Question

Prove that the tangents drawn from an external point to a circle are equal.

Answer 1

Statement: The tangents drawn from an external point to a circle are equal.

Given:

$PT$ and $QT$ are two tangents drawn from an external point $T$ to the circle $C(O,r)$.

To Prove: $PT=TQ$

Construction:

Join $OT$.

Solution:

We know that a tangent to a circle is perpendicular to the radius through the point of contact.

$\therefore \angle OPT=\angle OQT={90}^{\circ }$

In $△OPT$ and $△OQT,$

$\angle OPT=\angle OQT\left({90}^{\circ }\right)$

$OT=OT$ (common)

$OP=OQ$ (Radius of the circle)

$\therefore △OPT\cong △OQT$ (By RHS criterian)

So, $PT=QT$ (By CPCT)

Hence, the tangents drawn from an external point to a circle are equal.