Question
A cube. The top face has points C, A, B, D and the bottom face has points G, E, F, H.
Which are right triangles that can be formed using a diagonal through the interior of the cube? Select all that apply.
Triangle CEG
Triangle AEH
Triangle CFG
Triangle BCH
Triangle BFG
Triangle DEG
Which are right triangles that can be formed using a diagonal through the interior of the cube? Select all that apply.
Triangle CEG
Triangle AEH
Triangle CFG
Triangle BCH
Triangle BFG
Triangle DEG
Answers
GPT-4o mini
In a cube, a diagonal connects two opposite vertices. For a triangle formed with a diagonal of the cube to be a right triangle, it must include one of the endpoints of the diagonal as well as two other vertices forming a right angle at one of the points.
Let's analyze the triangles formed with the cube's vertices and the diagonals:
1. **Diagonal CEG** connects points C and G.
- Triangle **CEG** is formed by points C, E, and G.
- There is a right angle between C and E (since they are both in the same plane at the top and bottom faces, respectively). So, **Triangle CEG** is a right triangle.
2. **Diagonal AEH** connects points A and H.
- Triangle **AEH** consists of points A, E, and H.
- The right angle occurs between A and H. Thus, **Triangle AEH** is also a right triangle.
3. **Diagonal CFG** connects points C and F.
- Triangle **CFG** consists of points C, F, and G.
- Here, no right angle is formed, both C and G are on opposite vertical planes that do not contribute to a right angle. Thus, **Triangle CFG** is not a right triangle.
4. **Diagonal BCH** connects points B and H.
- Triangle **BCH** consists of points B, C, and H.
- B and C can form a right angle on the face of the cube, but since H connects vertically, it does not create a right triangle. Thus, **Triangle BCH** is not a right triangle.
5. **Diagonal BFG** connects points B and G.
- Triangle **BFG** consists of points B, F, and G.
- B and G can connect, but F does not maintain any right angle with them. Thus, **Triangle BFG** is not a right triangle.
6. **Diagonal DEG** connects points D and G.
- Triangle **DEG** consists of points D, E, and G.
- There is no right angle formed. Therefore, **Triangle DEG** is not a right triangle.
After analyzing all the triangles with respect to the diagonals of the cube, the following right triangles can indeed be formed:
- Triangle CEG
- Triangle AEH
So, the correct answer is **Triangle CEG** and **Triangle AEH**.
Let's analyze the triangles formed with the cube's vertices and the diagonals:
1. **Diagonal CEG** connects points C and G.
- Triangle **CEG** is formed by points C, E, and G.
- There is a right angle between C and E (since they are both in the same plane at the top and bottom faces, respectively). So, **Triangle CEG** is a right triangle.
2. **Diagonal AEH** connects points A and H.
- Triangle **AEH** consists of points A, E, and H.
- The right angle occurs between A and H. Thus, **Triangle AEH** is also a right triangle.
3. **Diagonal CFG** connects points C and F.
- Triangle **CFG** consists of points C, F, and G.
- Here, no right angle is formed, both C and G are on opposite vertical planes that do not contribute to a right angle. Thus, **Triangle CFG** is not a right triangle.
4. **Diagonal BCH** connects points B and H.
- Triangle **BCH** consists of points B, C, and H.
- B and C can form a right angle on the face of the cube, but since H connects vertically, it does not create a right triangle. Thus, **Triangle BCH** is not a right triangle.
5. **Diagonal BFG** connects points B and G.
- Triangle **BFG** consists of points B, F, and G.
- B and G can connect, but F does not maintain any right angle with them. Thus, **Triangle BFG** is not a right triangle.
6. **Diagonal DEG** connects points D and G.
- Triangle **DEG** consists of points D, E, and G.
- There is no right angle formed. Therefore, **Triangle DEG** is not a right triangle.
After analyzing all the triangles with respect to the diagonals of the cube, the following right triangles can indeed be formed:
- Triangle CEG
- Triangle AEH
So, the correct answer is **Triangle CEG** and **Triangle AEH**.