Can a line (not a line segment) have a perpendicular bisector?
No, It cannot. The end-points of a line are not defined.
Depends on how you rigorously define "perpendicular bisector". You could argue that any point on a line is the "midpoint".
Yes, a line can have a perpendicular bisector.
A perpendicular bisector is a line that cuts another line segment into two equal halves at a right angle. While a line segment is a portion of a line with two distinct endpoints, a line extends indefinitely in both directions.
To understand how a line can have a perpendicular bisector, consider the following steps:
1. Take any line, which does not have defined endpoints.
2. Choose any point on the line as the center point for constructing the perpendicular bisector.
3. From the center point, draw two lines in opposite directions, each extending indefinitely.
4. Make sure that both lines are perpendicular to the original line.
5. As these two lines extend in opposite directions, they will cross the original line at two points.
6. This perpendicular line passing through the center point of the original line cuts it into two equal halves and is therefore the perpendicular bisector of the original line.
So, while a line segment can have a perpendicular bisector that intersects the segment at its midpoint, a line can have a perpendicular bisector that cuts it into two equal halves without specific endpoints.