Question

In the figure, $PQRS$ is a parallelogram with $PQ=15cm$ and $RQ=10cm$. $L$ is a point on $RP$ such that $RL:LP=2:3$. $QL$ produced meets $RS$ at $M$ and $PS$ produced at $N$. Find the lengths of $PN$ and $RM$.

• A. $PN=15cm;RM=10cm$
• B. $PN=10cm;RM=10cm$
• C. $PN=25cm;RM=15cm$
• D. $PN=25cm;RM=10cm$

Answer 1

Answer: (A)

$PN=15cm;RM=10cm$

Given: $PQRS$ is a parallelogram. $PQ=15$ and $RQ=10$, $\frac{RL}{LP}=\frac{2}{3}$
In $△RLQ$ and $△NLP$
$\angle RLQ=\angle NLP$ (vertically opposite angles)
$\angle LRQ=\angle LPN$ (Alternate angles)
$\angle LQR=\angle LNP$ (Alternate angles)
Thus, $△RLQ\sim △PLN$ (AAA rule)
Hence, $\frac{RL}{PL}=\frac{RQ}{PN}$ (Corresponding sides of similar triangles)
$PN=\frac{3\times 10}{2}$
$PN=15cm$
Similarly, $△RLM\sim △PLQ$
$\frac{RL}{LP}=\frac{RM}{PQ}$ (Corresponding sides of similar triangles)
$RM=\frac{15\times 2}{3}$
$RM=10$ cm.