question_answer

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

Answer:

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.