To determine the minimum number of sides that the polygon at the base of the prism must have for a plane to intersect the prism and create a cross-section that is a polygon with five sides, consider the following:
A prism is a polyhedron with two congruent and parallel faces (called the bases) and other faces that are parallelograms (referred to as the lateral faces). The cross-section formed when a plane intersects a prism is dependent on the orientation of the plane with respect to the prism. However, one key property here is that a cross-section parallel to the base of the prism will produce a polygon that is congruent to the base.
Now, to produce a polygonal cross-section with five sides, there are several possibilities:
1. If the plane cuts through the prism parallel to the base, then the base polygon itself must be a pentagon with five sides to create a cross-section that also has five sides. This is because a parallel cut would yield a congruent polygon to the original base of the prism.
2. If the plane cuts through the prism at an angle, it could intersect more than one face of the prism. However, a minimum of five lateral faces would need to be intersected to produce a polygon with five sides; otherwise, if fewer than five sides are intersected, the resulting cross-section would have fewer than five sides.
Therefore, the minimum number of sides the polygon at the base of the prism must have is five to ensure that a cross-sectional cut by a plane would yield a polygon with five sides, whether it is a parallel cut or at an angle that intersects the lateral faces.
So, the minimum number of sides that the polygon at the base of the prism must have is 5.
