Question

In a triangle JKL, M and N are respectively the mid-points of JK and JL. If O is any point on KL such that

P and Q are respectively the mid-points of OK and OL, then prove that PMNQ is a parallelogram. ​

Answer 1

Given : In a triangle JKL, M and N are respectively the mid-points of JK and JL. If O is any point on KL such that  P and Q are respectively the mid-points of OK and OL,

To Find : prove that PMNQ is a parallelogram.

Solution:

In a triangle JKL, M and N are respectively the mid-points of JK and JL

Using converse of (BPT) Thales theorem  

MN ║ KL

P & Q are points of KL

=> MN || PQ

Join JO

in Δ KJO

M is mid point of JK  and P is mid point of  OK

Hence MP ║ JO

Similarly in Δ LJO

N is mid point of JL  and Q is mid point of  OL

Hence NQ ║ JO

MP ║ JO  & NQ ║ JO

=> MP || NQ

MN || PQ  & MP || NQ

Both pair of opposite sides are parallel

=> PMNQ is a parallelogram

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