In a regular quadrangular pyramid SABCD with vertex S, point M is marked – the midpoint of edge BC. SM = 3√86

In a regular quadrangular pyramid SABCD with vertex S, point M is marked – the midpoint of edge BC. SM = 3√86, the height of the pyramid is 13. Find the length of the line segment MD.

For the solution, you need to draw a picture where to designate:

At the base of a regular 4-angled pyramid is a square.
SO is the height of the pyramid and it is 13.
In a triangle SBC SM is the median that divides the BC into two equal segments.
Consider a triangle SMO, MO is half the side of the square ABCD and according to the Pythagorean theorem is:
MO ^ 2 = 3√86 ^ 2 – 13 ^ 2, where MO = 11√5

Consider a triangle МCD, in which MC = MO, and CD = 2 * MO.
Then the Pythagorean theorem:
МD ^ 2 = 11√5 ^ 2 + (2 * 11√5) ^ 2, where МD = 55.