 One very important formula in geometry is the midpoint formula. Imagine we have two points in the plane call one point a call the other point B and then consider the line segment that connects together the points a and b The midpoint which we'll call it m is the point. That's exactly halfway between a and b So that is the line segments a m is congruent to bm That is the distance from a to m is identical from the distance from m to b that we talk about right here So the midpoint is exactly halfway between the two points now the formula for the midpoint It's quite natural if we have coordinates for a and b. Let's say a has the coordinates x1 y1 and let's suppose b has the coordinates x2 y2 Then consider the right triangle that's associated to these two points. We see something like this now It turns out that the x-coordinate of the midpoint will be exactly halfway between This this side of the triangle that is this side is equal to this side In particular, we see that the x-coordinate of m is then just going to be the average of the x-coordinate of a with the x-coordinate of b Which looks something like this x1 plus x2 all over 2 and Similarly the y-coordinate of the midpoint is just going to be halfway between The y-coordinate of a and the y-coordinate of b that is this side right here it cuts in half And these two segments are exactly the same Therefore we can find the y-coordinate by averaging together the y-coordinates of a and b That is the y-coordinate of the midpoint is y1 plus y2 all over 2 So imagine we have the following picture Let's have the point. Let's take the point p to be negative 5 comma 5 Let's take the point q to be 3 comma 1 and let's find the midpoint between the two You can see it's already labeled here on the screen, but how does one find out that coordinate? Algebraically well the midpoint m will have the average of the x-coordinates negative 5 plus 3 over 2 Because we're adding them it doesn't matter who's the first point Who's the second coordinates averaging the x-coordinate? We have to also average the y-coordinate in which case then simplify these fractions negative 5 plus 3 is negative 2 over 2 You get 5 plus 1 which is 6 over 2 and then lastly if you simplify these fractions You're gonna get negative 1 over 3 which we then see is the midpoint of the line segment