Question

If the angle between two radii of a circle is 130°, the angle between the tangents at the ends of the radii is


(a) 65°
(b) 50°
(c) 40°
(d) 70°

Answer 1

(b) 50°

$$\begin{gathered} OA\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}OB\mathrm{are}\mathrm{}\mathrm{the}\mathrm{}\mathrm{two}\mathrm{}\mathrm{radii}\mathrm{}\mathrm{of}\mathrm{}\mathrm{a}\mathrm{}\mathrm{circle}\mathrm{}\mathrm{with}\mathrm{}\mathrm{centre}O.\mathrm{Also}AT\mathrm{and}\mathrm{}BT\mathrm{are}\mathrm{}\mathrm{the}\mathrm{}\mathrm{tangents}\mathrm{}\mathrm{to}\mathrm{}\mathrm{the} \\ \mathrm{circle}. \\ \mathrm{}\mathrm{Then}, \\ \angle TAO=\angle TBO={90}^0 \\ \Rightarrow \angle TAO+\angle TBO={180}^0 \\ \therefore \angle AOB+\angle ATB={180}^0 \\ \Rightarrow {130}^0+\angle ATB={180}^0 \\ \Rightarrow \angle ATB=\left({180}^0-{130}^0\right) \\ \Rightarrow \angle ATB={50}^0 \end{gathered}$$