Question

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. Show that :

(i) SR || AC and SR = \(\frac{1}{2}\)AC 

(ii) PQ = SR 

(iii) PQRS is a parallelogram.

Answer 1

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.