Finding an Endpoint From a Midpoint

A midpoint is just an average of coordinates: M = ((x1+x2)/2, (y1+y2)/2). If you know one endpoint and the midpoint, that's two known numbers and one unknown in each coordinate, so there's exactly one value that makes the average work. That's why one endpoint plus the midpoint always pins down the other endpoint with no ambiguity.

To solve it, flip the averaging around. If A = (x1, y1) and M = (m, n) is the midpoint of AB, then m = (x1+x2)/2, so x2 = 2m - x1. Same move for y: y2 = 2n - y1. You're not really doing new algebra — you're just undoing the average by doubling the midpoint and subtracting the known endpoint. Many students try to re-derive the formula from scratch each time; it's faster to just remember 'double the midpoint, subtract the endpoint.'

This skill sits right next to distance and slope on the coordinate plane because all three come from the same two points. Once you can recover an endpoint, you can extend the idea: instead of going exactly double the distance (to find the reflection point B), you can go any multiple of the step from A to M, which sets up ratio and section-formula questions where a point divides a segment in some other proportion, like 1:2 or 2:1.