A pair of intersecting planes ______ intersect in exactly one point?
A- always
B- sometimes
C- never
D- not enough information
I think A- always, but I do not know for sure.
When two planes intersect (such as the edge of a rectangular box), the intersection is a line (not a point).
When two planes do not intersect(such a the top and bottom faces of a rectangular box), then they don't intersect at exactly one point.
Ohh, ok. Is the answer never?
To determine if a pair of intersecting planes always intersect in exactly one point, here's how you can find the answer:
Step 1: Recall the definition of intersecting planes. Two planes are considered intersecting if they have at least one point in common.
Step 2: Analyze the possibilities for the intersection of two planes:
- If the two planes are non-parallel and not coincident, they will intersect in a line.
- If the two planes are parallel, they will not intersect (unless they are coincident planes).
- If the two planes are coincident, they will intersect in infinitely many points.
Step 3: Apply the above understanding to the given options:
- Option A ("always"): This option suggests that intersecting planes will always intersect in exactly one point. This is not accurate since, as we found in Step 2, the intersection can be a line or infinitely many points in specific cases.
- Option B ("sometimes"): This option suggests that intersecting planes can intersect in one point, a line, or infinitely many points. This aligns with what we discovered in Step 2.
- Option C ("never"): This option suggests that intersecting planes never intersect in exactly one point. Again, this is not entirely accurate since there are cases where they do intersect in one point.
- Option D ("not enough information"): This option implies that more information is needed to determine how intersecting planes behave. However, based on our analysis so far, we have sufficient information to conclude that the answer is not "always" (option A).
Therefore, the most suitable answer is B - "sometimes," as intersecting planes can intersect in one point, a line, or infinitely many points depending on their relative orientation.