
Problem 2. Let $A B C D E$ be a convex pentagon and let $M, P, N, Q$ be the midpoints of the segments $A B, B C, C D, D E$ respectively. If $K$ and $L$ are the midpoints of the segments $M N$ and $P Q$ respectively and the segment $A E$ is of length $a$, find the length of the segment $K L$.

Figure 1.

![](https://cdn.mathpix.com/cropped/2024_06_03_a20b863991b96ce15f00g-01.jpg?height=434&width=506&top_left_y=1893&top_left_x=615)

Figure 2 .

![](https://cdn.mathpix.com/cropped/2024_06_03_a20b863991b96ce15f00g-01.jpg?height=346&width=394&top_left_y=1963&top_left_x=1391)


Solution. Let $F$ be the midpoint of the segment $A D$ (Figure 1). Then the quadrilateral $M P N F$ is a parallelogram. Hence $K$ is midpoint of the segment $F P$. It follows from here that $K L=\frac{1}{2} \cdot F Q$. On the other hand $F Q=\frac{1}{2} \cdot a$ (because $F$ and $Q$ are midpoints of $E D$ and $A D$ respectively).

Therefore $K L=\frac{1}{4} \cdot a$.
