 In this video, I'm going to be talking about the Pythagorean theorem and how we can use the Pythagorean theorem to help us find the distance between two points. Usually, you would use the distance formula to do this, but in this case, we're actually going to use Pythagorean theorem. Now, first thing I'm going to do is I'm going to write down the Pythagorean theorem. A squared plus B squared is equal to C squared. This is one theorem that you need to need to memorize. You will use it throughout your high school career in every single math class. It's just one of those formulas that you're always going to use. A squared plus B squared is equal to C squared. This pertains to a right triangle. If you have side A, side B, and side C, A is a leg, B is a leg, C is the hypotenuse. This formula will always hold true as long as it's a right triangle. Make sure that you know what the Pythagorean theorem is. Now on to a couple of examples. I'll give myself a little bit more space down here, move down, actually get rid of the Pythagorean theorem. That's fantastic. I'll write it again. All right. Use a Pythagorean theorem to find the distance to the nearest tenth from R to S. Notice in the nearest tenth, we're going to be doing some rounding. We have these two points. Let's do this problem over here first. We have these two points. We're going to put them on the grid and find the distance between the two of them. Pythagorean theorem to find this distance instead of using the distance formula. This is for those who don't really like the distance formula, this is something else that you can use. So R is my first point, is 3, 2, so 1, 2, 3, 1, 2, so that is R which is 3, 2, and then I have my second point of S which is negative 3, negative 1, 1, 2, 3, negative 1. That is S, negative 3, negative 1. All right. Connect the dots. Now, what I'm going to do is I'm going to find the distance between those two. Now, again, instead of using distance formula, I'm going to treat this as if it is a right triangle. Remember your right triangle. We've got side A, side B, side C of a right triangle. Pythagorean theorem is A squared plus B squared is equal to C squared. All right. Now, looking at this, it doesn't look like a triangle. What do you mean make a triangle out of this? Well, treat this diagonal just like the diagonal of the right triangle. So what I'm going to do, let's use a different color here to denote these other two sides. Right here is going to be my base, and right here is going to be my height. So those are the other two sides of my right triangle that I'm going to use. All right. Now, what I need to do, though, is if I'm going to use the Pythagorean theorem, I need to find those two sides. Now, with this diagonal, I can't count the distance from point to point. It just doesn't work on a grid. But if I go horizontal or if I go vertical, I can actually count the distance of these lines. So I can know what they are. So in this case, 1, 2, 3, 4, 5, 6, so what's great about a grid is you can actually count those distances. That's a distance of 6. This one over here is 1, 2, 3, that is a distance of 3. So what that does is that tells me what my B and what my A is going to be for my Pythagorean theorem. And I can use this side. I don't know what that side is. So I can use that to help me find C squared. All right. So now what I'm going to do is I'm going to write that down. There we go. A squared plus B squared is equal to C squared. So a little bit of color coding here. Let's do the same as what I did up here. This vertical one is A. This horizontal one is B. So in this case, we have 3 squared plus, change colors here, 3 plus 6 squared is equal to C squared, just kind of showing you with a little bit of color where those numbers came from. All right. So again, the base is 6. The height is 3. We're going to use that to use the Pythagorean theorem because this is the right triangle. We're going to use that to find the third side, which is actually the distance from S to R. So this is just an alternate way of finding the distance between two points. So you don't have to use the distance formulae every time. All right. So 3 squared is 9 plus 6 squared, which is 36, which is equal to C squared. 9 plus 36 is 45, which is equal to C squared. Now, all that right there, we just do substitution, and then with a little bit of addition, this is just basic algebra stuff, but now it's a little bit difficult. You might have not have seen this for a while. C squared is equal to 45. I don't want to know what C squared is. I want to know what C is equal to, all right? So I want to get rid of that squared. How do I get rid of squared? Well, I'm going to square root both sides. I'm going to square root both sides. The square and square root are going to cancel, and so all I'm left with is C, and now I just need to take the square root of 45. Now one thing at the beginning of the problem is they told you to round to the nearest tenth. So when you take the square root of 45, plug that into a handy-dandy calculator, you get 6.708, which just rounds to 6.7. Now remember, this is not an answer. This right here, that's not an answer. This helps you to get the answer. What we're looking for is the distance from R to S. So if you want to write your answer, R S is equal to 6.7. Doesn't tell me inches yards feet, so I'm just going to put units. And there we are, rounding to the nearest tenth to help us find the distance between the two. Okay? Square root of 45 is 6.7. So that's how you use the Pythagorean Theorem to find the distance between two points. And again, this is an alternative method. You don't have to use distance formula every time you can use Pythagorean Theorem. And again, to reiterate Pythagorean Theorem, this is one formula that you just need to memorize. It's very useful in geometry. Your algebra, algebra 2's trigonometry, you're going to use it all throughout high school. So it's just one of those ones that you just need to memorize. Okay. All right. So what I'm going to do, let's get rid of that, give myself a little bit more room. Okay. So I'm going to do the same thing over here. I'm going to go a little faster through this second part just to get through just a little bit quicker. All right. So what I'm going to do is I'm going to put these points up here. So negative 45, so negative 45 is right there. There's R. Okay. Now I'm not going to write the points here because I didn't even use them last time. So I'm not going to bother to write them this time. So R is negative 45, yep, negative 45. This is 2 negative 1, so 1, 2, negative 1, and that is where S is. Connect the dots, and now what I'm going to do is I'm going to figure out the base of my triangle, and I'm going to figure out the height of my triangle. Okay. I'm going to count the height, 1, 2, 3, 4, 5, 6, and then I'm also going to figure out the base. 1, 2, 3, 4, 5, 6. Interesting. 6 and 6. Okay. So let's see what the, let's see what this, the, now we're going to find the hypotenuse of this triangle. And again, we're going to use the Pythagorean theorem to do that. Okay. I'm not going to color code it this time. I'm just going to solve it. So A squared plus B squared is equal to C squared. A and B are the legs. So that just means we have 6s in here, A squared, or C squared plus 6 squared is equal to C squared. This means 36 plus 36 is equal to C squared. True 36 is makes 6d72 is equal to C squared. Okay. Now, I actually showed, I showed over here the step of, of taking the square root of both sides. Okay. I showed that step over here. I'm not going to show it. I'm just going to simply say, okay, to get rid of this, I need to square root both sides. Just saves a little bit of time. So that is equal to C. And now in your handy dandy, all you do is take the square root of 72, which gives you 8.485 if you're following along, which rounds to 8.5. Okay. So I'm just, I'm looking at my calculator and I'm rounding to the nearest tenth because that's what they asked me to do in the directions. Okay. But again, that's not my answer. What they're asking for is find the distance from R to S. So I need to write that down, RS is equal to 8.5 units, 8.5 units. And again, I'm not, I don't know if we're talking about meters or centimeters or kilometers or whatever the deal is. So I just write units. So that's, I know what I'm using and that's a, that's a good tactic to use. It's a good strategy to use because then you're always looking for a label to put on all your numbers. So that's in case there's a label there. Anyway, those are two examples of using the Pythagorean theorem to find the distance between two points. I actually like this a lot better than distance formula because it uses geometric figures. I think the numbers are easier to work with and the algebra here and the algebra here can be, can be a lot easier to work with depending on if you like algebra or not. So I guess to each of their own, you can either use distance formula or Pythagorean theorem to, to, to solve this. But anyway, I think that's about it. And again, just to reiterate, make sure that you know the Pythagorean theorem. Just memorize this. It's very handy to know in every math class that you're going to take in high school. All right. So that is using the Pythagorean theorem to find the distance between two points. I hope that was helpful.