question_answer

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

Answer:

Given, a circle with centre O and external point P. Two tangents PA and PB are drawn.

To prove:                       \[PA=PB\]

Const.: Join radius OA and OB also join O to P.

Proof: In \[\angle OAP\] and \[\angle OBP\]

\[OA=OB\]                   (Radii)

\[\angle A=\angle B\]                             (Each \[90{}^\circ \])

\[OP=OP\]                               (Common)

\[\therefore \]      \[\Delta \,AOP\cong \Delta \,BOP\]            (RHS cong.)

\[\therefore \]      \[PA=PB\] (cpct)                                    Hence Proved.