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 the diagonal. Show that
(i) SR || AC and $SR=\frac{1}{2}AC$
(ii) PQ = SR
(iii) PQRS is a parallelogram.

Answer 1

(i) In $△ADC$, R is the mid-point of DC and S is the mid-point of DA.
Thus, by mid-point theorem, SR||AC and $SR=\frac{1}{2}AC.$

(ii) In $△BAC,$ P is the mid-point of AB and Q is the mid-point of BC.
Thus, by mid-point theorem, PQ || AC and $PQ=\frac{1}{2}AC.$
Also, $SR=\frac{1}{2}AC$
Hence, PQ = SR.

(iii) SR || AC ... From Question (i)
PQ || AC ... From Question (ii)
$\Rightarrow PQ||SR$
From (ii), PQ = SR
Since, one pair of opposides of the quadrilateral PQRS is parallel and equal, PQRS is a Parallelogram.
Hence Proved.