Question

In the given figure, PA and PB are the tangent segments to a circle with centre O. Show that the points A, O, B and P are concyclic.

Answer 1

$$\begin{gathered} \text{Given, PA and PB are tangents to the circle with centre O and OA is a radius}\text{.} \\ \text{Now, tangents drawn from an external point are perpendicular to the radius} \\ \text{at the point of contact}\text{.} \\ \therefore OA\perp PA\text{and}OB\perp PB. \\ \therefore \angle \mathit{\text{OAP}}={90}^0\text{and}\angle \mathit{\text{OBP}}={90}^0 \\ \therefore \angle \mathit{\text{OAP}}+\angle \mathit{\text{OBP}}={90}^0+{90}^0={180}^0 \\ \text{Thus, the sum of a pair of opposite angles of quadrilateral}\mathit{\text{AOBP}}{\text{is 180}}^0. \\ \text{Now,}\angle AOB+\angle APB+\angle PAO+\angle PBO={360}^0(\because {\text{Sum of all the angles of a quadrilateral is 360}}^0) \\ =>\angle AOB+\angle APB={180}^0(\because \angle \text{OAP}+\angle \text{OBP=}{180}^0) \\ \therefore \mathit{\text{AOBP}}\text{is a cyclic quadrilateral}\text{.} \end{gathered}$$