Clowns to the left of me, jokers to the right. Here I am, stuck at the midpoint with you…
At some point they’re going to give you a line segment on the coordinate plane and ask you to find the midpoint. You could just memorize the formula. Which would be fine, except you’ve got eight million other formulas memorized and when test time comes they’re all jumbled up in your noggin and you pick the wrong one. If you understand where the midpoint formula comes from though, you don’t need to memorize anything because you’ll know it. So here’s the deal.
Let’s say your line segment has endpoints A(x1,y1) and B(x2,y2) and you have to find the midpoint M of line segment AB. And let’s say you make a right triangle with AB as the hypotenuse and one of legs is parallel to the x axis and the other one to the y axis, like this:
Can you figure out what the coordinates of point C should be? It’s a little tricky, I’ll walk you through it. C lies on the same vertical line as point A, so it has the same x-coordinate as point A: x1. C is on the same horizontal line as point B, so C has to have the same y-coordinate that B does: y2. Cool, so now we know point C has coordinates (x1,y2). Right on, right on.
What’s the x coordinate of the point halfway between B and C? It’s halfway between x1 and x2, so it’s x1 + x2 divided by 2. And what’s the y coordinate of the point halfway between A and C? Same story, it’s halfway between y1 and y2, so the y coordinate of the point halfway between A and C is y1 + y2 divided by 2. Now let’s draw a horizontal and a vertical line connecting these two points to AC… and where those lines intersect… that’s the midpoint!
So to find the midpoint of a line segment, you pretty much take the average of your points’ x-coordinates to find the x-coordinate of the midpoint, then do the same thing for the y-coordinate. Now go forth, and find the midpoint like a boss!
If you need more help with the midpoint formula, be sure to check out our study videos on the topic and much more!