AB and BC are segments of tangents drawn from point B to a circle centered at point O and radius 6. Find BO if AOC = 60.

From the center of the circle, point O, we construct the radii OA and OС to the point of tangency. By the property of tangents drawn from one point, the radii OA and OС are perpendicular to the tangents BA and BC.

In right-angled triangles AOB and COB, the hypotenuse BO is common, and the leg OA is equal to the leg OC, then the right-angled triangles AOB and COB are equal in leg and hypotenuse, and therefore the angle BOA = BOC = AOC / 2 = 60/2 = 300.

In a right-angled triangle AOB Cos AOB = AO / VO.

ВO = AO / Cos30 = 6 / (√3 / 2) = 12 / √3 = 4 * √3 cm.

Answer: The length of the VO segment is 4 * √3 cm.