Question

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

• A. Proof Completed
• B. Quadrilateral is not Cyclic
• C. Angles do not sum to 360
• D. Cannot be Determined

Hint:

For any circle tangent configuration:
The angle between the tangents (\(\angle QPR\)) and the angle subtended by the contact radii at the center (\(\angle QOR\)) are always supplementary (\(\angle QPR + \angle QOR = 180^\circ\)).
This property is highly useful in circle geometry problems!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a circle with center \(O\). From an external point \(P\), two tangents \(PQ\) and \(PR\) are drawn to the circle, touching it at points \(Q\) and \(R\) respectively. We need to prove that the quadrilateral \(PQOR\) is a cyclic quadrilateral.

Step 2: Key Formula or Approach:
1. The radius of a circle is perpendicular to the tangent at the point of contact.
2. A quadrilateral is cyclic if the sum of either pair of opposite angles is \(180^\circ\) (supplementary angles).

Step 3: Detailed Explanation:
1. Let \(Q\) and \(R\) be the points of contact of the tangents \(PQ\) and \(PR\) on the circle.
2. The radius \(OQ\) is perpendicular to the tangent \(PQ\) at the point of contact \(Q\):
\[ \angle OQP = 90^\circ \quad \text{--- (Equation 1)} \]
3. Similarly, the radius \(OR\) is perpendicular to the tangent \(PR\) at the point of contact \(R\):
\[ \angle ORP = 90^\circ \quad \text{--- (Equation 2)} \]
4. Consider the sum of this pair of opposite angles in quadrilateral \(PQOR\):
\[ \angle OQP + \angle ORP = 90^\circ + 90^\circ = 180^\circ \]
5. Since the sum of the interior angles of a quadrilateral is \(360^\circ\):
\[ \angle QPR + \angle QOR + \angle OQP + \angle ORP = 360^\circ \]
\[ \angle QPR + \angle QOR + 180^\circ = 360^\circ \]
\[ \angle QPR + \angle QOR = 180^\circ \]
6. Since both pairs of opposite angles sum to \(180^\circ\), the quadrilateral \(PQOR\) satisfies the necessary and sufficient condition to be a cyclic quadrilateral.
The proof is complete.

Step 4: Final Answer:
The quadrilateral \(PQOR\) has been proven to be cyclic as its opposite angles are supplementary. Thus, the proof is completed.