Question

In the figure, $ A$, $ B$ and $ C$ are points on $ OP$, $ OQ$ and $ OR$ respectively such that $ AB \left|\right| PQ$ and $ AC \left|\right| PR$.

Show that $ BC \left|\right| QR$

Answer 1

Prove that $BC\parallel QR$

It is given that $AB\parallel PQ$ and $AC\parallel PR$

In $△OPQ,AB\parallel PQ$

By basic proportionality theorem,

$\frac{OA}{AP}=\frac{OB}{BQ}$ …………..(1)

In $△OPR,AC\parallel PR$

By basic proportionality theorem,

$\frac{OA}{AP}=\frac{OC}{CR}$ …………..(2)

Compare equation (1) and (2)

$\frac{OB}{BQ}=\frac{OC}{CR}$

$\therefore$By converse of basic proportionality theorem

In $△OQR,BC\parallel QR$

Hence it is proved that $BC\parallel QR$