Question

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

Answer 1

Given, Two tangents AM and AN are drawn from point A to a circle with centre O.



To Prove : AM = AN

Construction : Join OM, ON and OA.

Proof : Since, AM is a tangent and OM is a radius.

$\therefore OM\perp AM$

Similarly, $ON\perp AN$

Now, in $\Delta OMA$ and $\Delta ONA$

OA = OA (Common)

OM = ON (Radii of the circle)

$\angle OMA=\angle ONA={90}^{\circ }$

$\therefore \Delta OMA\cong \Delta ONA$ (By RHS congruence)

Hence, AM = AN Hence Proved.