Two circles touch each other at point A. Two common tangents of the circle meet at point X and neither of tangent passes through point A. These tangent touch the larger circle at point B and C. If radius of the largest circle is 15 cm and CX = 20 cm, then what is the radius of the smaller circle (in cm)?

3.75

Let radius of smaller circle = r

In ΔO'CX (right angle triangle, where \(\angle\)C = 90°)

O'X = \(\sqrt {(20)^2 + (15)^2}\) = \(\sqrt {625}\)

O'X = 25 cm

OX = O'X - O'A - OA

OX = 25 - 15 - (r)

OX = 10 - r

ΔOXD ∼ ΔO'XC

\(\frac{OX}{O'X}\) = \(\frac{OD}{O'C}\)

\(\frac{10 - r}{25}\) = \(\frac{r}{15}\)

⇒ 30 - 3r = 5r

⇒ r = \(\frac{30}{8}\) = 3.75 cm