In triangle ace, g is the centroid and be=9. find bg and ge.
(segments be and ad cross in the middle of the triangle, be overlaps ad and c is at the top of the triangle, and a is on the left corner and e is on the right)
you so ! ^0^

For the answer to the question, above BE is a median. Along with any median, the centroid of a triangle is always located 2/3 of the way from the vertex to the midpoint of the opposite side, so the centroid G is 2/3 of the way from E to B. 
So we have GE = (2/3)BE. 

Furthermore, by segment addition, BG + GE = BE. 

Since BE = 9, GE = (2/3)(9) = 6. 
Then BG + 6 = 9, which gives BG = 3. 
I hope this helps you.