Question

Prove that if the mid-points of the opposite sides of a quadrilateral are joined, they bisect each other.

Answer 1

Given: $ABCD$ is a quadrilateral $P,Q,R$ and $S$ are mid-points of $AB,BC,CD$ and $DA$ respectively.

To prove: $PR$ and $SQ$ bisect each other.

Construction: join $AC$.

In $\Delta ABC,PQ||AC$, and $PQ=\frac{1}{2}AC$ (mid-point theorem) ……. (1)

In $\Delta ADC,SR||AC$, and $SR=\frac{1}{2}AC$ (mid-point theorem) ……. (2)

Therefore, $PQ||SR$ and $PQ=SR$ (from (1) and (2) opposite sides equal and parallel)

Thus, $PQRS$ is a parallelogram and $PR$ and $SQ$ are diagonals

Hence, $PR$ bisect $SQ$.