To learn the whole story of a line, you sometimes need to crunch some numbers.
To learn the whole story of a line, you sometimes need to crunch some numbers.

The line is one of the simplest mathematical objects. Even so, finding all the information about a single line sometimes requires some number-crunching. When you know partial information about your line, you will often have to plug your known information into formulas to describe the rest of the line. The endpoint formula is one such formula, and it can help you find the unknown endpoint of an otherwise well-understood line.

The Problem: Where’s the Rest of My Line?

Often in math, you are left with pieces of information with which you must piece together the entire situation. This often occurs in geometry, a subject in which you must infer conclusions about lines and other shapes based on partial information about them. In some cases, you might be looking at information describing a line segment, but are missing the endpoint of the line. A line segment is defined by its two endpoints, the knowledge of which allows you to fully describe the line segment in mathematics. Thus, you need to know both endpoints before you can clearly label or define a line segment.

The Midpoint Formula: Half of the Story

Knowing the midpoint formula for a line segment will often help you derive the endpoint. Recall the midpoint formula being the mathematical formula that allows you to find the point lying in the center of a line segment, based on the two endpoints. Specifically, if your endpoints are (x0, y0) and (x2, y2), the midpoint will have an x-coordinate at (x0+x2)/2 and a y-coordinate at (y0+y2)/2. By plugging in the endpoints into the midpoint formula, you can find the midpoint of the segment. But what many students don’t notice is that you can reverse the midpoint equation to find the endpoint.

The Endpoint Formula: The Rest of the Story

If you have one endpoint and the segment’s midpoint, you can apply a modification of the midpoint formula to derive the other endpoint. The midpoint formula lets you find a coordinate by plugging the information you know into p1 = (p0 + p2)/2. If you don’t know one of the endpoints, either p0 or p2, but do know the midpoint, p1, you can solve this equation for the other endpoint. Using algebra, you can rewrite the midpoint formula as 2_p1-p0. Thus, your endpoint has an x-coordinate at 2_x1-x0 and a y-coordinate at 2*y1-y0, where (x1,y1) is the midpoint of the line segment and (x0,y0) is the endpoint of the segment.

A Quick Example

Assume you have a line segment with an endpoint at (-2,7) and a midpoint at (-9,2). You want to find the other endpoint through the endpoint formula, so you first label your variables: x0=-2, y0=7, x1=-9, and y1=2. Find the x-coordinate first: x2=2_x1-x0=2_(-9)-(-2)=-16. Find the y-coordinate next: y2=2_y1-y0=2_2-7=-3. Thus, your other endpoint is at (-16,-3).