AB is a diameter of a circle with center O.  CB is a tangent to the circle at B.  AC intersects the circle at G.  If the radius of the circle is 8 cm and AG = 12 cm, then the length of BC is?

\(\frac{16\sqrt {7}}{3}\)

In Δ ABG,

AB = 2 × 8 = 16,

AG = 12 and ∠AGB = 90°, therefore,

BG = \(\sqrt {16^2 - 12^2}\) = \(\sqrt {256 - 144}\) = \(\sqrt {112}\) = 4\(\sqrt {7}\)

BC = \(\frac{AB × BG}{AG}\) = \(\frac{16 × 4\sqrt {7}}{12}\)

= \(\frac{16\sqrt {7}}{3}\)