The height of a regular triangular pyramid is 20, and the median of its base is 6.

The height of a regular triangular pyramid is 20, and the median of its base is 6. Find the tangent of the angle that the side edge of the pyramid makes with the plane of the base.

Since the pyramid is regular, an equilateral triangle lies at its base. In an equilateral triangle, the medians, at the point of intersection, are divided by a ratio of 2/1, starting at the apex.

BO / HO = 2/1.

Then BO = 2 * HO.

BO + HO = BH = 6 cm.

2 * HO + HO = 6 cm.

3 * HO = 6 cm.

HO = 6/3 = 2 cm.

The height of the pyramid passes through the center of intersection of the medians, then the triangle DOH is rectangular.

Then tgDNO = DO / DO = 20/2 = 10.

Answer: The tangent of the angle is 10.