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:

A quadrilateral is cyclic if and only if its opposite angles sum to \( 180^\circ \).
Whenever you see tangents and radii meeting, identify the \( 90^\circ \) angles immediately, as they often lead directly to supplementary angle proofs.

Answer 1

Step 1: Recall the converse of the cyclic quadrilateral theorem.
A quadrilateral is cyclic if and only if one pair of its opposite angles adds up to $180^\circ$. This converse lets us prove a quadrilateral is cyclic by checking just one pair of opposite angles, without adding up all four angles first.

Step 2: Identify the opposite angle pair in PQOR.
Going around the quadrilateral $P,Q,O,R$ in order, $Q$ and $R$ sit opposite each other, so the relevant angles are $\angle PQO$ and $\angle PRO$.

Step 3: Use the tangent-radius property to evaluate these two angles.
$PQ$ and $PR$ are tangents at $Q$ and $R$, and a tangent is perpendicular to the radius at the point of contact:
\[ \angle OQP = 90^\circ, \qquad \angle ORP = 90^\circ \]

Step 4: Add the pair and apply the converse theorem.
\[ \angle OQP+\angle ORP = 90^\circ+90^\circ = 180^\circ \]
Since this one pair of opposite angles is supplementary, the converse of the cyclic quadrilateral theorem tells us at once that $PQOR$ is cyclic, with no need to work out $\angle QPR$ and $\angle QOR$ separately.

Final Answer:
Since $\angle OQP+\angle ORP=180^\circ$, the quadrilateral $PQOR$ is cyclic.