The rectangular prism below has two congruent square faces and four congruent rectangular faces.

Question

A diagram of a cube with 8 vertices labelled as A,B,C,D,E,F,G,H. vertices H,C,A  joined to form a triangle inside cube.
Note: Figure is not drawn to scale.

The line segment formed by connecting vertices E and B is 18 units long, the line segment formed by connecting vertices G and E is 13 units long, and the line segment formed by connecting vertices H and A is 18 units long.

If the rectangular prism is sliced by a plane that passes through vertices H, C, and A, which of the following best describes the resulting cross-section of the prism?

Answer 1

To determine the resulting cross-section of the rectangular prism when sliced by a plane that passes through vertices \( H \), \( C \), and \( A \), we need to analyze the given configuration of the cube and the positions of these vertices.

1. Understanding the vertices of the rectangular prism/cube:

   • The vertices of a cube can be labeled as follows:
     • \( A \), \( B \), \( C \), \( D \) (top face)
     • \( E \), \( F \), \( G \), \( H \) (bottom face)
   • In a standard orientation, the vertices can be assumed to have the following positions:
     • \( A \) = (0, 0, 0)
     • \( B \) = (L, 0, 0)
     • \( C \) = (L, L, 0)
     • \( D \) = (0, L, 0)
     • \( E \) = (0, 0, H)
     • \( F \) = (L, 0, H)
     • \( G \) = (L, L, H)
     • \( H \) = (0, L, H)

2. Identifying the coordinates of \( H \), \( C \), and \( A \):

   • With the cube's vertices defined as above, we find:
     • \( H \) corresponds to (0, L, H)
     • \( C \) corresponds to (L, L, 0)
     • \( A \) corresponds to (0, 0, 0)

3. Analyzing the slice:

   • The plane formed by points \( H \), \( C \), and \( A \) will effectively cut through the rectangular prism.
   • The three points are not collinear, indicating that the slice will create a polygonal shape.

4. Determining the resulting cross-section:

   • The triangle formed by points \( H \), \( C \), and \( A \) will intersect the top and bottom faces of the rectangular prism.
   • The resulting shape on the face that the cut intersects will be a triangle. Since it is defined by the three points which form a triangle, the resulting cross-section will be triangular.

Thus, the best description of the resulting cross-section when the prism is sliced by a plane passing through vertices \( H \), \( C \), and \( A \) is a triangle.