The Perpendicular Bisector Theorem: Geometry And Its Applications In Circles And Triangles

Perpendicular Bisector Theorem

If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

The Perpendicular Bisector Theorem is a geometric theorem which states that for any given segment AB, its perpendicular bisector (a line that intersects the segment at its midpoint and is perpendicular to it) will also pass through the point equidistant from A and B. In other words, any point on the perpendicular bisector of a segment is equidistant from its endpoints.

Mathematically, this can be written as follows:

Given segment AB with midpoint M and perpendicular bisector l, any point P on l satisfies the following equation:

AP = BP

where AP and BP denote the distance between point P and points A and B respectively.

This theorem is commonly used in geometry problems involving circles, triangles, and other shapes, where determining the position of the perpendicular bisector can provide important information about the properties of the figure. For example, in a triangle, the perpendicular bisectors of the sides intersect at a single point known as the circumcenter, which is the center of the triangle’s circumcircle.

More Answers: