Data: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a diagonal.
To Prove: (i) SR || AC and SR = \(\frac{1}{2}\)AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
Proof: (i) In ∆ADC, S and R are mid points of AD and DC sides. As per mid-point theorem,
SR || AC and SR = AC.
(ii) In ∆ABC, P and Q are mid-points of AB and BC. As per mid-point theorem,
PQ || AC
and PQ = \(\frac{1}{2}\)AC
But, SR = \(\frac{1}{2}\)AC (Proved)
∴ PQ = SR
(iii) PQ = SR (Proved)
SR || AC and PQ || AC
∴ SR || PQ
Opposite sides of a quadrilateral PQRS are equal and parallel, hence PQRS is a parallelogram.