Okay, I've realized that you were actually correct, I limited myself with my thinking. Due to what the problem appears to be asking, we only need to know the length of the tangent line, we do not need to know the point. And because it is a tangent line, it will have a 90º angle, so you were correct, we can use Pythagorean's theorem.

$\displaystyle L = \sqrt{4^2-2^2} = \sqrt{12}$

Then summing all our distances together (from my previous post), we get $\displaystyle L = 3\pi + 4\pi + \sqrt{12} = 7\pi + \sqrt{12}$ units.

This answer is assuming that our assessment of the problem is correct, and that I have not screwed up somewhere along the way