Question

How do I solve for x?

Answer 1

Tangents to a circle from the same point are equal length. Thus, for example, the two segments at the upper left (to A and to D) are both length 4. The ones at lower left are both length 6.

Then
  x = 4+6
  x = 10

The perimeter is twice the sum of the lengths shown, so is
  perimeter = 2(4 + 8 + 3 + 6)
  perimeter = 42