Geometry

Triangle ABC is a right triangle because it has it's hypotenuse on a diameter of a circle. (point C is backwards on my graph, but that doesnt affect the result because its a reflection of the original image)

Finding AB with pythagorean theorem, we get $AB = \sqrt{4^2 + 8^2} = 4\sqrt{5}$.

We also Get ACM=MCB = 45 degrees 

Because interior angles of a circle are congruent, we get AMB = BAM = 45. Now we see that ABM is a right isoceles triangle. Also, CAMB is a cyclic quadrilateral. We can calculate AM and BM too. $AM = BM = \frac{4\sqrt{5}}{\sqrt{2}} = 2\sqrt{10}$

Because ACBM is a cyclic, Ptolemy's theorem can be applied. 

Set the length of CM as X

then $4*2\sqrt{10}+8*2\sqrt{10}=x*4\sqrt{5}$ and solving the equation we get x = $x = 6\sqrt{2}$