A plane is drawn through the midpoints of the side edges of the triangular pyramid. Prove that it is parallel

A plane is drawn through the midpoints of the side edges of the triangular pyramid. Prove that it is parallel to the plane of the base of the pyramid. Base area 64 cm2, find the cross-sectional area.

Let’s designate the sides of the base of the pyramid a, b, c. Its area is S.

The sides of the triangle formed by the midpoints of the side edges are respectively a1, b1, c1. Its area is S1.

The plane formed by the midpoints of the side edges is parallel to the base plane based on the parallelism of the planes: The planes are parallel to each other if two intersecting straight lines lying in one plane are respectively parallel to two intersecting straight lines lying in another plane.

This condition is fulfilled: a || a1; b || b1, and a∩b; a1∩b1.

The ratio of the areas of two similar triangles is equal to the square of the similarity coefficient.

The midpoints of the edges are the midlines of the triangles that are the faces of the pyramid. Therefore, the coefficient of similarity is k = 2 (the middle line of the triangle is half the base).

S / S1 = k ^ 2;

S1 = S / k ^ 2;

S1 = 64/4 = 16 (cm ^ 2)

Answer. 16 cm ^ 2.