Question

Prove that the midpoints of the adjacent sides of a quadrilateral will form a parallelogram. [4 MARKS]

Answer 1

Diagram : 1 Mark
Concept : 1 Mark
Proof : 2 Marks

Let $ABCD$ be the quadrilateral and let $P,Q,R$ and $S$ be the midpoints of sides $AB,BC,CD$ and $DA$ respectively.

In $\Delta CDB,RQ\parallel DB$ and $RQ=\frac{1}{2}DB$......(i)
[By mid point theorem]

In $\Delta ADB,SP\parallel DB$ and $SP=\frac{1}{2}DB$.......(ii)
[By mid point theorem]

By (i) and (ii),

$\Rightarrow RQ\parallel SP$ and $RQ=SP$

$\therefore$ a pair of opposite sides of $PQRS$ are equal and parallel.

$\Rightarrow PQRS$ is a parallelogram.