Cimple Circumradius

Using cosine rule, we can find the cosine of the angle $\theta$ opposite to the side of length $9$ .

$\cos\theta=\frac{4^2+10^2-9^2}{2*4*10}=\frac 7{16}$ Thus $\sin\theta=\frac{\sqrt{207}}{16}$

Now using sine rule, we can say that $\frac 9{\sin\theta}=2R$ Where $R$ is the circumradius of the triangle.

Substuting the value of $\sin\theta$ and simplifying, we get $R=\sqrt{\frac{576}{23}}$

The formula for the circumradius of a triangle of sides $a$ , $b$ and $c$ is:

\begin{align} R = \frac{abc}{\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}} \end{align}

Plugging the values into the formula, we get that $R = \frac{24}{\sqrt{23}} \Rightarrow R = \sqrt{\frac{576}{23}}$ . Thus, $m+n = \boxed{599.}$