Question

In the given figure, $O$ is the centre of a circle. If $\angle AOB={100}^{\circ }$ and $\angle AOC={90}^{\circ }$ , then $\angle BAC=?$

(a) ${85}^{\circ }$

(b) ${80}^{\circ }$

(c) ${95}^{\circ }$

(d) ${75}^{\circ }$

Answer 1

In figure,

$\angle AOB={100}^{\circ }$

$\angle AOC={90}^{\circ }$

In figure, $\angle BOC,\angle BOA$ and $\angle AOC$ formed the complete angle

$\Rightarrow \angle BOC+\angle BOA+\angle AOC={360}^{\circ }$

$\Rightarrow \angle BOC+{100}^{\circ }+{90}^{\circ }={360}^{\circ }$

$\Rightarrow \angle BOC+{190}^{\circ }={360}^{\circ }$

$\Rightarrow \angle BOC={360}^{\circ }-{190}^{\circ }$

$\Rightarrow \angle BOC={170}^{\circ }$ ---(1)

In a circle, angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining arc of the circle.

i.e., in Figure,

$\angle BAC=\frac{1}{2}\angle BOC$

$\Rightarrow \angle BAC=\frac{1}{2}\angle BOC$

$\Rightarrow \angle BAC=\frac{{170}^{\circ }}{2}$

$\Rightarrow \angle BAC={85}^{\circ }$

Hence, Option A is correct.