Lines AB and AC touch the circle with center O, at points B and C. Find BC if the angle OAB = 30 degrees and AB = 5 cm.

Given:
Surround (O; R)
angle ОАВ = 30 degrees.
AB = 5 cm.
Find: ВС
Decision:
In tr. AOB: angle B = 90 degrees.
angle A = 30 degrees, therefore BO = 1/2 AO
Let BO – x, then AO – 2x
AO ^ 2 = BO ^ 2 + AB ^ 2
4x ^ 2 = x ^ 2 + 25
x = 5 √3 / 3
In tr. ВOС: angle O = 120 degrees.
angle BOE = 60 degrees.
angle OBE = 30 degrees.
BE = ВO * cosB
BE = 5 √3 / 3 * cos 30 = 5 cm.
Answer: 5 cm.