Question

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

Answer 1

Given: A circle with centre O; PA and PB are two tangents to the circle drawn from an external point P.

To prove: PA = PB

Construction: Join OA, OB, and OP.

It is known that a tangent at any point of a circle is perpendicular to the radius through the point of contact.

$OA\perp PA$

$OB\perp PB$

In $△OPA$ and $△OPB$

$\angle OPA=\angle OPB$ (Using (1))

$OA=OB$ (Radii of the same circle)

$OP=OP$ (Common side)

Therefor $△OPA\cong △OPB$(RHS congruency criterion)

$PA=PB$

(Corresponding parts of congruent triangles are equal)

Thus, it is proved that the lengths of the two tangents drawn from an external point to a circle are equal.

The length of tangents drawn from any external point are equal.

So statement is correct..