Question

M is the midpoint of side PQ of a parallelogram PQRS. A line through Q parallel to MS meets SR at N and PS produced at L. Prove that QL = 2QN.

Answer 1

In triangle PLQ,
M is the mid point of PQ and MS ||QL
therefore S is the mid point of PL ---------(1)
[converse of mid-point theorem]
Given, ABCD is parallelogram
therefore SR || PQ or SN || PQ
[Opposite sides of a parallelogram are parallel]
In triangle LPQ, S is the mid point of PL (from 1)
and SN || PQ.
therefore N is the midpoint of QL
[converse of mid-point theorem]
$\Rightarrow QL=2QN$