Question

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Answer 1

Proving that the line segments joining the mid-points of opposite sides of a quadrilateral bisect each other.

Assume $ABCD$ is a quadrilateral, where $P,Q,R,\mathrm{and}S$ are the mid-points of sides $AB,BC,CD,\mathrm{and}DA$ respectively.

Draw the diagonal $AC$

Apply the mid-point theorem in $∆ABC$ and in $∆ADC$.

In $∆ABC$, $PQ\parallel AC$ and $PQ=\frac{1}{2}AC\cdots \left(1\right)$.

In $∆ADC$, $SR\parallel AC$ and $SR=\frac{1}{2}AC\cdots \left(2\right)$.

From equations $\left(1\right)$ and $\left(2\right)$,

$PQ\parallel SR$ and $PQ=SR$.

Since one set of opposite sides in quadrilateral $PQRS$ is equal and parallel to each other, $PQRS$ is a parallelogram.
It is known that the diagonals of a parallelogram bisect each other,

So, se conclude that $PR$ and $QS$ also bisect each other.

Hence, it is proved that the line segments connecting the mid-points of opposite sides of a quadrilateral bisect each other.