Question

A pair of perpendicular tangents are drawn to a circle from an external point. Prove that length of each tangent is equal to the radius of the circle.

Answer 1

It is given that a pair of perpendicular tangents are drawn to the circle that means $\angle APB={90}^0$ as shown in the above figure:

Therefore,

$PA=PB$ (Tangents from the external point $P$)

$\angle PAO=\angle PBO={90}^0$ (Radius and tangents are perpendicular)

$OA=OB$ (Radii of same circle)

Thus, $AOBP$ is a square.

Now we have $AP=PB=OA=OB$

Hence, the length of tangent is equal to radii of the circle.