Start by finding the radius/diameter: if the circumference is 12pi*sqrt3, then this means pi*d = 12pi*sqrt3 or d = 12sqrt3.
Furthermore, if AC is a diameter, we know the angle at B to be 90 degrees. This means that the entire triangle is a 30-60-90 triangle with proportions x:xsqrt3:2x.
That means that, in this case, our x would be opposite the 30--that's the one we're looking for.
We know of course that the 2x, our AC, is 12sqrt3. This implies of course that x, our AB, is 6sqrt3.
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