Question

BC is chord of a circle with centre O. A is a point on major arc BC. Find the total measure of $\angle BAC$ and $\angle OBC$.

• A. ${90}^{\circ }$
• B. ${100}^{\circ }$
• C. ${120}^{\circ }$
• D. ${150}^{\circ }$

Answer 1

The correct option is A ${90}^{\circ }$

$\text{In}\Delta \text{OBC, OB=OC = radius of the circle}.$
$\Rightarrow \angle OBC=\angle OCB=y$
Using angle sum property, $z+y+y={180}^{\circ }.$
$\Rightarrow z={180}^{\circ }-2y$ ....(1)
Also, since the angle subtended by an arc at the center is double the angle subtended by it any part of the circle,
$\angle BOC=2\angle BAC.$
$\Rightarrow z=2x$ ....(2)
From (1) and (2),
$2x+2y={180}^{\circ }\Rightarrow x+y={90}^{\circ }$
i.e., $\angle BAC+\angle OBC={90}^{\circ }$