Question

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that $\angle QPR={120}^{\circ }$, prove that 2PQ = PO.

Answer 1

Let us draw the circle with extend point P and two tangents PQ and PR.

We know that the radius is perpendicular to the tangent at the point of contact.
$\therefore \angle OQP={90}^{\circ }$
We also know that the tangent drawn to a circle from an external point are equally inclined to the joining the centre to that point.
$\therefore \angle QPO={60}^{\circ }$

Now, in ∆QPO:

$\Rightarrow Cos60°=\frac{PQ}{PO}$

$\Rightarrow \frac{1}{2}=\frac{PQ}{PO}$

⇒2PQ = PO