AB and BC are segments of tangents drawn to a circle with a center at point O and a radius of 3 cm.

AB and BC are segments of tangents drawn to a circle with a center at point O and a radius of 3 cm. Find the angle AOC if OB = 6 cm.

From the center of the circle, point O, we construct the radii OA and OS to the points of tangency.

Since the radii drawn to the tangency points are perpendicular to the tangents, the triangles AOB and COB are rectangular.

In triangles AOB and COB, the hypotenuse BO is common, legs OA = OA = R, then these triangles are equal in leg and hypotenuse, and then the angle OBA = OBC.

In a right-angled triangle AOB, SinOBA = OA / OB = 3/6 = 1/2.

Then the angle OBA = 30, and the angle ABC = 2 * 30 = 60.

Then the angle AOC = (360 – 90 – 90 – 60) = 120.

Answer: The AOC angle is 120.