VTMO 2026 - Grade 8 - Final - SET 8 - Self-study Question 3

In rectangle $ABCD$, $AB=5$ and $BC=2$. Points $M$ on $AB$ and $N$ on $DC$ satisfy $AM=1$ and $DN=2$. Let $E$ be the reflection of $D$ in $M$, and let $F=EN\cap CM$. (a) Prove that $CM\perp DM$. (b) Let $K=DF\cap CE$. Find $DK$.