In the cube ABCDA1B1C1D1, a section is drawn through the vertices A, C1 and the middle of the edge DD1.

In the cube ABCDA1B1C1D1, a section is drawn through the vertices A, C1 and the middle of the edge DD1. Find the length of the edge if the cross-sectional area is 50√6.

From point C1, draw a segment parallel to AK, and from point A, parallel to KC1. The point of their intersection M is the point of the section.

Section AMC1K – rhombus. Let’s define his side.

Let the side of the square be X cm.

We define the diagonals of the rhombus MK = DD, then MK ^ 2 = AB ^ 2 + AD ^ 2 = X ^ 2 + X ^ 2 = 2 * X ^ 2.

MK = X * √2 cm.

AC1 ^ 2 = AC ^ 2 + CC1 ^ 2 = KM ^ 2 + CC1 ^ 2 = 2 * X ^ 2 + X ^ 2 = 3 * X ^ 2.

AC1 = X * √3 cm.

By the condition Ssech = 50 * √6 = MK * AC1 / 2 = X * √2 * X * √3 / 2.

X ^ 2 = 2 * 50 = 100.

X = 10 cm.

Answer: The side of the cube is 10 cm.