**Contents** (Click to skip to that section):

- Line Segment Definition
- Directed Line Segment Definition
- Equivalent Directed Line Segments
- Endpoints
- Problem Solving: Length of a Line Segment

## 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).

## Problem Solving: Length of a Line Segment & The Distance Formula

The length of a line segment between two points is calculated with the **distance formula**:

This formula only works for linear (straight line) functions. If you have a **curved function**, use the arc length formula instead.

## Length of a Line Segment: Example

**Example question:** What is the length of the line segment for f(x) = 5x + 2 on the interval [0,1]?

Step 1: (Optional) **Draw a Graph of the function**. This step can help you see where the points you need for the formula are (x_{1}, x_{2}, y_{1}, y_{2}). I used Desmos.com to create this graph:

A quick look at the line segment between these two points and I’m going to guess that the length should be around 5. That gives me a way to check that my final answer is reasonable.

Step 2: Plug your two (x, y) coordinates into the distance formula.

If you’re having trouble deciding which point goes where, the following graph has the x and y points labeled.

Step 3: **Simplify and solve**:

- (1 – 0)
^{2}= 1 - (7 – 2)
^{2}= 25 - 1 + 25 = 26
- √(26) ≈ 5.1

So my guess at 5 was fairly close.

## Length of a Line Segment & Pythagorean Theorem

You might notice that the distance formula is really an application of the Pythagorean theorem. And in fact, you *could* get to the same answer by using the Pythagorean formula a^{2} + b^{2} = c^{2}.

The following image shows that the hypotenuse of the triangle lies on the line segment you’re trying to find the length of:

You may be wondering why we use the distance formula to find the length of a line segment instead of the more simple Pythagorean formula. The answer is that while a, b, and c work perfectly well in geometry, it doesn’t translate well to the Euclidean plane of x’s and y’s, especially once you get to variations on the distance formula for curves. You can’t fit a triangle to a curve, but you can fit a series of small lines. This process of fitting small lines to approximate a curve uses integration, and (along with derivatives) it’s one of the two fundamental areas of calculus.

## 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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