Question

Two tangents $PQ$ and $PR$are drawn from an external point to a circle with centre $O.$ Prove that $QORP$ is a cyclic quadrilateral.

Answer 1

To prove $QORP$ is a cyclic quadrilateral:

We know that, radius $\perp$tangent $=OR\perp PR$

Hence $\angle ORP=90°$

Similarly

$OQ\perp PQ$

$\angle OQP=90°$

In quadrilateral $ORPQ,$

Th sum of all interior angles of a quadrilateral $=360º$

$$\begin{gathered} \angle ORP+\angle RPQ+\angle PQO+\angle QOR=360º \\ 90º+\angle RPQ+90º+\angle QOR=360º \\ Hence,\angle O+\angle P=180° \end{gathered}$$

Since, the sum of the opposite angles of the quadrilateral $PROQ$ is $180°$. Therefore, it is a cyclic quadrilateral.

Hence, proved.