Perpendicular lines are always, sometimes , never coplanar
The Correct Answer and Explanation is:
The correct answer is always.
Explanation:
Perpendicular lines are always coplanar. This means that if two lines are perpendicular, they will lie in the same plane. Let’s break down this concept and understand why this is true.
1. Definition of Perpendicular Lines:
Perpendicular lines are two lines that intersect at a right angle, i.e., at 90 degrees. For two lines to be perpendicular, they must meet at some point in space.
2. Coplanar Lines:
Coplanar lines are lines that lie within the same plane. A plane is a flat, two-dimensional surface that extends infinitely in all directions. For two lines to be coplanar, they must be in the same plane, meaning they must lie on the same flat surface.
3. Why Perpendicular Lines are Always Coplanar:
- When two lines are perpendicular, they intersect at a common point.
- Since the two lines meet at this common point, they must be part of the same plane. Any two lines that intersect at a point (whether perpendicular or not) always lie on the same plane because you can draw a flat surface that contains both lines.
- For example, imagine two perpendicular lines on a flat piece of paper. These lines meet at a point on the paper, and both lines lie on the same flat surface (the paper). This is an illustration of how perpendicular lines are coplanar.
- Additionally, in three-dimensional space, even if lines are not horizontal or vertical, any two lines that intersect at a right angle will always lie within a single plane that passes through the intersection point. This is a fundamental geometric property.
4. Conclusion:
Because two perpendicular lines must intersect at a right angle and can always be drawn in a single plane that passes through their intersection point, they are always coplanar. There are no exceptions to this rule.