Two tangents \(PQ\) and \(PR\) are drawn from an external point \(P\) to a circle with centre \(O\). Prove that \(QORP\) is a cyclic quadrilateral.
Cyclic.
Step 1: We know that the radius of a circle is always perpendicular to the tangent at the point of contact.
So, \(OQ \perp PQ\) and \(OR \perp PR\).
Step 2: This means angle \(OQP = 90^\circ\) and angle \(ORP = 90^\circ\).
Step 3: Now, look at quadrilateral \(QORP\). Its vertices are \(Q, O, R, P\).
In this quadrilateral, the opposite angles are:
Step 4: Add these opposite angles:
\(90^\circ + 90^\circ = 180^\circ\).
Step 5: A property of a cyclic quadrilateral is: If a pair of opposite angles is supplementary (i.e., adds up to \(180^\circ\)), then the quadrilateral is cyclic.
Step 6: Since in quadrilateral \(QORP\), one pair of opposite angles is supplementary, we can say that \(QORP\) is a cyclic quadrilateral.