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

In ΔRLQ and ΔPLN,

⇒ ∠RLQ = ∠PLN [Vertically opposite angles are equal]

⇒ ∠LRQ = ∠LPN [Alternate angles are equal]

∴ ΔRLQ ∼ ΔPLN (By A.A. axiom)

Since, corresponding sides of similar triangles are proportional we have :

$\Rightarrow \dfrac{RL}{LP} = \dfrac{RQ}{PN} \\[1em] \Rightarrow \dfrac{2}{3} = \dfrac{10}{PN} \\[1em] \Rightarrow PN = \dfrac{30}{2} = 15 \text{ cm}.$

In ΔRLM and ΔPLQ,

⇒ ∠RLM = ∠PLQ [Vertically opposite angles are equal]

⇒ ∠LRM = ∠LPQ [Alternate angles are equal]

∴ ΔRLM ∼ ΔPLQ (By A.A. axiom)

Since, corresponding sides of similar triangles are proportional we have :

$\Rightarrow \dfrac{RM}{PQ} = \dfrac{RL}{LP} \\[1em] \Rightarrow \dfrac{RM}{15} = \dfrac{2}{3} \\[1em] \Rightarrow RM = \dfrac{30}{3} \\[1em] \Rightarrow RM = 10 \text{ cm.}$

Hence, PN = 15 cm and RM = 10 cm.