Question

use the diagram to answer the questions
what two - dimensional figure will be created by a
horizontal cross section?

what two - dimensional figure will be created by a
vertical cross section through the apex?

Explanation:

Step1: Analyze horizontal cross - section

A horizontal cross - section of a pyramid (assuming the base is a polygon, here we consider a general pyramid) cuts parallel to the base. If the base is a polygon (for example, if it's a square - based pyramid, pentagonal - based pyramid etc.), a horizontal cross - section will be similar to the base. In the case of a pyramid with a polygonal base, a horizontal cross - section is a polygon. If we assume the base is a square (a common case in basic geometry problems about pyramids), a horizontal cross - section is a square. In general, for a pyramid with a polygonal base, the horizontal cross - section is a polygon. But if we consider the most basic non - degenerate case (like a square - based pyramid which is often used in textbooks for such problems), a horizontal cross - section is a square.

Step2: Analyze vertical cross - section through the apex

A vertical cross - section through the apex of a pyramid. Let's consider a pyramid with a polygonal base. When we take a vertical cross - section through the apex, we can think of the two slant edges and the height. If the base is a polygon (say a square), the vertical cross - section through the apex is a triangle. The two equal sides of the triangle are the slant heights (or the edges from the apex to the base vertices) and the base of the triangle is the side of the base polygon (for a square - based pyramid).

Answer:

1. Polygon (or more specifically, if base is square - square)
2. Triangle