Question

In the adjoining figure, $PQRS$ is a trapezium in which $PQ||SR$ and $M$ is the midpoint of $PS$. A line segment $MN||PQ$ meets $QR$ at $N$ show that $N$ is the midpoint of $QR$.

Answer 1

Construct a line to join diagonal $QS$

Diagonal $QS$ intersect the line $MN$ at point $O$

It is given that $PQ\parallel SR$ and $MN\parallel PQ$

We can write it as
$PQ\parallel MN\parallel SR$

Consider $△SPQ$

We know that $MO\parallel PQ$ and $M$ is the midpoint to the side $SP$

$O$ is the midpoint of the line $QS$

We know that $MN\parallel SR$

In $△QRS$ we know that $ON\parallel SR$

$O$ is the midpoint of the diagonal $QS$

Hence, based on the converse mid-point theorem we know that $N$ is the midpoint of $QR$

therefore it is proved that $N$ is the midpoint of $QR$