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

 Given, Two tangents AM and AN are drawn from a point A to the circle with centre O. To prove: $AM=AN$ Construction: Join $OM,ON$ and $OA$. Proof: Since AM is a tangent at M and OM is radius $\therefore \,OM\bot AM$ Similarly, $ON\bot AN$ Now, in $\Delta \text{ }OMA$and $\Delta \text{ }ONA$ $OM=ON$                       (Radii of the circle) $OA=OA$          (Common) $\angle OMA=\angle ONA-90{}^\circ$ $\therefore \,\,\Delta \,OMA\cong \,\Delta \,ONA$ (By RHS congruence) Hence,                   $AM=AN$ (by c/p.c.t)                       Hence Proved.