Calculate the length of a direct common tangent of two circles of radii 3cm and 8cm with thier centres 13cm apart????

Consider two circles of radii 8 cm and 5 cm with center D and C respectively.

It is given that, distance between the center of the circle is 13 cm.

∴ DC = 13 cm

Let AB be the length of common tangent.

From above figure, it is quite clear that AB = CE = x cm (let) and 

DE = AD - AE = 8 cm - 3 cm = 5 cm 

In right triangle DCE, by pythagoras theorem,

DC² = DE² + CE²

⇒ 13² = 5² + x²

⇒ x² = 169 - 25

⇒ x² = 144

⇒ x = √144 cm = 12 cm

Hence, AB = CE =  x cm = 12 cm

 Hence, the length of direct common tangent is 12 cm.