Question

In Fig. 10.86, PQL and PRM are tangents to the circle with centre O at the points Q and R respectively and S is a point on the circle such that $\angle$SQL = 50⁰ and $\angle$SRM = 60⁰. Then , find $\angle$QSR.
figure

Answer 1

In the given figure,
PL and PM are the tangents to the circle with centre O.
$\angle \mathrm{SQL}=50°,\angle \mathrm{SRM}=60°$
$\mathrm{PL}\perp \mathrm{OQ}\mathrm{and}\mathrm{PM}\perp \mathrm{OR}$
Thus,
$$\begin{gathered} \angle \mathrm{ORS}=90°-60°=30° \\ \angle \mathrm{OQS}=90°-50°=40° \end{gathered}$$
OQ = OS
So, $\angle \mathrm{OQS}=\angle \mathrm{OSQ}=40°$
Similarly.
OS = OR
So, $\angle \mathrm{ORS}=\angle \mathrm{OSR}=30°$
$$\begin{gathered} \angle \mathrm{RSQ}=\angle \mathrm{QSO}+\angle \mathrm{RSO} \\ \Rightarrow \angle \mathrm{RSQ}=40°+30°=70° \end{gathered}$$