Calculus How To

Line Segment: Definition, Endpoints, Examples, Equivalent

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Contents (Click to skip to that section):

  1. Line Segment Definition
  2. Equivalent Directed Line Segments
  3. Endpoints

1. Line Segment Definition

Simply put, a line segment is just a piece of a given straight line. Each end of the line is called an endpoint. The two endpoints are usually denoted by letters, like A B or C D.

line segment

Two endpoints A and B on a line segment.

While AB is a common choice for endpoint labels, you could theoretically name them with any variable you like (e.g. FG or PQ). The usual notation is to write out the two endpoint labels with a line above them. For example: AB.

Directed Line Segment

directed line segment

A directed line segment has, like the name suggests, a direction. The straight line is drawn with an arrow pointing in a definite direction. Many quantities, like acceleration, force, or velocity, involve a magnitude and a direction, so directed line segments, like lines (2) and (3) in the above image, are used to represent them. To put this another way, directed line segments are vectors (Kishan, 2007).

The initial point is where the line begins. In the above image, the initial points are point B (image 2) and point A (image 3). The terminal point is where the line ends: point A (image 2) and point B (image 3). The initial and terminal points are not interchangeable, so ab segmentand ba segmentare not the same.

2. Equivalent Directed Line Segments

Directed line segments are equivalent if they have the same length and direction. You can show that two segments are equivalent with the distance formula (an application of the Pythagorean theorem):
distance formula
and slope formula (rise over run).

Example question: Are these two line segments equivalent?

  1. A(0, 0) to B(3, 2)
  2. C(1, 2) to D(4, 4).

Step 1: Apply the distance formula to both line segments (Note: the double lines || indicate length):
distance formula step 1
. Both segments are the same length (radic;13).

Step 2: Use the slope formula for both segments:
slope of both segments
Both segments have the same slope.

The two line segments have the same length and slope, so they are equivalent.

3. Endpoints

An endpoint is a point at the boundary of one end of a closed interval, ray, or line segment. It’s literally the point where the interval, ray, or line ends.

Endpoint on a Ray and Line Segment

A ray only extends indefinitely in just one direction. A ray is denoted with an arrow above the two endpoints. The initial point is where the line is limited (i.e. ends abruptly) and the terminal point is where the ray continues indefinitely. In ray notation, an arrow is placed above the two endpoints; the initial point comes first.

For example, the following image shows the ray ray:


The ray here has an initial point A and terminal point B.

line segment

Two endpoints A and B on a line segment.

Use of Endpoints

Endpoints are primarily used to find Riemann sums. The right-hand rule uses right endpoints for the calculation; The left-hand rule uses left endpoints.

A left hand Riemann sum.

The term is also used define points where a function simply ends.

inverse sine function graph

Graph of cos x (red) and the inverse cosine function cos-1x (blue). The inverse cosine has two endpoints, at x = -1 and x =1.

An important note though, is that an endpoint in calculus isn’t usually a “point” in the usual sense of the word. It’s defined by a directional derivative or limit (i.e. values leading up to the endpoint, rather than the value at the endpoint itself).


Blank, B. & Krantz, S. (2006). Calculus: Multivariable, Volume 2. Key College Pub.
Kishan, H. (2007). Vector Algebra and Calculus. Atlantic Publishers & Distributors (P) Limited.
Stewart, J. (2015). Single Variable Calculus. Cengage Learning.
Introduction to Geometry: Rays and Angles

Stephanie Glen. "Line Segment: Definition, Endpoints, Examples, Equivalent" From Calculus for the rest of us!

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