AB and AC are segments of tangents drawn to a circle of radius 9 cm. Find the lengths of segments AC and AO if AB = 12 cm.

Since the segments of tangents drawn from one point to one circle are equal, then AB = AC. Therefore, AC = 12 cm.

Consider a triangle OBA: the segment OВ is equal to the radius of the circle, OВ = 9 cm. AB = 12 cm (by condition).

The ABO angle is 90 ° (the tangent to the radius is at right angles). Hence, the triangle OBA is rectangular.

By the Pythagorean theorem: AO² = AB² + BO² = 12² + 9² = 144 + 81 = 225.

Hence AO = √225 = 15 (cm).

Answer: AC = 12 cm, AO = 15 cm.