Question

$PQRS$ is a trapezium in which $PQ\parallel RS$ and its diagonal intersect each at the point $O$.
Prove that $\frac{PO}{QO}=\frac{RO}{SO}$

Answer 1

As $PQ||RS$ and $PR$ and $QS$ are transversal

$\angle OSR=\angle OQP$ (alternate angles)

$\angle ORS=\angle OPQ$ (alternate angles)

$\therefore △SOR\sim △POQ\Rightarrow \frac{SO}{PO}=\frac{RO}{QO}=\frac{SR}{PQ}\Rightarrow \frac{SO}{PO}=\frac{RO}{QO}\Rightarrow \frac{QO}{PO}=\frac{RO}{SO}$