In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10 cm. L is a point Q 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.

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Solution

In the given figure,
In $△$ RLQ and $△$ PLN,

∠ RLQ = ∠ PLN
∠ LRQ = ∠ LPN

Hence, it is proved that $△$ RLQ ~ $△$ PLN (by AA criterion)

=> $fraction numerator Q R over denominator P N end fraction equals fraction numerator R L over denominator L P end fraction$

= $fraction numerator 2 x over denominator 3 x end fraction$

=> $fraction numerator 10 over denominator P N end fraction equals 2 over 3$ (QR = 10)

=> PN = 15cm

Now,
Let RL = 2x and LP = 3x

As triangle RLM ~ triangle PLQ (proved by AA criterion)

=> $fraction numerator R M over denominator P Q end fraction equals fraction numerator L M over denominator Q L end fraction equals fraction numerator R L over denominator L P end fraction$
=> $fraction numerator R M over denominator 16 end fraction equals fraction numerator 2 x over denominator 3 x end fraction$

=> RM = $fraction numerator 2 cross times 16 over denominator 3 end fraction$ cm.

=> RM = cm

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In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10 cm. L is a point Q 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.

In the given figure, In △ RLQ and △ PLN, ∠ RLQ = ∠ PLN ∠ LRQ = ∠ LPN Hence, it is proved that △ RLQ ~ △ PLN (by AA criterion) => = => (QR = 10) => PN = 15cm Now, Let RL = 2x and LP = 3x As triangle RLM ~ triangle PLQ (proved by AA criterion) => => => RM = cm. => RM = cm