Question

In parallelogram PQRS, L is mid-point of side SR and SN is drawn parallel to LQ which meets RQ produced at N and cuts side PQ at M. Prove that $SP=\frac{1}{2}RN$.

Answer 1

Given:

$ABCD$ is a parallelogram

$SL=LR$ $\left(L\text{is mid-point}\right)$

$LQ\parallel SN$

In $△SRN,$ $L$ is mid-point of $SR$ and $LQ$ parallel to $SN$

$\therefore RQ=QN\text{(by mid-point theorem)}$

$\frac{RL}{SR}=\frac{LQ}{SN}$

$\frac{RL}{2RL}=\frac{LQ}{SN}$

$\therefore SN=2LQ$

$\because RQ=QN$

$RQ=\frac{1}{2}RN$

similarly $SP=\frac{1}{2}RN$