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

Answer 1

The correct option is B

${90}^{\circ }$

In $\Delta OBC$, OB = OC (radius)
$\Rightarrow \angle OBC=\angle OCB=y$
Now, $z+y+y={180}^{\circ }$ [Sum of angles of a triangle is ${180}^{c}irc$]
$\Rightarrow z={180}^{\circ }-2y$ .... (1)
Also, $\angle BOC=2\angle BAC$ [The angle subtended by an arc at the center is double the angle subtended by it any part of the major arc]
$\Rightarrow z=2x$ .... (2)
From (1) and (2),
$\Rightarrow 2x+2y={180}^{\circ }\Rightarrow x+y={90}^{\circ }$
$\therefore \angle BAC+\angle OBC={90}^{\circ }$