# In the regular hexagonal pyramid SABCDEF, the sides of the base of which are 4, and the side edges are three roots out of six

**In the regular hexagonal pyramid SABCDEF, the sides of the base of which are 4, and the side edges are three roots out of six, find the angle between the straight lines BG and AD, where G is a point on the edge SC, with SG: GC = 1: 2.**

Since the hexagon at the base of the pyramid is correct, the diagonal of the AD is parallel to the BC side, and then the angle between the straight lines BG and AD is equal to the angle between BC and BC. The lateral edge ВSC is an isosceles triangle; let us draw its height SK, which divides the side ВС in half. VK = SC = BC / 2 = 4/2 = 2 cm Point G divides the edge SC in the ratio of 1/2, then SG = SC / 3 = 3 * √6 / 3 = √6 cm, CG = 2 * SC / 3 = 2 * 3 * √6 / 3 = 2 * √6 cm.

In a right-angled triangle SKC, by the Pythagorean theorem, SK2 = SC2 – KC2 = 54 – 4 = 50.

SK = 5 * √2 cm.

Draw a perpendicular from point G to the BC side. Triangles SKC and GMC are similar in two angles, then SC / KC = GC / MC.

MC = KC * GC / SC = (2 * 2 * √6 / 3 * √6) = 4/3. Then BM = BC – MC = 4 – 4/3 = 8/3.

SK / GM = KC / MS.

GM = SK * MC / KC = 5 * √2 * (4/3) / 2 = 10 * √2 / 3.

Then tgGBM = GM / BM = (10 * √2 / 3) / (8/3) = 5 * √2 / 4.

Answer: Angle GBM = arctan (√2 / 4).