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:

In any tangent-radius geometry, the angle between the two tangents and the angle subtended by the radii at the center are always supplementary (\(180^\circ\)).

Answer 1

Step 1: Concept of Cyclic Quadrilateral:
A cyclic quadrilateral is one where the sum of the opposite angles is \( 180^\circ \). Additionally, the radius of the circle is always perpendicular to the tangent at the point of contact.
Step 2: Method to Prove the Cyclic Nature:
Prove that \( \angle Q + \angle R = 180^\circ \) or \( \angle P + \angle O = 180^\circ \) by using properties of tangents and angles.
Step 3: Alternative Approach:
In quadrilateral PQOR:
1. The line \( OQ \) is the radius, and \( PQ \) is the tangent to the circle at \( Q \). Since a radius is perpendicular to the tangent, we have \( \angle OQP = 90^\circ \).
2. Similarly, \( OR \) is the radius, and \( PR \) is the tangent to the circle at \( R \). Thus, \( \angle ORP = 90^\circ \).
3. Since both \( \angle OQP \) and \( \angle ORP \) are right angles, their sum is \( 180^\circ \):
\[ \angle OQP + \angle ORP = 90^\circ + 90^\circ = 180^\circ \] 4. As the sum of opposite angles is \( 180^\circ \), we can conclude that quadrilateral PQOR satisfies the condition of a cyclic quadrilateral.
Step 4: Conclusion:
Quadrilateral PQOR is cyclic because the sum of its opposite angles is \( 180^\circ \), proving the cyclic property.