# Line segment explained

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).

## Definition

If

V

is a vector space over

R

or

C

, and

L

is a subset of

V,

then

L

is a line segment if

L

can be parameterized as

L=\{u+tv\midt\in[0,1]\}

for some vectors

u,v\inV

, in which case the vectors

u

and

u+v

are called the end points of

L.

Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a closed line segment as above, and an open line segment as a subset

L

that can be parametrized as

L=\{u+tv\midt\in(0,1)\}

for some vectors

u,v\inV

.

An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two points.

## Properties

V

is a topological vector space, then a closed line segment is a closed set in

V.

However, an open line segment is an open set in

V

if and only if

V

is one-dimensional.
• More generally than above, the concept of a line segment can be defined in an ordered geometry.

## In Proofs

In geometry, the segment addition postulate states that if B is between A and C, then segment AB + segment BC = segment AC.