The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a
Given that,
ABCD is a quadrilateral and P, Q, R and S are the mid points of the sides AB, BC, CD and DA respectively
To prove: PQRS is a parallelogram
Construction: Join A with C
Proof: In ,
P and Q are the mid-points of AB and BC respectively
Therefore,
PQ ‖ AC and PQ = AC (Mid-point theorem) (i)
Again,
In ,
R and S are mid-points of sides CD and AD respectively
Therefore,
SR ‖ AC and SR = AC (Mid-point theorem) (ii)
From (i) and (ii), we get
PQ ‖ SR and PQ = SR
Hence, PQRS is a parallelogram (One pair of opposite sides is parallel and equal)