The height CK of triangle ABC divides side AB into segments AK and BK. Find the BC side if AC = 6 cm, BK = 3 cm, ∠ A = 60 °.

Consider triangle AKC. It is rectangular (СK is the height of the ABC triangle), its hypotenuse AC = 6 cm, and ∠ A = 60 °. The leg, opposite an angle of 60 °, is equal to half the hypotenuse. СK leg = 3 cm.

Consider a ВKС triangle. It is rectangular (СK – the height of the ABC triangle), its legs ВK = СK = 3 cm – it is isosceles. The hypotenuse BC is calculated according to the Pythagorean theorem: BC = (BK ^ 2 + CK ^ 2) ^ 1/2.

BC = (3 ^ 2 + 3 ^ 2) ^ 1/2 = 18 ^ 1/2 = 3 * 2 ^ 1/2 cm.

Answer: the side BC in triangle ABC is 3 * 2 ^ 1/2 cm.