Prove that the figure formed by joining the mid-points of the pairs of consecutive sides of a quadrilateral is a parallelogram.

Explanation

Given :`squareABCD` in which P,Q,R and S are respectively the mid-points AB, BC ,CD and DA.
To prove :` squarePQRS` is a parallelogram.
Construction: Join AC.
Proof: In `triangleABC`, since P and Q are the mid-points of AB and BC respectively.
`therefore PQ"||"AC and PQ=(1)/(2)AC" "` (mid-point theorem)...(1)
Now, in `triangleADS`, since S and R are the mid-point of AD and DC respectively.
`therefore SR||AC and SR=(1)/(2)AC" "` (mid-point theorem)...(2)
`therefore` From eqs. (1) and (2) , we get
`PQ "||"SR and PQ=SR`
Thus, in quadrilateral PQRS, one pair of opposite sides is parallel and equal.
Hence, `squarePQRS` is a parallelogram. Hence proved.