If \(AB\) is the diameter of the circle with center \(X\) and \(C\) is a point on the circle such that \(AC = AX = 3,\)

Since AB is the diameter, and X is the center
=> AX is the radius.

It is given that AX = 3
=> AB = 6

Also, angles in a semi-circle are 90 degrees.
=> \angle ACB = 90 degrees
So, in \triangle ABC,
AB^2 = AC^2 + BC^2
6^2 = 3^2 + BC^2
BC = 3\sqrt{3}

=> Perimeter of \triangle ABC = 3 + 3 + 3\sqrt{3} = 6 + 3\sqrt{3}