# End Point: Definition & Formula

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• 0:00 What Is an Endpoint?
• 1:45 The Midpoint Formula
• 3:17 Endpoint Formula
• 4:57 Examples
• 6:08 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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

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