Question

In the given figure, $O$ is the centre of a circle and $BD$ is a diameter. $ABandAC$ are tangents touching the circle at $B\&C$ respectively. If $\angle BAC={70}^{\circ }$ then $\angle OBC$ is

• A. ${30}^{\circ }$
• B. ${35}^{\circ }$
• C. ${40}^{\circ }$
• D. ${45}^{\circ }$

Answer 1

The correct option is B ${35}^{\circ }$

Given that: $\angle BAC=70°$

$\angle BDC=90°$ (Diameter of the circle subtends an angle of 90°)

$\angle OBA=90°$ ($AB$ and $AC$ are tangents to the circle)

In $△ABC$:

$AB=AC$ (Tangents from a point are equal)

$\angle ACB=\angle ABC=\frac{1}{2}\times (180°-70°)$ (angles opposite to equal sides are equal)

Thus, $\angle ACB=\frac{110}{2}=55°$

Now, $\angle ABD=\angle DBC+\angle ABC=90°$

Hence, $\angle DBC=\angle OBC=90°-\angle ABC=90°-55°=35°$

Answer: $\angle OBC=35°$