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 Cand 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.
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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.