection of $C H$ and $B G$, and let $L$ be the intersection of $D E$ and $C H$. We want to find the area of square $I J K L$. The area of $A B F$ is $\frac{1}{4}$, which is equal to $\frac{1}{2} A F \cdot B J=\frac{\sqrt{5}}{4} B J$, so $B J=\frac{1}{\sqrt{5}}$. Using similar triangles, $G K=J F=\frac{1}{2} B J$. Thus the length of a side of $I J K L$ is $J K=\frac{\sqrt{5}}{2}-\frac{1}{\sqrt{5}}-\frac{1}{2} \frac{1}{\sqrt{5}}=\frac{1}{\sqrt{5}}$, and the area of $I J K L$ is $\frac{1}{5}$.