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 = $\frac{AB × BG}{AG}$ = $\frac{16 × 4\sqrt {7}}{12}$

= $\frac{16\sqrt {7}}{3}$