91Ó°ÊÓ

Prisms. Take any polygon \(A B C D E\) (Figure 39 ), and through its vertices, draw parallel lines not lying in its plane. Then on one of the lines, take any point \(\left(A^{\prime}\right)\) and draw through it the plane parallel to the plane \(A B C D E\), and also draw a plane through each pair of adjacent parallel lines. All these planes will cut out a polyhedron \(A B C D E A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\) called a prism. The parallel planes \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\) are intersected by the lateral planes along parallel lines (\$13), and therefore the quadrilaterals \(A A^{\prime} B^{\prime} B, B B^{\prime} C^{\prime} C\), etc. are parallelograms. On the other hand, in the polygons \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\), corresponding sides are congruent (as opposite sides of parallelograms), and corresponding angles are congruent (as angles with respectively parallel and similarly directed sides). Therefore these polygons are congruent. Thus, a prism can be defined as a polyhedron two of whose faces are congruent polygons with respectively parallel sides, and all other faces are parallelograms connecting the parallel sides. The faces \(\left(A B C D E\right.\) and \(\left.A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\right)\) lying in parallel planes are called bases of the prism. The perpendicular \(O O^{\prime}\) dropped from any point of one base to the plane of the other is called an altitude of the prism. The parallelograms \(A A^{\prime} B^{\prime} C, B B^{\prime} C^{\prime} C\), etc. are called lateral faces, and their sides \(A A^{\prime}, B B^{\prime}\), etc., connecting corresponding vertices of the bases, are called lateral edges of the prism. The segment \(A^{\prime} C\) shown in Figure 39 is one of the diagonals of the prism. A prism is called right if its lateral edges are perpendicular to the bases (and oblique if they are not). Lateral faces of a right prism are rectangles, and a lateral edge can be considered as the altitude. A right prism is called regular if its bases are regular polygons. Lateral faces of a regular prism are congruent rectangles. Prisms can be triangular, quadrangular, etc. depending on what the bases are: triangles, quadrilaterals, ete.

Parallel cross sections of pyramids. Theorem. If a pyramid (Figure 49) is intersected by a plane parallel to the base, then: (1) lateral edges and the altitude \((S M)\) are divided by this plane into proportional parts; (2) the cross section itself is a polygon \(\left(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\right)\) similar to the base; (3) the areas of the cross section and the base are proportional to the squares of the distances from them to the vertex.

In a rectangular parallelepiped with a square base and the altitude \(h\), a cross section through two opposite lateral edges is drawn. Compute the total surface area of the parallelepiped, if the area of the cross section equals \(S\). e.

Prove that if all altitudes of a tetrahedron are concurrent, then each pair of opposite edges are perpendicular, and vice versa.

Compute the volume and lateral surface area of a regular hexagonal pyramid whose altitude has length \(h\) and makes the angle \(30^{\circ}\) with the apothem.

Describe the cross section of a cube by the plane perpendicular to one of the diagonals at its midpoint.

Show that each of the 12 rotations of a regular tetrahedron permutes the four vertices, and that to different rotations there correspond different permutations of the set of vertices.

How many planes of symmetry does a regular tetrahedron have?

Show that a cube has nine planes of symmetry.

Find all axes of symmetry (of any order) of an icosahedron, and show that there are in total 60 ways (including the trivial one) to superimpose the icosahedron onto itself by rotation.