How many triangles whose vertices are chosen from the vertices of a cube are equilateral?

• A. 3
• B. 2
• C. 4
• D. 8

To determine how many equilateral triangles can be formed using the vertices of a cube, we first need to analyze the geometry of a cube. A cube has 8 vertices.

Understanding the Cube's Structure

A cube consists of 12 edges and 6 faces. The vertices are positioned in such a way that not all combinations of three vertices will form an equilateral triangle.

Identifying Possible Triangles

For a triangle to be equilateral, all sides must be of equal length. In a cube, the only way to achieve this is by selecting vertices that are equally spaced. Let's consider the following:

• Triangles formed by vertices on the same face will not be equilateral.
• Triangles formed by vertices from different faces can be checked for equal lengths.

Counting Equilateral Triangles

After examining the vertices, we find that there are two distinct sets of equilateral triangles:

• One set consists of triangles formed by selecting vertices from three different faces.
• The other set includes triangles that use the midpoints of edges, but these do not correspond to the vertices of the cube.

Final Count

Ultimately, the only equilateral triangles that can be formed using the vertices of a cube are:

• Two triangles that can be formed by selecting vertices from the cube.

Thus, the answer to the question is B. 2.