ABCD is a quadrilateral in which P,Q,R and S are mid points of sides AB BC CD DA

show that PQRS IS A PARALELLOGRAM 

ABCD is a quadrilateral.

 

P, Q, R, S are the mid points of sides AB, BC, CD and DA.

 

Join Diagonal AC.

 

Consider triangle ABC.

 

Here, P and Q are the mid points of sides AB and BC respectively.

 

So, by using the mid point theorem we can say that PQ is parallel to AC.

 

Similarly RS can be shown parallel to AC.

 

So, we have PQ parallel to RS (as both are parallel to AC)

 

In the same way you can join diagonal BD and show that PS is parallel to QR.

 

So we have a quadrilateral PQRS in which both pairs of opposite sides are parallel. SO, it becomes a parallelogram.