A point lies on the of a line segment if and only if the point is equidistant from the endpoints of the segment.


angle bisector

trisector

altitude

perpendicular bisector

perpendicular bisector

A point lies on the perpendicular bisector of a line segment if and only if the point is equidistant from the endpoints of the segment.

To determine the perpendicular bisector of a line segment, follow these steps:

1. Take the given line segment and identify its two endpoints.
2. Draw a line that passes through the midpoint of the line segment and is perpendicular to it. This line is the perpendicular bisector.
3. To verify if a point lies on the perpendicular bisector, measure the distance between the point and each of the endpoints of the line segment. If the distances are equal, the point lies on the perpendicular bisector.

Regarding the other terms mentioned:

- Angle bisector: An angle bisector is a line or ray that divides an angle into two equal angles. To find the angle bisector of an angle, draw two rays starting from the vertex of the angle, then draw a line that passes through the vertex and divides the angle in half.

- Trisector: A trisector is a line or ray that divides an angle into three equal angles. To find the trisector of an angle, divide the angle into three equal parts by drawing two lines that pass through the vertex and divide the angle evenly.

- Altitude: An altitude is a line segment drawn from a vertex of a triangle perpendicular to the opposite side (or an extension of it). Each triangle has three altitudes. To find the altitude of a triangle, draw a line segment from a vertex, perpendicular to the opposite side or its extension.

So, in summary, a point lies on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of the segment.