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(D) The angle subtended by an arc at the centre is double the angle subtended by the arc in the remaining part of the circle.Given that $\angle ACB =

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$\sqrt{2}$

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(D) The angle subtended by an arc at the centre is double the angle subtended by the arc in the remaining part of the circle.Given that $\angle ACB = \frac{\pi}{4}$,the angle subtended by the arc $AB$ at the centre $O$ is $\angle AOB = 2 	imes \angle ACB = 2 	imes \frac{\pi}{4} = \frac{\pi}{2}$.In $	riangle AOB$,$OA = OB = 1$ (radii of the circle) and $\angle AOB = \frac{\pi}{2}$.Using the Pythagorean theorem in $	riangle AOB$:$AB^2 = OA^2 + OB^2 = 1^2 + 1^2 = 2$.Therefore,$AB = \sqrt{2}$.

Let $A, B, C$ be three points on a circle of radius $1$ such that $\angle ACB = \frac{\pi}{4}$. Then,the length of the side $AB$ is

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