# AB and BC are segments of tangents drawn from point B to a circle with center O. AB = 6cm

AB and BC are segments of tangents drawn from point B to a circle with center O. AB = 6cm, OB = 12. What is the angle ABC?

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

In a right-angled triangle AOB, through the leg and hypotenuse, we determine the value of the angle OBA.

CosABO = AB / BO = 6/12 = 1/2.

Angle ABO = arcos (1/2) = 60.

In right-angled triangles AOB and COB, the hypotenuse BO is common, and the leg OA is equal to the leg OS, then the right-angled triangles AO and COB are equal in leg and hypotenuse, and therefore the angle ABO = CBO = 60.

Then the angle ABC = 2 * ABO = 2 * 60 = 120.

Answer: Angle ABC is 120.

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