The problem states that $PQ = 20$ cm and asks to find the length of $MN$ based on the given figure. In the figure, $MN$ is parallel to $QR$, and $N$ and $M$ are the midpoints of $PR$ and $PQ$ respectively. Also, $QL$ is perpendicular to $QR$.

1. Problem Description

The problem states that $PQ = 20$ cm and asks to find the length of $MN$ based on the given figure. In the figure, $MN$ is parallel to $QR$, and $N$ and $M$ are the midpoints of $PR$ and $PQ$ respectively. Also, $QL$ is perpendicular to $QR$.

2. Solution Steps

Since $M$ and $N$ are the midpoints of $PQ$ and $PR$ respectively, $MN$ is parallel to $QR$, and $MN = \frac{1}{2}QR$.

Given that $MN$ is parallel to $QR$.

Also, since $M$ is the midpoint of $PQ$, $PM = MQ$. Similarly, since $N$ is the midpoint of $PR$, $PN = NR$. Therefore $MN$ is parallel to $QR$, and $MN = \frac{1}{2}QR$.

We know that $PM = MQ$ and $PN = NR$, thus $MN = \frac{1}{2} QR$.

Consider triangle $PQR$.

Since $M$ and $N$ are midpoints of $PQ$ and $PR$ respectively, $MN$ is parallel to $QR$ and $MN = \frac{1}{2}QR$.

Since $MN$ is the midsegment of triangle $PQR$, $MN = \frac{1}{2}QR$.

We are given $PQ = 20$ cm.

Let's look at similar triangles $PMN$ and $PQR$.

$\frac{PM}{PQ} = \frac{PN}{PR} = \frac{MN}{QR}$

Since $PM = \frac{1}{2} PQ$, $\frac{PM}{PQ} = \frac{1}{2}$.

Therefore, $\frac{MN}{QR} = \frac{1}{2}$, which means $MN = \frac{1}{2}QR$.

In triangle $PQR$, the length of $PQ$ is known, $PQ = 20$. However, we don't have any information on $QR$, so we can't find $MN = \frac{1}{2}QR$.

However, the figure suggests that $PQ = PR$. In such a case, $PQR$ is an isosceles triangle. Also, since $MN \parallel QR$ then $\triangle PMN$ and $\triangle PQR$ are similar triangles. Also, since $PM = \frac{1}{2} PQ$ and $PN = \frac{1}{2} PR$, then $MN = \frac{1}{2} QR$.

If we also assume that $PQR$ is an equilateral triangle, then $PQ = PR = QR = 20$ cm. In such a case, $MN = \frac{1}{2}QR = \frac{1}{2}(20) = 10$ cm.

But with the information provided, it is not possible to find the length of $MN$. If we assume that triangle $PQR$ is equilateral, i.e. $PQ = QR = RP$, then $MN = \frac{1}{2} QR = \frac{1}{2} PQ = \frac{1}{2} (20) = 10$.

3. Final Answer

Assuming $PQR$ is an equilateral triangle, $MN = 10$ cm. Otherwise, the length of $MN$ cannot be determined from the given information.

The intended answer is likely $MN = 10$ cm, assuming $PQR$ is an equilateral triangle.