Question

ABCD is a quadrilateral in which P, Q,R and S are mid-points of the sides AB, BC, CD and DA ( see the given figure). AC is a diagonal. Show that:

PQRS is a parallelogram

Answer 1

In $\Delta ADC$, S and R are mid - points of sides AD and DC respectively.
Therefore, by using mid-point theorem, we get
$SR||ACandSR=\frac{1}{2}AC...\left(1\right)$

In $\Delta ABC$, P and Q are mid - points of sides AB and BC respectively.
Therefore, by using mid-point theorem, we get
$PQ||ACandPQ=\frac{1}{2}AC...\left(2\right)$

Using equations (1) and (2) , we obtain
$PQ||SRandPQ=SR$
$\therefore PQ=SR$

We obtained,
PQ || SR and PQ = SR

Clearly, one pair of opposite sides of quadrilateral PQRS is parallel and equal.
Hence , PQRS is a parallelogram.