Question

ABCD is a quadrilateral in which P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that PQRS is a parallelogram.​

Answer 1

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Step-by-step explanation:

Given : A quadrilateral ABCD in which P, Q, R, and S are respectively the mid- points of the sides AB, BC, CD and DA. Also AC is its diagonal.

To prove :

(i) SR || AC and SR=12AC

(ii) PQ = SR

(iii) PQRS is a parallelogram.

Proof : (i) In ΔACD, we have S is the mid- point of AD and R is the mid- point of CD.

Then SR||ACandSR=12AC [Mid- point theorem]

(ii) In Δ ABC, we have P is the mid- point of the side AB and Q is the mid- point the side BC.

Then, PQ || AC

and, PQ=12AC [Mid- point theorem]

Thus, we have proved that :

PQ||ACSR||AC}⇒PQ||SR

Also PQ=12ACSR=12AC}⇒PQ||SR

(iii) Since PQ = SR and PQ|| SR

⇒ One pair of opposite sides are equal and parallel.

⇒ PQRS is a parallelogram