Question

if the cube shown above is sliced by a plane to create a triangle, which sets of vertices could the plane pass through?
a. 1, 3, and 4
2, 3, and 8
1, 3, and 7
b. 4, 5, and 7
1, 2, and 7
2, 3, and 7
c. 3, 5, and 8
1, 2, and 6
2, 4, and 7
d. 4, 5, and 7
1, 3, and 8
2, 4, and 7

Explanation:

Step1: Recall the property of a plane - triangle intersection

A plane passing through three non - collinear points of a cube can form a triangle. We need to check each set of vertices to see if they are non - collinear and can be intersected by a single plane.

Step2: Analyze option A

For the set 1, 3, and 4: Points 1, 3, and 4 are on the same face of the cube. A plane passing through them will give a straight - line segment (since they are collinear on the face), not a triangle.

Step3: Analyze option B

For the set 4, 5, and 7: Points 4, 5, and 7 are non - collinear. But when considering the cube's structure, we can visualize that a plane passing through them can form a triangle. Also, for 1, 2, and 7: these points are non - collinear and a plane can pass through them to form a triangle. And for 2, 3, and 7: they are non - collinear and a plane can intersect them to form a triangle.

Step4: Analyze option C

For the set 3, 5, and 8: These points are non - collinear, but it is difficult to visualize a single plane that can pass through them to form a triangle due to the cube's structure. For 1, 2, and 6: they are on the same face and will not form a triangle. For 2, 4, and 7: it is hard to find a single plane to form a triangle.

Step5: Analyze option D

For the set 4, 5, and 7: As mentioned before, they can form a triangle. But for 1, 3, and 8: these points are in such a position that it is not possible to have a single plane passing through them to form a triangle.

Answer:

B. 4, 5, and 7; 1, 2, and 7; 2, 3, and 7