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

Step 1: Understanding the Concept:
A quadrilateral is cyclic if the sum of either pair of opposite angles is \( 180^\circ \).
Step 2: Detailed Explanation:
In quadrilateral PQOR:
1. \( \angle OQP = 90^\circ \) (Radius OQ is perpendicular to tangent PQ).
2. \( \angle ORP = 90^\circ \) (Radius OR is perpendicular to tangent PR).
Sum of this pair of opposite angles:
\[ \angle OQP + \angle ORP = 90^\circ + 90^\circ = 180^\circ \]
In any quadrilateral, the sum of all angles is \( 360^\circ \):
\[ \angle QOR + \angle QPR = 360^\circ - 180^\circ = 180^\circ \]
Since opposite angles are supplementary, PQOR is a cyclic quadrilateral.
Step 3: Final Answer:
The quadrilateral PQOR is cyclic. Hence Proved.