Question

If the given figure, $l$ & $m$ are two parallel tangents to a circle with centre $O$, touching the circle at $A$ and $B$ respectively. Another tengent at $C$ intersects the lone $l$ and $D$ and lone $m$ at $E$. Find $\angle DOE$

Answer 1

DC is the tangent and CO is the normal

$\Rightarrow \angle DCO={90}^0.Similarly,\angle OAD={90}^0.In△DCO\&△ADO:\Rightarrow OA=OC\left(radius\right)OD=OD\left(common\right)\therefore △DCO\cong △ADObyR.H.S.Hence\angle AOD=\angle DOC=x\left(let\right)\because AB\parallel DE\Rightarrow \angle AOC=\angle DCO={90}^0\Rightarrow \angle DOC+\angle DOA={90}^0\Rightarrow x={45}^{0}Similarly,△DOC\cong △EOC(\because DC=CE(Tangentfromsameexternalpoint).\angle DCO=\angle OCE={90}^{0}OC=OC(common)Hence\angle COE=\angle COD=y(let).\Rightarrow x=y={45}^0.\therefore \angle DOE=x+y={90}^0.$