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Q: Is The midpoint of a given line segment must lie on the given line segment?

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No. Since a line is infinite, it has no mid-point. A bisector must go through a midpoint so nothing can bisect a line (not even a segment).

With a straight-edge and a compass:Swing arcs from each end of the segment with the compass (without changing the settings)Connect the intersections of these arcs.The resultant is a perpendicular bisector of the segment.

That means that it is not a line segment.

A line segment can be defined as having two endpoints

Can't be sure what you're asking, but it would be two circles with equal radii longer that half the length of the segment, using the endpoints as the origins of the two circles.

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It bisects the line segment at midpoint at 90 degrees and its slope is the reciprocal of the line segment's slope plus or minus.

No. Since a line is infinite, it has no mid-point. A bisector must go through a midpoint so nothing can bisect a line (not even a segment).

Yes

Equilateral triangles

With a straight-edge and a compass:Swing arcs from each end of the segment with the compass (without changing the settings)Connect the intersections of these arcs.The resultant is a perpendicular bisector of the segment.

That means that it is not a line segment.

A line segment can be defined as having two endpoints

A line of symmetry must go from one vertex to the midpoint of the opposite side.

Can't be sure what you're asking, but it would be two circles with equal radii longer that half the length of the segment, using the endpoints as the origins of the two circles.

A segment has two end points. If a line has one end point, then it must be called ray.

The perpendicular bisector of the line XY will meet it at its midpoint at right angles.