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: Understanding the Concept:
A quadrilateral is cyclic if the sum of its opposite angles is \(180^\circ\). Also, the radius is perpendicular to the tangent at the point of contact.
Step 2: Key Formula or Approach:
Show that \(\angle Q + \angle R = 180^\circ\) or \(\angle P + \angle O = 180^\circ\).
Step 3: Detailed Explanation:
In quadrilateral PQOR:
1. \(OQ\) is the radius and \(PQ\) is the tangent at \(Q\). Therefore, \(OQ \perp PQ \Rightarrow \angle OQP = 90^\circ\).
2. \(OR\) is the radius and \(PR\) is the tangent at \(R\). Therefore, \(OR \perp PR \Rightarrow \angle ORP = 90^\circ\).
3. Now, find the sum of opposite angles \(\angle OQP\) and \(\angle ORP\):
\[ \angle OQP + \angle ORP = 90^\circ + 90^\circ = 180^\circ \]
4. Since the sum of one pair of opposite angles is \(180^\circ\), the quadrilateral PQOR must be cyclic.
Step 4: Final Answer:
Quadrilateral PQOR is cyclic because its opposite angles are supplementary.