Attachment:
GRE circle (2).png [ 68.63 KiB | Viewed 74 times ]
We know that the length of the chord in a circle is $
4√3=AB$ (as shown in the figure above).
As perpendicular drawn from the centre of the circle to the chord bisects the chord, we get $
AC=$ $
CB=12(4√3)=2√3$
Now, let us assume the measure of angle $
OBC=θ∘$, so in triangle $
BOC$, we get $
Cosθ=BaseHypotenuse=2√34=√32⇒θ=30∘$
Finally in triangle BOC , we get angle $
COB=180∘−(90∘+30∘)=180∘−120∘=60∘(Sum$ of the angles of a triangle is 180 degrees)
As $
△AOC≅△BOC$, we get $
∠AOC=∠BOC=60∘⇒∠AOB=120∘=$ angle subtended by the chord at the centre of the circle.
Hence the answer is (E).