Question

If two tangents are drawn to a circle from an external point,show that they subtend equal angles at the centre.

Answer 1

Given : A circle with centre O and a point A outside it. Also, AP and AQ are the two tangents to the circle.
:$$\begin{gathered} \mathrm{To}\mathrm{prove}:\angle AOP=\angle AOQ\mathrm{}. \\ \mathrm{Proof}:\mathrm{In}\mathrm{}∆AOP\mathrm{}\mathrm{and}∆AOQ,w\mathrm{e}\mathrm{}\mathrm{have}: \\ AP=AQ\left[\mathrm{tangents}\mathrm{}\mathrm{from}a\mathrm{n}\mathrm{}\mathrm{external}\mathrm{}\mathrm{point}\mathrm{}\mathrm{are}\mathrm{}\mathrm{equal}\right] \\ OP=OQ\left[\mathrm{radii}\mathrm{}\mathrm{of}\mathrm{}\mathrm{the}\mathrm{}\mathrm{same}\mathrm{}\mathrm{circle}\right] \\ OA=OA\left[\mathrm{common}\mathrm{side}\right] \\ \therefore ∆AOP\cong ∆AOQ\left[\mathrm{by}\mathrm{}\mathrm{SSS}-\mathrm{congruence}\right] \\ \mathrm{Hence},\angle AOP=\angle AOQ(\mathrm{c}.\mathrm{p}.\mathrm{c}.\mathrm{t}). \end{gathered}$$