Perpendicular Bisector
GeometryA 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 line segment at its exact middle point and forms a right angle ($90^\circ$) with the segment. It both bisects (cuts in half) and is perpendicular (at $90^\circ$).
Example
If segment $AB$ is $10$ cm long, the perpendicular bisector crosses it at the $5$ cm mark and stands straight up at a right angle. Any point on this bisector line is the same distance from $A$ as it is from $B$.
Key Insight
The perpendicular bisector is the set of all points that are exactly the same distance from both endpoints of a segment. This property is used to find the center of a circle passing through two points.
Definition
The perpendicular bisector of a segment is the line perpendicular to the segment passing through its midpoint. Perpendicular Bisector Theorem: a point is on the perpendicular bisector of a segment if and only if it is equidistant from the two endpoints. The three perpendicular bisectors of a triangle's sides are concurrent at the circumcenter.
Example
Segment from $A(1,2)$ to $B(5,6)$: midpoint $= (3, 4)$. Slope of $AB = (6-2)/(5-1) = 1$. Perpendicular slope $= -1$. Perpendicular bisector: $y - 4 = -1(x - 3)$, so $y = -x + 7$.
Key Insight
The circumcenter of a triangle (intersection of the three perpendicular bisectors) is the center of the circumscribed circle. It is equidistant from all three vertices. For acute triangles it is inside; for obtuse triangles, outside; for right triangles, at the midpoint of the hypotenuse.
Definition
The perpendicular bisector of segment $AB$ is the set $\{P \in \mathbb{R}^2 : |PA| = |PB|\}$, which simplifies to the linear equation $2(B - A) \cdot P = |B|^2 - |A|^2$. It is the perpendicular hyperplane to $(B - A)$ passing through the midpoint $(A + B)/2$. In Voronoi diagrams, the perpendicular bisector of two sites is the boundary between their Voronoi cells.
Example
Given $A = (1, 1)$ and $B = (5, 3)$: midpoint $= (3, 2)$, direction $B-A = (4, 2)$. Perpendicular bisector equation: $4(x-3) + 2(y-2) = 0$, or $4x + 2y = 16$, or $2x + y = 8$.
Key Insight
Perpendicular bisectors generate Voronoi diagrams: given n points, the Voronoi region of each point is the intersection of half-planes defined by perpendicular bisectors with all other points. Voronoi diagrams arise in computational geometry, spatial statistics, and modeling natural patterns like cell structures.