Question

The diagonals of a quadrilateral PQRS intersect each other at the point O such that $\frac{PO}{QO}=\frac{RO}{SO}$. Show that PQRS is a trapezium.

Answer 1

Given:

$PQRS$ is a quadrilateral with its diagonals intersecting at $O$

$\frac{PO}{QO}=\frac{RO}{SO}$

To prove that:

$PQRS$ is a trapezium

Proof:

$\frac{PO}{RO}=\frac{QO}{SO}$

$\angle POQ=\angle ROS$ Vertically opposite angles are equal to each other.

$⟹\Delta POQ\sim \Delta ROS$ by $SAS$ similarity

$⟹\angle OPQ=\angle ORS$ by property of similar triangles

$⟹PQ||RS$ by property of alternate angles.

Hence, $PQRS$ is a trapezium.

Hence proved.