The radius of the circle is 2. We know this because the top half of AC starts at the circle's center and goes to the edge of the circle. That means that AB is 7 long (the known 5 + the 2 from the radius of the circle).
We also know that AC and BC are tangent to the circle which allows us to deduct that the bottom part of AC is equivalent to the bottom part of BC. So AC is 5.
Now we can use law of sines to find the measure of angle ABC.
sin(C)/c = sin(B)/b.. We know what angle ACB and the side opposite of it is. We also know the side opposite to ABC. So now wee can solve for ABC.
sin(60)/7 = sin(B)/5
Angle ABC = sin^-1(5sin(60)/7) = 38.2132..
Finally, to find the length of side BC, we can find angle BAC.. 180 - 60 - 38.2132 = 81.7868.
Now do law of sines once more..
sin(60)/7 = sin(81.7868)/a
BC = 8
So AB = 7, AC = 5, and BC = 8.. The perimeter is 20.
This was a pretty neat challenge! Hopefully everything makes sense and is correct :) Let me know if you'd like a diagram to explain the process I used.