The side faces of the triangular pyramid form angles of 60 degrees

The side faces of the triangular pyramid form angles of 60 degrees with the base plane. Find the lengths of the side ribs if the height of the pyramid is 12 cm.

The ABCM pyramid is correct. At the base lies the equilateral triangle ABC.

The right-angled triangle of the МOК is formed by the apothema MK and the height of the pyramid MO, the angle MKO = 60. KO = 12 * √ (MKO) = 12 * (3 ^ (1/2)) / 3 = 4 * 3 ^ (1/2) cm, MK ^ 2 = MO ^ 2 + KO ^ 2 = 144 + 48 = 192.

The AOC triangle is isosceles with the KO height and the angles OAС = OCA = 30. From the OCВ triangle: OВ = 12 cm, KВ ^ 2 = OВ ^ 2 – OK ^ 2 = 144 – 48 = 96.

From the MKВ triangle: MВ ^ 2 = MK ^ 2 + KB ^ 2 = 192 + 96 = 288, MV = 12 * 2 ^ (1/2).