The perpendicular bisector of a line segment is exactly those points that are the same distance from both endpoints of the line segment. This idea can be broken down into two statements:

• If a point is on the perpendicular bisector of a segment, then it must be the same distance from both endpoints of the line segment.
• If a point is the same distance from both endpoints of a line segment, then it must be on the perpendicular bisector of the segment.

These statements are converses of each other. Two statements are converses if the “if” parts and the “then” parts are swapped. The converse of a true statement isn’t always true, but in this case, both statements of the Perpendicular Bisector Theorem are true.

A line of reflection is the perpendicular bisector of segments connecting points in the original figure with corresponding points in the image. Therefore, the following three lines are all the same:

• The perpendicular bisector of a segment.
• The set of points equidistant from the 2 endpoints of a segment.
• The line of reflection that takes the 2 endpoints of the segment onto each other and the segment onto itself.

A figure: A large quadrilateral with a small triangle inside. The two upward facing sides of the quadrilateral each have two tick marks, the downward facing sides each have three tick marks. Two sides of the downward facing small triangle each have one tick mark. A perpendicular bisector is drawn through the figure, intersecting at three points. A horizontal line segment is drawn in the center of the figure.  

It is useful to know that the perpendicular bisector of a line segment is also all the points which are the same distance from both endpoints of the line segment, because then:

• If 2 points and are both equidistant from the endpoints of a segment, then the line through and must be the perpendicular bisector of the segment (because 2 points define a unique line).
• If 2 points and are both equidistant from the endpoints of a segment, then the line through and must be the line of reflection that takes the segment to itself and swaps the endpoints.
• If a point is on the line of reflection, then the distance from to a point on the original figure is the same as the distance from to its corresponding point on the image.
• If a point is on the perpendicular bisector of a segment, then it is the same distance from to both endpoints of the segment.