 Once we have two points we might want to calculate the distance between so how can we do that? Well, let's have our two points So I have one point here with the coordinates x1 y1 and I have another point here with coordinates x2 y2 and I'd like to travel from here to there So what can I do? Well, the important thing to remember is that the coordinates tell us how to get to the point from the origin So what I might do is I might navigate back to my starting point and then navigate out to the new location Now this isn't particularly efficient But it is a good starting point because we know how to do that So let's take a look at this these coordinates here are starting point x1 y1 In order to get to this point I had to have gone out horizontally x1 and then up Vertically y1 I had to go out some distance up some distance. So if I want to go back to the origin I'm going to reverse those directions. I'm going to go down some distance back some distance so from that point I'll go ahead and make those moves and I'll record the total distances so I can figure out what I Did as a net Displacement so here I start at my point because I went up Distance y1 to get back down I'm going to go down distance y1 and I'll represent that by saying my distance is minus y1 Now I'm here since I got out Distance x1 to get to that point. I went out x1. I'm going to go back x1 and that'll take me right back to the origin and that is a horizontal distance Well, now I'm back at my starting point So now I want to get over to here and here I can just use the coordinates directly So the coordinates tell me that to get to this point. I need to go horizontally x2 and Vertically y2 so I'm at the origin. I'm going to go horizontally x2 and I'm going to add that to my horizontal distance So I've gone over x2 and so far I've traveled minus x1 Plus x2 and now I'm going to travel vertically Distance y2 to get there and so I'm going to add that vertical distance y2 to how far I've traveled So that's going to be minus y1 plus y2 Now remember the real question that we really wanted to know was how do I get from here to there? I'd rather not have to walk all the way back to the origin and then all the way back up to the point I'd like to go the direct route. So let's see what I can do here Well, the thing to notice here is that my horizontal distance minus x1 plus x2 Well, I can rearrange that to make it look a little bit neater. That's x2 minus x1 There's my horizontal distance and my vertical distance minus y1 plus y2 again I'll rearrange that to make it look a little neater. That's y2 minus y1 And so my vertical distance y2 minus y1 my horizontal distance x2 minus x1 and I can get to this point from this point by undergoing a Vertical shift of this far and a horizontal shift of that far Now what about the actual distance between the two one thing to notice here is that that distance that straight line Distance if I take into account this Vertical and horizontal distance here that vertical distance is a hypotenuse of a right triangle And what this means is we can then use the Pythagorean theorem And I know the two sides x2 minus x1 y2 minus y1 This is the hypotenuse of the right triangle so I can use a Pythagorean theorem and The square of the hypotenuse is the square of one side plus the square of the other side of the triangle And I want to do that difficult algebraic step of solving for D I'll take the square root both sides and this gives us a way of calculating the distance between two points So for example, let's say I have the point 3 5 and the point 2 negative 3 well paper is cheap So I'll go ahead and write down the distance formula Then I'll substitute in the values that I have So let's take a look at those values. I have x2. That's my second x-coordinate. That's two x1 That's my first x-coordinate. That's three Y2 that's my second y-coordinate. That's negative three. I'll substitute that in Y1 that's my first y-coordinate. That's five. I'll substitute that in and Now I have a bunch of computations to perform parentheses. They do the stuff inside first two minus three squared That's negative one squared minus three minus five squared negative eight squared The square root symbol is like a set of parentheses. It says do whatever is inside first So I have squaring I have adding I've got to do the squaring first minus one squared is one negative eight squared is 64 The square root symbol still a grouping symbol still says do what's inside first one plus 64 65 And now I'm going to take the square to 65 and since 65 is not a perfect square. I'll just leave it in that form