The circle centered at O touches the sides of the 60-degree angle BAC at points B and C.

The circle centered at O touches the sides of the 60-degree angle BAC at points B and C. Find the length of line segment AO. if BO = 2 cm

The OB radius is the perpendicular to the tangent AB (by the tangent and radius theorem).
Consider a right-angled triangle ABO.
By the theorem on tangents drawn from one point, we obtain:
∠ OAB = 1/2 * ∠ BAC = 1/2 * 60 ° = 30 °.
The leg ВO given by the condition of the problem lies opposite the angle OAB, the degree measure of which is 30 °. We conclude that the hypotenuse AO = 2 * BO = 4 cm.
Answer: the length of the ВO segment is 4 cm.