Question

Assertion: The measure of $\angle AOC={60}^{\circ }$

Reason: Angle subtended by an arc of a circle at the center of the circle is double the angle subtended by the arc on the circumference.

Which of the following is correct?

• A. Both Assertion and Reason are true and Reason is the correct explanation of Assertion
• B. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is D

Assertion is false but Reason is true

Join BO.
In $\Delta AOB$, we have
OA = OB [radius]
$\Rightarrow \angle OBA=\angle OAB$ [Angle opposite to equal sides of a triangle are equal]
$\Rightarrow \angle OBA={30}^{\circ }$$\dots$(1)
Similarly, in $\Delta BOC$, we get OB = OC
$\Rightarrow \angle OCB=\angle OBC$ [Angle opposite to equal sides of a triangle are equal]
$\Rightarrow \angle OBC={40}^{\circ }$$\dots$(2)
$\therefore \angle ABC=\angle OBA+\angle OBC={30}^{\circ }+{40}^{\circ }={70}^{\circ }$ [Using (1) and (2)]
Since angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by the arc on the circumference.
$\therefore \angle AOC=2\times \angle ABC=2\times {70}^{\circ }={140}^{\circ }$
As the Assertion states that $\angle AOC={60}^{\circ }$, it is false.