>> We can actually develop a formula for calculating the distance between any two points. And this formula then would be called the distance formula. I can just give it to you here, but you might remember it a little bit better if we show how this formula is developed. I won't take a great deal of time with this but here's the gist of the idea. Suppose we're talking about the point 2, 3 and the point 10, 9. Alright. So these are the coordinates of those two points. We want the distance between the two points, the idea is this. I'd like to calculate this distance and the way that the distance formula is developed, it actually comes from the Pythagorean relationship which is a relationship between the sides of right triangles. And if we drop a vertical segment down here and then a horizontal segment across to that one then we have a right triangle formed and in this right triangle configuration, if we call this side A, and this side B, and this side C, then we know that by the Pythagorean relationship C squared is equal to A squared plus B squared. So we have C squared is A squared plus B squared. Now, we're gonna develop a distance formula, this distance along--between the two points, so this distance C is what we're after here. So I wanna solve this for C. Taking the square root on both sides, I find C to be the square root of A squared plus B squared. Now, let's talk about the distances A and B. Now this distance A is a horizontal distance. The coordinates of this point 10, 3. Now, horizontal distance, distance along the number line you may recall from before is calculated as the absolute value of the difference in values--the real number values along the horizontal number line. Now, here we're talking about horizontal distances so the components for horizontal distances are the 2 and the 10. So, this distance A is the absolute value of 2 minus 10 or 10 minus 2 whichever. But, at any rate that A distance is this 2 minus 10 and it's being squared here. In a similar way the distance from this point to this point, these are vertical distances now, so the second component comes in to play here in this coordinate. So, it's the absolute value of 3 minus 9 for that distance. So, absolute value of 3 minus 9, or 9 minus 3 if you like. And that value is squared. Now, absolute value was put into distance idea when we were talking about distance on the number line because we wanted to make sure that the value was always positive. Well, it doesn't really matter in this context whether the value is positive or negative because we're squaring and whether we square a positive number or a negative number, we get a positive result. So the absolute value symbols are unnecessary here. So, let's take them off. We come to this situation. Now, I'm calling this the distance formula so we'll label the C value D. We'll just change the labeling on it, D for distance you see between any two points. So, this is the general idea. Now here's what I want you to notice. You see the 2 and the 10 here, the 3 and the 9 here. Now, those are actually, those came from this configuration of the right triangle. But, the 2 here and the 10 here, gee, I used this 10 and that 2 but I didn't need to use this 10, I could use that 10. You see I can just use the horizontal components of the two given points. And in a similar way I used the 3 here and the 9. Well, instead of using this 3, I could have used that one. See, I have two vertical components here given those two original points. So, I don't really need this--the coordinates of this point in order to use this distance idea. I can just use the two points given. Now, to generalize a situation whether I was using the points 2, 3, 10, 9 or simply X, Y or X1, Y1, X2, Y2. You see, I could general points to make this kind of development or to develop this distance formula. And the distance formula then emerges like this. This is just a difference in X-components or a distance in horizontal distances and this is used a difference in Y components or a difference in vertical distances. Now, the subscripted numbers here, the subscript numbers just indicate that these are different X-components. And these just say that these are different Y-components and that's all and because this is a 2 and that's a 2, this X-Y pair, they're related to one another and the ones indicate that those rascals are related to one another as well. But, that's really all there is to it. Now, to make the calculation we would presumably go from the distance formula, plug the information into a step that looks like this and perform the indicated operations to find that distance to be, I think in this case, it turns out to be 10. ==== Transcribed by Automatic Sync Technologies ====