Question

MN and MQ are tangents from a point M outside the given circle with center $O$. If $\angle NOQ={120}^{\circ }$ then which of the following rlations holds true:

• A. $NQ=MN=MQ$
• B. $NQ=OM$
• C. $OQ=OM$
• D. $ON=MN$

Answer 1

The correct option is A $NQ=MN=MQ$

From the above figure it is clear that $\angle ONM=\angle OQM={90}^o$

and also $\angle QMN={60}^o$

Now draw $OM$ and $NQ$

Now in $\Delta ONQ$ , $\angle ONQ=\angle OQN={30}^o$ ($ON=OQ=r$, hence opposite angles are same )

Now In $\Delta MQN$ ,$\angle MNQ=\angle NQM=\angle QMN={60}^o$

$\therefore \Delta MQN$ is an equilateral triangle.

Hence $MN=NQ=MQ$