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Question

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. 


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 = 32 over 3 cm

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