Which is a right triangle formed using a diagonal through the interior of the cube?

Question

A cube. The top face has points G, B, C, F and the bottom face has points H, A, D, E.
triangle AEH
triangle CGE
triangle DGH
triangle HFB

Answer 1

To determine which triangle is a right triangle formed using a diagonal through the interior of the cube, first note that a cube has diagonals that connect opposite vertices.

The vertices of the cube are labeled as follows:

• Top face: G, B, C, F
• Bottom face: H, A, D, E

Now, let's analyze each triangle proposed:

1. Triangle AEH:

   • Points A, E, and H are corners of the bottom face of the cube. This triangle lies entirely on the bottom face and does not involve any interior diagonals.

2. Triangle CGE:

   • Points C, G, and E are composed of one vertex from the top face (C), one vertex from the bottom face (E), and one vertex from the top face (G). Considering the positions in 3D, CG and GE are not perpendicular, so it is not a right triangle.

3. Triangle DGH:

   • Points D, G, and H include D from the bottom face, G from the top face, and H from the bottom face. To see if it has a right angle:
     • DG represents the vertical edge, while DH and GH are lateral edges that meet at point G, forming a right angle at G. Thus, this triangle contains a right angle.

4. Triangle HFB:

   • Points H, F, and B involve one corner from the bottom (H) and two corners from the top face (F and B). In triangle HFB, the angles do not create a scenario where there is a 90° angle as they all connect through edges.

Based on this analysis, triangle DGH is the only triangle that can be identified as having a right angle. Thus, the answer is triangle DGH.