question_answer

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

Answer:

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

To prove: \[AM=AQ\]

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 \[\Delta \,OMA\] and \[\Delta \,ONA\].

\[OM=ON\]            (radii of same circle)

\[OA=OA\]                        (common)

\[\angle OMA=\angle ONA=90{}^\circ \]

\[\therefore \Delta \,OMA\cong \Delta \,ONA\]         (By RHS congruence)

Hence,              \[AM=AN\]  (By cpct)                 Hence Proved.