jmc

Since the square has side length $3$, the area of the entire square is $9$. The segments divide the square into 3 equal parts, so the area of each part is $9÷3=3$. Since $△CBM$ has area $3$ and base $CB=3$, using the area formula for a triangle: $A_{tri}=\frac{1}{2}bh$ $3=\frac{1}{2}3h$ $h=2$ Thus, height $BM=2$. Since $△CBM$ is a right triangle, $CM=\sqrt{B{M}^2+B{C}^2}=\sqrt{2^2+3^2}=\boxed{\sqrt{13}}$.