Distance Formula

Using what we know about the Pythagorean theorem, we are able to derive the distance formula which is used to find the straight distance between two points in a coordinate plane. The distance formula is a standard formula that allows us to plug a set of coordinates into the formula and easily calculate the distance between the two.

If you have two points, let's call them A and B, somewhere in a coordinate plane, and we call A X1 and Y1. That's the ordered pair A and we say B has ordered pair X2 and Y2. We can calculate the direct straight line distance between them. using what we know about the Pythagorean theorem.

You might say Mr. Mccall how are we going to use a Pythagorean on a line that's diagonal like that, you don't even have a triangle. Well, what I'm going to do, I'm going to draw in one leg of that triangle that's going to be parallel to my X axis. And we're going to draw in another leg of that triangle which is parallel to the Y axis.

I know the X axis and Y axis are perpendicular to each other which means that this must be a right triangle. If we want to find out the distance between A and B, first we need to say, well, what are the lengths of my legs. The reason why that's important is because we're going to use A squared plus B squared equals C squared.

So A is going to be one of my legs. And let's call it the leg that's parallel to the X axis. Well, this point right here is going to be the point not X 1, but it's going to be X2 and Y1. Because notice the only thing that's changed from A to this corner is my value of X. If these two lines are parallel, then Y1 will stay the same. So if I want to find the distance between these two, all I need to do is subtract my Xs.

So this distance is X2 minus X1. That difference will tell me how far away those points are. So I'm going to say that A is X2 minus X1. If I find B, B is going to be the other leg of this triangle. So just like I said that this horizontal distance was the difference of our axis, the vertical distance will be the vertical distance of our Yes. So this will be Y2 minus Y1. So B is going to equal Y2 minus Y1. And the hypotenuse C we could say is D, our distance. Or I guess if you want to, you could say that this is line segment AB. Either way, you're trying to find your hypotenuse here.

So let's substitute in what we know. Well, we said -- if I use a different marker -- we said we were going to use the Pythagorean theorem, and A is X2 minus X1. I'm going to say we're going to have X2 minus X1 squared. So all I'm doing is substituting in here. B we said was Y2 minus Y1, starting to add Y2 minus Y1 squared. And C we said is our distance, AB. And that's going to be squared. So if you want to know the square of the distance, in your coordinate plane, you're going to subtract your Xs square them. Subtract your Y square them and add them up.

Well, that's not quite useful. So we're going to take the square root of both sides, because the square of a distance doesn't help me that much. So I'm going to say that the square root of X2 minus X1 squared plus Y2 minus Y1 squared is equal to this distance AB. And, voila, we have our distance formula.

So the distance between any two points in space is going to be the difference of your Xs squared plus the difference of your Ys squared. Now, some of you might be thinking, Mr. McCall, I know that the square root of something squared is whatever that base term is. Now, you cannot say that either of these squares are going to come out.

The reason is we have this expression by this plus sign. So if this whole thing was being squared, then, yes, something could come out of this square root. But since we have this plus sign it's going to stay the way this is.

So the keys to using this formula are subtracting your Xs, subtracting your Ys, squaring those and then taking the square root. We got this formula by using the Pythagorean theorem.