Question

The measure of $\angle AOC=$ ___

• A. ${130}^{\circ }$
• B. ${120}^{\circ }$
• C. ${100}^{\circ }$
• D. ${140}^{\circ }$

Answer 1

The correct option is D

${140}^{\circ }$

In $\Delta AOB$, we have
OA = OB [radius of the circle].
$\Rightarrow \angle OBA=\angle OAB$ [Angles opposite to equal sides of a triangle are equal]
$\Rightarrow \angle OBA={30}^{\circ }$$\dots$(1)
Similarly, in $\Delta BOC$, we have OB = OC.
$\Rightarrow \angle OCB=\angle OBC$ [Angles opposite to equal sides of a triangle are equal]
$\Rightarrow \angle OBC={40}^{\circ }$$\dots$(2)
$\therefore \angle ABC=\angle OBA+\angle OBC={30}^{\circ }+{40}^{\circ }={70}^{\circ }$ [Using (1) and (2)]
Since the angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by the arc on the remaining part of the circle,
$\therefore \angle AOC=2\times \angle ABC=2\times {70}^{\circ }={140}^{\circ }.$