CBSE2026Class Class 10 · Mathematics2 Marks · Short✅ Verified
In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.
*(Figure : PQ and PR are tangents touching the circle at Q and R respectively, drawn from external point P.)*
✅ Answer & Solution
**Step 1 — Use the tangent–radius property.**
The radius drawn to the point of contact is perpendicular to the tangent.
$$OQ \perp PQ \ \Rightarrow\ \angle OQP = 90^\circ$$
$$OR \perp PR \ \Rightarrow\ \angle ORP = 90^\circ$$
**Step 2 — Add these two opposite angles of quadrilateral PQOR.**
$$\angle OQP + \angle ORP = 90^\circ + 90^\circ = 180^\circ$$
**Step 3 — Use the angle sum property of a quadrilateral.**
$$\angle QPR + \angle QOR + \angle OQP + \angle ORP = 360^\circ$$
$$\Rightarrow\ \angle QPR + \angle QOR = 360^\circ - 180^\circ = 180^\circ$$
**Step 4 — Apply the converse of the cyclic-quadrilateral theorem.**
Both pairs of opposite angles of quadrilateral PQOR are supplementary.
$\therefore$ **PQOR is a cyclic quadrilateral. Hence shown.**
