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,

∠RLQ = ∠PLN  ...(Vertically opposite angles)

∠LRQ = ∠LPN  ...(Alternate angles)

∆RLQ ~ ∆PLN  ...(AA Similarity)

∴ `(RL)/(LP) = (RQ)/(PN)`

`2/3 = (10)/(PN)`

PN = 15 cm

In ∆RLM and ∆PLQ

∠RLM = ∠PLQ   ...(Vertically opposite angles)

∠LRM= ∠LPQ   ...(Alternate angles)

∆RLM ~ ∆PLQ  ...(AA Similarity)

∴ `(RM)/(PQ) = (RL)/(LP)`

`(RM)/16 = 2/3`

`RM = 32/3 = 10 2/3 cm`