Question

ABCDA1B1C1D1-куб, ребро которого 4см. Постройте сечение куба плоскостью, проходящей через точки А, D1 и М, где М – середина ребра ВС. Вычислите периметр сечения. С рисунком, пожалуйста.

Answer 1

Building a Section of the Cube

To build a section of the cube, we'll start by finding the coordinates of points A, D1, and M. Then, we'll use these coordinates to determine the equation of the plane passing through these points. Finally, we'll calculate the perimeter of the resulting section.

Finding the Coordinates of Points A, D1, and M

Given: ABCDA1B1C1 is a cube with an edge length of 4 cm.

1. Coordinates of A, D1, and M: - A: (0, 0, 0) - D1: (4, 0, 0) - M: (2, 2, 0)

Determining the Equation of the Plane

The equation of the plane passing through points A, D1, and M can be found using the point-normal form of the equation of a plane, which is given by: \[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \] where (x_0, y_0, z_0) is a point on the plane, and (a, b, c) is the normal vector to the plane.

The normal vector can be found using the cross product of two vectors lying in the plane. Let's denote the vector AD1 as \(\vec{v_1}\) and the vector AM as \(\vec{v_2}\). Then, the normal vector \(\vec{n}\) is given by: \[ \vec{n} = \vec{v_1} \times \vec{v_2} \]

After finding the normal vector, we can use point A (0, 0, 0) as a point on the plane to determine the equation of the plane.

Calculating the Perimeter of the Section

Once the equation of the plane is determined, we can find the intersection of this plane with the cube to obtain the shape of the section. The perimeter of this section can then be calculated using the coordinates of the intersection points.

I'll now provide a diagram to illustrate the section of the cube and its perimeter.