its C- 12 +12sqrt(3)
triangle OAC is a 30-60-90 triangle
so AC will be 6sqrt(3)
now take this length and calculate the length
length of side will be 6sqrt(3) +6+6+6sqrt(3)
it is C

I second that. Its C due to the same reasons although I have to tell you that this is not likely in GMAT.

tohandb,

Is it a rule that the line from the centre of the circle to the meeting point of the tangents should be an angle bisector(as the two triangles are similar)

winnerhere wrote:tohandb,

Is it a rule that the line from the centre of the circle to the meeting point of the tangents should be an angle bisector(as the two triangles are similar)

one way to understand is that you can see in the image :
1) both triangles have a common side
2) both triangles have a side of same length(the radius)
3) and both triangles have the third side also of the same length. ( two tangents drawn from any point outside the circle are of same length)

4) line drawn from circle to the point of tangent is perpendicular to the tangent. so the two angles shown are same i.e. 90 a
so yes it must be an angle bisector.

but I don't think about these rules. I got by the symmetry. In geometry, using symmetry you can solve a lot of questions.
for example in this question,
1)triangle is equilateral- symmetric
2) three circles with equal radius inside an equilateral triangle all touching the sides.
from this you can easily say that the angles must be symmetric.