G is the centroid of the triangle ABC, where AB, BC and CA are 7 cm, 24 cm and 25 cm respectively, then BG is:

As per the given question,

AB=7cm, BC=24cm and CA=25cm

We can see, from the pythagoras theorem$$BC^2+BA^2=AC^2$$

$$\Rightarrow 7^2+24^2=25^2$$

Now, 

Let the point B at the origin so the co-ordinate of the point B =(0,0)

The co-ordinate of the point C =(7,0)

The co-ordinate of the point A =(0, 24)

Co-ordinates G $$ =(\dfrac{0+0+7}{3}),(\dfrac{0+0+24}{3})=(\dfrac{7}{3},8)$$

Hence the length of BG $$=\sqrt{(\dfrac{7}{3})+(8)^2}=\sqrt{\dfrac{625}{9}}=\dfrac{25}{3}$$

Or we can write it as $$=8\dfrac{1}{3}$$