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## 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*.

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

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 and *are 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):

and slope formula (rise over run).

**Example question: **Are these two line segments equivalent?

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

Step 1: Apply the distance formula to both line segments (Note: the double lines **||** indicate length):

. Both segments are the same length (radic;13).

Step 2: Use the slope formula for 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 :

## 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.

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

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).

## References

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

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**Stephanie Glen**. "Line Segment: Definition, Endpoints, Examples, Equivalent" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/line-segment-equivalent/

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