Question

Show that the line segments joining the midpoints of the opposite sides of a quadrilateral and bisect each other

Answer 1

Given $ABCD$ is quadrilateral , $S$ and $R$ are the midpoints of $AD$ and $DC$ respectively.

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 $SR\parallel AC$ $SR=\frac{1}{2}AC$........ (1)

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

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

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

$PQ\parallel SR$ $SR=PQ$................(3)

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

Hence $PQRS$ is a parallelogram

we know that the diagonals of parallelogram $PQRS$ bisect each other.

Thus $PR$ and $QS$ bisect each other.