Line Segment
GeometryA line segment is a part of a line with two definite endpoints, having a measurable length.
Formula
\text{Length} = \text{distance between endpoints}
Definition
A line segment is a straight path between two endpoints. Unlike a line, it does not go on forever - it starts at one point and stops at another. We can measure its length.
Example
The side of a triangle is a line segment. The edge of your notebook, a piece of string pulled tight between two thumbtacks, or the distance from your house to school can all be modeled as line segments.
Key Insight
The key difference between a line and a line segment is the endpoints. A line goes forever; a segment has a definite start and a definite stop - and that is why you can measure it.
Definition
A line segment is the set of all points between two endpoints, including the endpoints themselves. Segment $AB$ (written with a bar above $AB$) consists of points $A$ and $B$ and all points between them. Its length, denoted $AB$ or $|AB|$, is the distance between the two endpoints.
Example
If $A = (1, 1)$ and $B = (4, 5)$, then $|AB| = \sqrt{(4-1)^2 + (5-1)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. The midpoint of $AB$ is $((1+4)/2, (1+5)/2) = (2.5, 3)$.
Key Insight
Congruent segments have equal lengths (written $AB = CD$). The midpoint divides a segment into two congruent halves. These ideas are foundational to triangle congruence and similarity proofs.
Definition
A line segment in $\mathbb{R}^n$ is the convex hull of two distinct points: segment $AB = \{A + t(B - A) : t \in [0, 1]\}$. Its length is the Euclidean norm $|B - A|$. In metric geometry, segments are geodesics of finite length between two points.
Example
The parametric form $A + t(B - A)$ for $t \in [0,1]$ traces the segment from $A$ ($t=0$) to $B$ ($t=1$). Setting $t \in (-\infty, +\infty)$ gives the full line through $A$ and $B$. This parametric view connects segments to vector algebra and affine geometry.
Key Insight
The segment is the simplest example of a geodesic - the shortest path between two points. In non-Euclidean spaces (e.g., on a sphere), geodesics are arcs of great circles, not straight segments, revealing how curvature alters the nature of "shortest path."