jmc

First, we notice that $8:15:17$ is a Pythagorean triple, so we have a right triangle. Let's draw a sketch: Since we know that $BM=\frac{1}{2}\cdot AB=4,$ we simply apply the Pythagorean theorem to right triangle $△MBC.$ $$\begin{array}{rl}C{M}^2& =B{M}^2+B{C}^2\\ C{M}^2& =4^2+15^2=241\\ CM& =\boxed{\sqrt{241}}.\end{array}$$