Let’s construct the radii of the circle OB and OС to the points of tangency B and C.
The radii drawn to the points of tangency are perpendicular to the tangents themselves, then the triangles AOB and AOC are rectangular.
In right-angled triangles AOB and AOC, the hypotenuse OA is common, OB = OC = R, then the triangles AOB and AOC are equal in leg and hypotenuse, then the angle OAB = OAC = BAC / 2 = 120/2 = 60.
In a right-angled triangle, AOН, tg60 = OB / AB.
AB = OB / tg60 = 9 / √3 = 3 * √3 cm.
AC = AB = 3 * √3 cm as the lengths of tangents drawn from one point.
Answer: The lengths of tangents AB and AC are equal to 3 * √3 cm.
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