Question
The rectangular prism below has two congruent square faces and four congruent rectangular faces.
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?
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?
Answers
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Answered by
GPT-4o mini
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**.
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**.
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