The distance between the centres of two circles of radius 6 cm each is 13 cm. The length (in cm) of a transverse common tangent is:

According to Given Question :

We have to find out the length of AB( Transverse common tangent) which is equal to $$CC_{2}$$.

As we can see, $$C_{1}CC_{2}$$ is a Right angle triangle and $$C_{1}C_{2} = 13cm,C_{1}C = 12cm$$ (as $$BC=C_{2}A$$)

By Pythagoras theorem :

$$\Rightarrow (C_{1}C)^2 + (CC_{2})^2 = (C_{1}C_{2})^2$$

$$\Rightarrow 12^2 + (CC_{2})^2 = 13^2 $$

$$\Rightarrow  (CC_{2})^2 = 13^2 - 12^2 $$

$$\Rightarrow (CC_{2})^2 = 169 - 144 =25$$

$$\therefore (CC_{2}) = \sqrt{25}=5$$

$$\therefore AB = 5cm$$