In $$\triangle$$ABC with sides 6 cm, 7 cm and 8 cm,the angle bisector of the largest angle divides the opposite side into two segments. What is the length of the shorter segment?

As per the given in the question,

AB=6cm, BC=8cm and CA=7cm

AD is the angle bisector of $$\angle$$ BAC

As per the angular bisector theorem,

$$\Rightarrow \dfrac{BD}{DC}=\dfrac{AB}{AC}$$,

Now, substituting the values,

$$\Rightarrow \dfrac{BD}{DC}=\dfrac{6}{7}$$,

Let, 

$$\Rightarrow \dfrac{BD}{DC}=\dfrac{6}{7}=k$$,

So, BD=6k and DC=7k

It is given that, $$BC=8cm =BD+DC$$

$$\Rightarrow 6k+7k=8$$

$$\Rightarrow 13k=8$$

$$\Rightarrow k=\dfrac{8}{13}$$

Hence, $$BD=\dfrac{8}{13}\times 6=\dfrac{48}{13}$$

And, $$DC=\dfrac{8}{13}\times 7=\dfrac{56}{13}$$

Hence the required answer is $$=\dfrac{48}{13}$$