Two circles of radii 8 cm and 3 cm have their centres 13 cm apart. Find the length of a direct common tangent to the two circles.

Let there be two circles with centre A and B with radius 8 cm and 3 cm respectively.

Let TT' be the length of common tangent.

Construct a right-angled triangle ΔADB by drawing a line from center B parallel to TT' to intersect radius AT at D.

From figure,

DT = BT' = 3cm.

AD = AT - DT = 8 - 3 = 5 cm.

In right angled triangle ADB,

⇒ AB² = AD² + DB²

⇒ 13² = 5² + DB²

⇒ DB² = 13² - 5²

⇒ DB² = 169 - 25

⇒ DB² = 144

⇒ DB = 12 cm

Since, TDBT' is a rectangle,

So, TT' = DB = 12 cm.

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