This lesson will familiarize you with endpoints, how to identify them, and how to find them algebraically. We’ll go over what endpoints look like on a graph and what information we need to identify endpoints of a line segment.

## What Is an Endpoint?

Before getting to the definition of an endpoint, let’s first learn what a line segment is and what a ray is. In mathematics, a **line segment** is just what the name sounds like – a segment of a line. More formally, a line segment is a line that connects two points and does not extend past either of the points. A **ray** is a line that starts at a point and extends forever in one direction.

Anywhere you see a line in the environment around you, if you consider just a piece of that line between two distinct points, then you have a line segment, and if you consider a line starting at one point and then continuing on forever in one direction, then you have a ray. For example, this image shows lines on a field, with points *A*, *B*, and *C* added in.

If we only consider the line between points *A* and *B*, and nothing extending past them, then we have the line segment *AB*. If we consider the line starting at *C* and going on forever in one direction (indicated by the arrow), then we have a ray.

**Endpoints** are the points on either end of a line segment or on one end of a ray. In a line segment, the line does not extend past either of its endpoints that it connects. Similarly, in a ray, a line has one endpoint, and the line goes in one direction away from that point and does not extend past that endpoint in the other direction. Therefore, we can think of endpoints as a point where a line ends (or stops). Thus, line segment *AB* in the image has endpoints *A* and *B*, and the ray has the endpoint *C*.

## The Midpoint Formula

On every line segment, there is a point that lies halfway between the endpoints. This point is called the **midpoint**, and it lies on the line segment equal distance from each of the endpoints. In simpler terms, the midpoint lies in the middle of the line segment. The graph shows a line segment and its midpoint.

The midpoint *M* has coordinates (5, 3), and lies halfway between *A* and *B*. In general, when we have the endpoints of a line segment (*x1*, *y1*) and (*x2*, *y2*), we can find the coordinates of the midpoint by finding the average of each of the coordinates. The x coordinate of the midpoint is found by adding the two x coordinates, *x1* and *x2*, and dividing them by 2. Similarly, the y coordinate of the midpoint is found by adding the two y coordinates, *y1* and *y2*, and dividing by 2. This gives us the midpoint formula.

In our graph of line segment AB with midpoint M, our endpoints are given as (2, 2) and (8, 4), so we have *x1* = 2, *x2* = 8, *y1* = 2, and *y2* = 4. We plug these values into our midpoint formula to get:

*M* = ((2 + 8) / 2, (2 + 4) / 2) = (10 / 2, 6 / 2) = (5, 3)

Thus, our midpoint is (5, 3) as shown in the graph.

## Endpoint Formula

When we are given one endpoint and the midpoint of a line segment, we can determine the other endpoint using the endpoint formula. The endpoint formula is derived from the midpoint formula.

If the midpoint of a line segment is (*m1*,*m2*) and the endpoints are (*x1,* *y1*) and (*x2,* *y2*), then the midpoint formula is:

*M1* = (*x1* + *x2*) / 2

*M2* = (*y1* + *y2*) / 2

Solving each of these for *x2* and *y2*:

*x2* = 2(*m1*) – *x1*

*y2* = 2*(m2*) – *y1*

(*x2*,*y2*) = (2(*m1*) – *x1*, 2(*m2*) – *y1*)

We see that if we are given one endpoint and the midpoint of a line segment, then we can use the endpoint formula to find the other endpoint . Let’s look at our graph example again.

Suppose we are given the endpoint *A*, which has coordinates (2, 2), and the midpoint *M*, which has coordinates (5, 3), and we want to find the other endpoint of our line segment.

*x1* = 2

*y1* = 2

*m1* = 5

*m2* = 3

We plug these values into our endpoint formula to get the following:

(*x2*, *y2*) = (2*5 – 2, 2*3 – 2) = (10 – 2, 6 – 2) = (8, 4)

Thus, our other endpoint is (8, 4) as shown in the graph.

## Examples

Let’s take a look at a couple example problems using what we just learned.

1) In the graph, identify any line segments, rays, endpoints, and midpoints.

We have line segments *AB*, *AC*, and *BC*. There is a ray that starts at point *D*. All of the points can be endpoints, because *A* and *B* are endpoints of line segment *AB*, *C* is an endpoint of line segment *AC*, and *D* is the endpoint of the ray. Point *B* is the midpoint of line segment *AC*.

2) If a line segment has one endpoint (1, 5) and midpoint (-2, 7), find the other endpoint.

We see that

*x1* = 1

*y1* = 5

*m1* = -2

*m2* = 7

Plugging these values into the endpoint formula’

(*x2*, *y2*) = (2*m1* – *x1*, 2*m2* – *y1*) = (2*-2 – 1, 2*7 – 5) = (-5, 9)

Thus, our other endpoint is (-5, 9).

## Lesson Summary

A **line segment** is just what the name sounds like – a segment of a line. A **ray** is a line that starts at a point and extends forever in one direction. **Endpoints** are the points on either end of a line segment or on one end of a ray. The **midpoint** of a line segment is the point that lies halfway between the endpoints.

The formula for determining the midpoint of a line segment is:

(*m1*, *m2*) = ((*x1* + *x2*) / 2, (*y1* + *y2*) / 2)

The formula for determining an endpoint when you know the midpoint is:

(*x2*, *y2*) = (2(*m1*) – *x1*, 2(*m2*) – *y1*)

## Learning Outcomes

This video should help you to:

- Explain what a line segment is
- Identify the endpoint of a ray or line segment
- Demonstrate how to use the midpoint formula
- Understand how to use the endpoint formula