Question

In figure, l and m are two parallel tangents at P and R. The tangent at Q makes an intercept ST between l and m. Find $\angle SOT$(in degrees)

Answer 1

In triangle PSO and SOQ,

$SP=SQ$ [tangents drawn from external points are equal]

$SO=SO$ [common]

$OP=OQ$ [radii of the same circle]

So, by SSS- criterion of congruency $\Delta SPO\cong \Delta SQO$

$\Rightarrow PSO=\angle QSO$

$\angle PSQ=2\angle QSO$ .....(i)

Similarly, we can prove that $\angle RTO=\angle QTO$

$\Rightarrow \angle QTR=2\angle QTO$

Now, $\angle PSQ+\angle QTR={180}^{\circ }$

$\Rightarrow 2\angle QSO+2\angle QTO={180}^{\circ }$

$\Rightarrow QSO+\angle QTO={90}^{\circ }$

In triangle $SOT$

$\Rightarrow {180}^{\circ }-\angle SOT={90}^{\circ }$ (In $\Delta SOT,\angle QSO+\angle QTO+\angle SOT={180}^{\circ }$)

$\Rightarrow \angle SOT={90}^{\circ }$