Question

In the figure, PN, and RT are medians of $△$ PQR and NM || RT. Prove that $QM=\frac{1}{4}PQ$.

Answer 1

It is given that $RT$ is the median of triangle $PQR$.

Now, in triangle $QRT$

N is the midpoint of $QR$ (It is given that PN is the median. So $QN=RN$)

And $NM$ is parallel to $TR$ (also given)

So by converse of mid point theorem

M is the mid point of $TQ$

Hence, $QM=\frac{1}{2}TQ$

So, $QM=\frac{1}{2}\left(\frac{1}{2}PQ\right)$ (Since $TQ=\frac{1}{2}PQ$)

So, $QM=\frac{1}{4}\left(PQ\right)$

Hence proved