In a regular quadrangular pyramid SABCD, the height SO is 9, the diagonal of the base BD is 8.

In a regular quadrangular pyramid SABCD, the height SO is 9, the diagonal of the base BD is 8. Points K and M are the midpoints of the edges CD and BC, respectively. Find the tangent of the angle between the SMK plane and the base plane.

Consider the triangle MCK, as it is known that the points K and M are the midpoints of the edges, then we can say that the length of the height CL will be equal to half the height of the CO, in turn, the length of the CO is 4 cm, since this is half of the diagonal of the square, based on this reasoning , let’s say that the lengths CL and LO are equal to each other and equal to 2 cm each. Consider a right-angled triangle SOL, we can immediately find the tangent of the angle SLO:

tgSLO = SO / OL;

tgSLO = 9/2 = 4.5.

Answer: The tangent of the angle between the SMK plane and the base plane is 4.5.