Perpendicular Bisector of a Line Segment

The perpendicular bisector of a line segment is a straight line that is perpendicular to the segment and divides it into two equal halves.

In simple terms, it’s the line that bisects the segment, passing through its midpoint.

This concept is fundamental in geometry and is applied in both plane and solid geometry.

For instance, consider a line segment AB:

The perpendicular bisector of this segment is a straight line that intersects AB at its midpoint and forms a right angle (90 degrees) with it.

This means that the two segments created by the bisector will have equal lengths.

The perpendicular bisector is a precise way to find the center of a line segment and divide it evenly.

It also serves as the axis of symmetry for the segment. In other words, if you were to fold the segment along its bisector, the two halves would align perfectly.

In geometric shapes like triangles or quadrilaterals, the perpendicular bisectors of the sides are often used to locate points such as the circumcenter or incenter.
For example, in a triangle, the point where the perpendicular bisectors of the sides intersect is the circumcenter.

To draw the perpendicular bisector of a segment, start by finding the midpoint M of segment AB. Then, draw a line perpendicular to the segment through this midpoint.

Here’s a method: Draw two circles, one centered at point A and the other at point B, with radii greater than half the length of the segment.

$$ r > \frac{ \overline{AB} }{2} $$

Next, connect the points where the two circles intersect.

This line will intersect AB at its midpoint M and form a 90-degree angle, making it the perpendicular bisector of segment AB.