If G is the centroid of  $\Delta ABC$ and $BC=5,CA=12,AB=13$ then $BG=$

$a=BC=5,b=CA=12,c=AB=13$

$BG=\frac{2BE}{2+1}=\frac{2BE}{3}$

$BE=$ length of the median through B

$= \frac{1}{2}\sqrt{2a^2+2c^2-b^2}$

$=\frac{1}{2}\sqrt{50+338-144}$

$=\frac{1}{2}\sqrt{244}$ $=\sqrt{61}$

 $\therefore$  $BG=\frac{2\sqrt{61}}{3}$