question_answer

In Fig. 1, PA and PB are tangents to the circle with centre O such that \[\angle APB=50{}^\circ \], Write the measure of\[\angle OAB\].

Answer:

Since PA and PB are tangents to the circle with centre O then,

\[PA=PB\]

and            \[\angle APO=\angle BPO=25{}^\circ \]

Join OP and \[OA\bot PA\].

In \[\Delta \,APO\],

\[\angle APO+\angle POA+\angle OAP=180{}^\circ \]

\[25{}^\circ +\angle POA+90{}^\circ =180{}^\circ \]

\[\angle POA=65{}^\circ \]

Join OB, then

In \[\Delta \,AOB\]

\[\angle OAB+\angle OBA+\angle AOB=180{}^\circ \]

\[2\angle OAB+2\angle POA=180{}^\circ \]                  \[[\because \,\angle OAB=\angle OBA\,\,OA\,\,\And \,\,OB\,\,\text{are}\,\,\text{radii}]\]

\[2\angle OAB+2\times 65{}^\circ =180{}^\circ \]

\[\angle OAB=90{}^\circ -65{}^\circ \]

\[\therefore \angle OAB=25{}^\circ \]