Overview

In elementary geometry, a line segment is the straight portion of a line bounded by two distinct endpoints. It contains every point on the straight path between those endpoints and does not extend beyond them. A common notation places a short horizontal bar over the two endpoint labels (for example, AB with a bar above it). The figure below illustrates a typical segment and its endpoints:

Basic characteristics

A line segment has several simple but important features: it has two endpoints, a finite length, and an interior made up of all points between the endpoints. The length of a segment is the distance between its endpoints and is a nonnegative real number. Segments are the shortest path connecting two points in Euclidean space and are straight, meaning they lie entirely on a single line.

Common properties and notation

  • Endpoints: named by their letters, e.g., A and B for segment AB.
  • Midpoint: the point equidistant from both endpoints; it divides the segment into two equal subsegments.
  • Directed segment (vector): when an orientation is given from one endpoint to the other, the segment represents a vector.
  • Concatenation: adjacent segments sharing an endpoint can be joined to form a longer segment if they are collinear.

Types and examples

Segments appear in many basic shapes and constructions. A side of a polygon or of a triangle is a line segment. When both endpoints lie on a curve (for example, a curve such as a circle), the segment is called a chord; a diameter is a chord passing through the center of a circle. In coordinate geometry a segment between points (x1,y1) and (x2,y2) can be described algebraically and its length computed with the distance formula.

Historical and mathematical context

The idea of a line segment dates back to the origins of geometry. Classical treatments, including Euclidean geometry, consider the segment a primitive or fundamental object used to build more complex figures. With the development of analytic and vector geometry, segments also gained algebraic descriptions and became central to constructions, proofs, and computations.

Uses, applications, and distinctions

Line segments are foundational in construction, design, computer graphics, surveying, and computational geometry (for example, algorithms that check segment intersection). It is important to distinguish a segment from a line (which extends infinitely in both directions) and a ray (which extends infinitely in one direction). In many problems the closed nature of a segment (including its endpoints) matters; in others, only its interior is considered. Understanding segments and their properties is essential for measuring distances, defining shapes, and formal geometric reasoning.