The angle subtended by the chord DC at the major arc is half the angle subtended at the centre of the circle.

Hi, thank you for pointing out the error. We have updated the question.

Hi Ashwin Felix,
Because of the hut, the cow will not be able to graze as shown in the field, so out of rope length 2L, only the length L can turn around the hut, as shown in the figure. The quarter circle is shown in the figure.

And similarly, on the other side of the hut, the cow can turn around the hut with rope length L; this area is the quarter circle of rope length L.
Thus, a semicircle area or 2 times the quadrant's area is also where the cow can graze. (2 times area of quadrant = $$2\times\dfrac{\pi\ r^2}{4}$$)
I hope this helps!

HI Chinmay Dwivedi,
We have not assumed AC pass through center. Please refer to the image. The line OC and the line AC are not concurrent.
I hope this helps!!