Question

Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

Answer 1

In
$△ADC$ , S and R are the midpoints of AD and DC respectively.

Recall that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and half of it.

Hence $SR\parallel ACandSR=\frac{1}{2}AC$ → (1)

Similarly, in $△ABC$, P and Q are midpoints of AB and BC respectively.

$\to PQ\parallel ACandPQ=\frac{1}{2}AC$ → (2) [By midpoint theorem]

From equations (1) and (2), we get

$PQ\parallel SRandPQ=SR$ → (3)

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

Hence PQRS is a parallelogram

Hence the diagonals of parallelogram PQRS bisect each other.

Thus PR and QS bisect each other.