 In this video, I'm going to talk about the distance formula. I'm going to do just one example, and I have to do the distance formula a couple of times on this example. I'm just going to do one example of the distance formula, how to use it, and one thing that I can actually be used for. All right, so as you can see, one of the things that we have here is we have a Cartesian coordinate system. We have an XY coordinate system. We have a couple of segments on here, JK and FG. What we're going to do is we're going to find FG and JK. Now notice there's no lines or anything out here. Find the length FG and find the length JK, that's what those two mean. Find those two lengths, and then determine whether the segments FG is congruent to the segment JK. So what we want to do is we want to find these two segments, how long are they, and are they congruent? Okay, so a couple of different questions here. All right, so first thing that we need to do is we need to know what the distance formula is. So I'm going to write down the distance formula up here. Distance, usually determined by little d, is equal to big square root sign. What you want to do is you want to take your x1 minus x2 in parentheses, square it, plus y1 plus y2 in parentheses, square it. Okay, now what these y1s and y2s mean is it simply means our first x coordinates and first x, second x, or first y, second y. Because what we're doing is we're finding the distance between two points. So we have to have a first point and a second point, and that's a distance formula. Oh, I made a little bit of a mistake here. Let's go back real quick. It is not plus, it is subtract for both of them, my mistake. Sorry about that. All right, so take the x's, subtract them, square them, take the y's, subtract them, square them, and add those two results together, and then take the square root of that. Okay, so find fg, find the length of fg, and find the length jk, okay, then determine whether they can grow. We'll worry about that later. Okay, now one thing I'm going to do here is that on this grid, I don't have the coordinates here. So I'm actually going to write out those coordinates real quick. So this j here is a negative 4, 0, that's what the coordinates for that one is. This k down here is a negative 1, negative 3, negative 1, negative 3, this f over here is 1, 2, 1, 2, and then this g over here is 5, 5, 5, this g is a 5, 5. All right, so what we're going to do is we're going to find the distance of these two segments. Let's just start with fg, okay, let's start with fg, since that's the one we want to start with. Okay, so I'm going to use my distance formula. So what I'm going to do, now when you're using distance formula, I know it's intimidating because it's a very large formula, but one thing you can do to make this go a little bit faster is kind of write everything out to begin with, but don't write these x's and don't write the y's, okay? Write everything else, the symbol, the parentheses, the squares, write all that stuff out first, but then don't write the x's and the y's, this actually makes things go a little bit faster. So here's my parentheses, my parentheses, okay, and then I'm going to subtract these numbers, okay? Write that out first. Now all I have to do is plug in some of the numbers. So f is going to be my first point, g is going to be my second point. So as I come over here, I'm going to look at my first point and second point, okay? So what I'm going to do is I'm going to take the x's and subtract them. So I've got to go over here and find the x's. So here's an x of one, plug it in, all right? Here over here for my second point is an x of five, plug it in, okay? No sign changes or anything like that, so it's just one minus five. Pretty simple. All right, so now for the second one over here, this is for the y's. So now I've got to go over to my points. I have a two for my first y and I have a five for my second y. Okay, again no sign changes or anything like that, that's kind of handy. Now again, I think this is an easier way to do this because now, now you just subtract the number, square them, and you've got everything already set up. I think it's a lot easier to put in the numbers this way. But anyway, you might have a different way that's easier for you. I would go with whatever's easier for you. This is easy for me, so this is what I do. All right, one minus five is negative four, quantity squared, plus negative three, quantity squared. I use quantity to kind of close up parentheses. Anyway, negative four squared is sixteen, negative three squared is nine. So when you add those together, you get, let's go off to the right here. The square root of twenty-five, which is five. So when you know fg is equal to five, I'm going to write that over here, fg is equal to five. Okay, that's one thing that we do know. Great. Okay, so that's using the distance formula. What I suggest to do is write out the square root symbol, write out the parentheses and everything first, and then fill in your numbers after that. So I'm going to do that for the second one. I'm going to go a little bit faster through the second one since you just saw me do the first one. So now this one, what is the second one? jk. So I got to do the second one of jk. Let me actually, let me move that down just a little bit. That's what's nice about working on a computer is that you can move your text that you draw down a little bit, which is nice. Okay, so jk, I'm going to write my big square root symbol, put my parentheses squared plus a parentheses squared. All right, now I'm just going to put in my x's here and my y's here. So let's go over to my point. I have an x of negative four and an x of negative one. Negative four minus a negative one that actually turns into plus one. Negative four minus a negative one that turns into negative four plus one. So there is a sign change for that one. Make sure you keep track of those sign changes. That can be very tricky. There were no sign changes for this first one, this one, there's probably going to be sign changes. Now notice where our points are. These points were both positive, so we didn't have any sign changes. These points here, some of those numbers are negative, so there's probably going to be some sign changes. Just be aware of that. All right, so now I've got to put in the y's, so I have a y of zero, pretty easy, zero minus a negative three. So there's a sign change, zero minus negative three becomes a positive three, a little sign change there. Okay, now what I need to do is add these together, so this gets negative three, bad three, negative three squared plus three squared, squared symbol over that, and that's nine plus nine, which gives me the square root of eighteen. Now one of the things I'm not going to do is I'm not going to actually plug that into a calculator. I don't need to. Look at this right here, jk is equal to the square root of eighteen. These do not match up. Why am I noting that they don't match up? Look at the second part of our question. Determine whether f, the segment fg is congruent to the segment jk. These, since they're not the same length, since they're not the same length, okay, they're not going to be congruent. That was basically what we're trying to get in the first part. We'll have a little bit of work, a couple of examples with the distance formula, but we also want to figure out if they're the same length, and they're not going to be the same length, okay? So that's one thing to note as you're going through the problem, make sure you keep track of everything that you're supposed to find. Try to keep track of everything you're supposed to find. One other thing that I want to note here before I'm done with this problem is that this person right here, the square root of eighteen, we can actually simplify that. The square root of eighteen, I can split that up into the square root of nine, and the square root of two, okay, because nine times two gives me back to eighteen. Now the reason I do this is because I actually know what the square root of nine is, that's going to be three. That's going to be three. Square root two, I don't know what that is. I'm just going to leave it as a square root of two. So a lot of times when you see these radicals, when you see these square roots, a lot of times you'll see them simplified like this. Instead of the square root of eighteen, eighteen's kind of a big number, we can simplify it to three root two. And so this is still the square root of eighteen, but it's just been simplified a little bit. But still, this does not match with what we had up here. So another part of my answer is segment fg is not congruent to segment jk, run out of room here on the bottom. That's also part of my answer, okay? And you know this because these numbers don't match up. They're not the same length. All right, all right, hopefully that example helps you with the distance formula. Hopefully it gave you a couple of strategies to use to make the distance formula easier to use. Don't be intimidated by the length of the formula or by all these x's and y's. It's pretty simple once you use it a few times. Again, don't be intimidated by any of these. They're all pretty simple, at least at this level. Anyway, if you have any questions or anything like that, make sure you get those answered by either me or your teacher. This can be a difficult formula to use if you haven't used it very much. So anyway, hopefully those strategies will help you for the distance formula.