Answer
$$BM=AM=CM=\frac{\sqrt{a^2+b^2}}{2}$$
Work Step by Step
Let's calculate the distances using $a$ and $b$. But first we will need $M$ coordinates (We can calculate it using midpoint method of $AB$).
$M=(\frac{a+0}{2}, \frac{0+b}{2})=(\frac{a}{2}, \frac{b}{2})$
$BM=\sqrt{(\frac{a}{2}-0)^2+(\frac{b}{2}-b)^2}=\sqrt{(\frac{a}{2})^2+(-\frac{b}{2})^2}=\sqrt{\frac{a^2}{4}+\frac{b^2}{4}}=\frac{\sqrt{a^2+b^2}}{2}$
$AM=\sqrt{(\frac{a}{2}-a)^2+(\frac{b}{2}-0)^2}=\sqrt{(-\frac{a}{2})^2+(\frac{b}{2})^2}=\sqrt{\frac{a^2}{4}+\frac{b^2}{4}}=\frac{\sqrt{a^2+b^2}}{2}$
$CM=\sqrt{(\frac{a}{2}-0)^2+(\frac{b}{2}-0)^2}=\sqrt{(\frac{a}{2})^2+(\frac{b}{2})^2}=\sqrt{\frac{a^2}{4}+\frac{b^2}{4}}=\frac{\sqrt{a^2+b^2}}{2}$
As we can see above $BM=AM=CM=\frac{\sqrt{a^2+b^2}}{2}$