When bisectors are perpendicular when triangles are isosceles. By definition, the triangle being isosceles will have the height (that's perpendicular) and the side bisector being the same segment.
Because of that, we can conclude the three smaller triangles in the picture are isosceles triangles.
Isosceles triangles have two congruent sides. That said, BG is congruent to GC.
We have FG=6cm and CF=8cm. Those are the two legs of the right triangle GFC.
So, using pythagorean theorem:
GC²=6²+8²
GC²= 36+64
GC²=100
GC=10
But, GC=BG= 10cm
So, BG = 10cm.