Question

In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.

Hint:

The angle between two tangents from an external point is always supplementary to the central angle subtended by the radii at points of contact.

Answer 1

Given:
O is the centre of the circle.
PQ and PR are tangents to the circle from point P.

To Prove:
Quadrilateral PQOR is cyclic.

Step 1: Use the tangent–radius property
A radius drawn to the point of contact of a tangent is ⟂ to the tangent.

Therefore:
\[ OQ \perp PQ \quad \Rightarrow \quad \angle OQP = 90^\circ \]
\[ OR \perp PR \quad \Rightarrow \quad \angle ORP = 90^\circ \]

Step 2: Add the two angles
Consider the angles at Q and R inside quadrilateral PQOR:
\[ \angle OQP + \angle ORP = 90^\circ + 90^\circ = 180^\circ \]

Step 3: Use the cyclic quadrilateral criterion
If a pair of opposite angles in a quadrilateral are supplementary (sum to 180°),
the quadrilateral is cyclic.

Since: \[ \angle OQP + \angle ORP = 180^\circ \] The quadrilateral PQOR satisfies the condition.

Final Conclusion:
\[ \boxed{\text{PQOR is a cyclic quadrilateral.}} \]