Perpendicular Bisector
A perpendicular bisector is a line that crosses a segment at its midpoint at a right angle, and every point on it is equidistant from the segment's endpoints.
Formula
\text{Any point } P \text{ on bisector: } PA = PB \text{ (equidistant from endpoints)}
Definition
A perpendicular bisector is a line that crosses a segment at its exact midpoint at a right angle, both bisecting the segment and being perpendicular to it. The Perpendicular Bisector Theorem states that a point is on the perpendicular bisector of a segment if and only if it is equidistant from the two endpoints; formally, the perpendicular bisector of segment $AB$ is the set $\{P \in \mathbb{R}^2 : |PA| = |PB|\}$, which simplifies to the linear equation $2(B-A)\cdot P = |B|^2 - |A|^2$, the perpendicular hyperplane to $(B-A)$ through the midpoint $(A+B)/2$.
Example
If segment $AB$ is $10$ cm long, its perpendicular bisector crosses at the $5$ cm mark at a right angle, and any point on it is equally far from $A$ and $B$. For $A(1,2)$ to $B(5,6)$: midpoint $=(3,4)$, slope of $AB=1$, perpendicular slope $=-1$, giving bisector $y=-x+7$. For $A=(1,1)$, $B=(5,3)$: midpoint $(3,2)$, direction $(4,2)$, bisector equation $4(x-3)+2(y-2)=0$, or $2x+y=8$.
Key Insight
This equidistance property is used to find the center of a circle passing through two points, and the three perpendicular bisectors of a triangle's sides are concurrent at the circumcenter, equidistant from all three vertices (inside for acute triangles, outside for obtuse, at the hypotenuse's midpoint for right triangles). Perpendicular bisectors also generate Voronoi diagrams: the Voronoi region of each of several points is the intersection of half-planes defined by perpendicular bisectors with all other points, a construction used in computational geometry, spatial statistics, and modeling natural cell patterns.