• # question_answer 22) Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Answer:

 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. $\therefore OM\bot AM$ Similarly,           $ON\bot AN$ 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.

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