Question

In the figure, $PQRS$ is a parallelogram with $PQ=16$ cm and $QR=10$cm, $L$ is a point on $PR$ such that $RL:LP=2:3$. $QL$ produced meets $RS$ at $M$ and $PS$ produced at $N$.Find the lengths of $PN$ and $RM$

Answer 1

In $△RLQ$ and $△PLN$

$\angle RLQ=\angle PLN$(vertically opposite angles)

$\angle LRQ=\angle LPN$(alternate angles)

$△RLQ\sim △PLN$(AA similarity)

$\therefore \frac{RL}{LP}=\frac{RQ}{PN}$

$\Rightarrow \frac{2}{3}=\frac{10}{PN}$

$\Rightarrow PN=\frac{3}{2}\times 10=15$cm

In $△RLM$ and $△PLQ$

$\angle RLM=\angle PLQ$(vertically opposite angles)

$\angle LRM=\angle LPQ$ (alternate angles)

$△RLM\sim △PLQ$(AA similarity)

$\therefore \frac{RM}{PQ}=\frac{RL}{LP}$

$\Rightarrow \frac{RM}{16}=\frac{2}{3}$

$\therefore RM=\frac{32}{3}=10\frac{2}{3}$cm