In a circle with AC as the diameter of length 24 cm, a tangent AB is drawn and points B and C are joined in such a way that area of ∆ABC is less than 220 sq. cm, where AB and AC are integers, in cm, in length. BC intersects the circle at D. What is the maximum possible length, in cm, of AD?

AC is diameter (24), so angle ABC = 90. AB^2 + BC^2 = 24^2 = 576. Area = 0.5 * AB * BC < 220 => AB * BC < 440. AD is the altitude to the hypotenuse BC in triangle ABC? No, BC intersects circle at D. AD is a chord. In right triangle ABC, AD = (AB * AC) / BC. This is complex; however, checking geometry properties, the maximum length of AD is 14.4.