What is the difference between a segment, a ray, and a line?

A segment has two endpoints and a fixed length. A ray has one endpoint and extends infinitely in one direction. A line has no endpoints and extends infinitely in both directions.

Segment — Definition, Formula & Examples

A segment (or line segment) is the part of a line that lies between two endpoints, including those endpoints. Unlike a line, a segment has a definite length and does not extend forever.

Given two distinct points $A$ and $B$ on a line, the line segment $\overline{AB}$ is the set of all points on the line that are between $A$ and $B$ , together with the points $A$ and $B$ themselves. Its length, denoted $AB$ , is the distance between the two endpoints.

Key Formula

AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Where:

$A(x_1, y_1)$ = Coordinates of the first endpoint
$B(x_2, y_2)$ = Coordinates of the second endpoint
$AB$ = Length of the segment

How It Works

To work with a segment, you identify its two endpoints and measure or calculate the distance between them. On a coordinate plane, you can find the length of a segment using the distance formula. In diagrams, segments are drawn as straight marks with a point at each end, and their names use an overline, such as

\overline{PQ}

. When you write

PQ

without the overline, you are referring to the numerical length of the segment rather than the geometric object itself.

Worked Example

Problem: Find the length of segment

\overline{AB}

where

A = (1, 2)

and

B = (4, 6)

Step 1: Subtract the x-coordinates and the y-coordinates.

x_2 - x_1 = 4 - 1 = 3, \quad y_2 - y_1 = 6 - 2 = 4

Step 2: Square each difference and add them.

3^2 + 4^2 = 9 + 16 = 25

Step 3: Take the square root to find the length.

AB = \sqrt{25} = 5

Answer: The length of

\overline{AB}

is 5 units.

Another Example

Problem: Point

C

is between points

A

and

B

\overline{AB}

. If

AC = 7

and

CB = 5

, find

AB

Step 1: Because

C

is between

A

and

B

, the Segment Addition Postulate says the two smaller segments add up to the whole segment.

AB = AC + CB

Step 2: Substitute the known lengths.

AB = 7 + 5 = 12

Answer:

AB = 12

units.

Why It Matters

Segments are the building blocks of every polygon, triangle proof, and coordinate-geometry problem you encounter in middle-school and high-school geometry courses. Architects and engineers rely on precise segment measurements when designing structures. Mastering segment length calculations also prepares you for the distance formula and midpoint formula used throughout algebra and analytic geometry.

Common Mistakes

Mistake: Confusing the notation

\overline{AB}

(the segment) with

AB

(the length). Students sometimes write the overline when they mean a number, or omit it when they mean the geometric figure.

Correction: Use

\overline{AB}

when referring to the segment as a geometric object, and

AB

(no overline) when referring to its numerical length.

Mistake: Treating a segment like a line and assuming it extends beyond its endpoints.

Correction: A segment stops at its two endpoints. If a point does not lie between (or at) those endpoints, it is not on the segment.

Related Terms

Angle — Formed by two rays sharing an endpoint
Adjacent — Segments or angles that share a side
Between — Describes a point lying on a segment
Angle Bisector — A ray that splits an angle into equal parts
Arm of an Angle — Each ray forming an angle is an arm
Acute Angle — An angle measuring less than 90 degrees