12. A regular tetrahedron is a polyhedron with four faces, each of which is an equilateral triangle, as shown. A solid regular tetrahedron is cut into two pieces by a single plane cut.
Which of the following could not be the shape of the section formed by the cut?
A a pentagon
B a square
C a rectangle that is not a square
D a trapezium
E a triangle that is not equilateral
Solution
A

When the regular tetrahedron is cut by a single plane cut, each of its four faces is cut at most once. The place where each face is cut becomes the edge of the newly formed section. Therefore a pentagon, with five edges, cannot be formed.

However, each of the other four options is possible, as the following diagrams show.