Geometry

First, let's use the Law of Cosines. It states that \( cos PMQ = - cos PMR \)

Thus, we can write the equations

\(PQ^2 = QM^2 + PM^2 - 2(QM * PM) * (-cos PMR) \\ PR^2 = RM^2 + PM^2 - 2 (RM * PM) * ( cos PMR)\)

Plugging in the values we already know from the problem, we get

\(5^2 = 5.5^2 + PM^2 + 2(5.5 * PM)cos(PMR) \\ 8^2 = 5.5^2 + PM^2 - 2(5,5 * PM) cos (PMR)       \)

Now, add these two equations. We get

\(5^2 + 8^2  = 2 * 5.5^2 + 2PM^2 \\ 89 = 60.5 + 2PM^2 \\ 28.5 / 2 = PM^2 \\ 14.25 = PM^2 \)

Thus, we have \(\sqrt{14.25}  = PM ≈  3.77\)

The answer is 3.77. 

Thanks! :)