 And welcome back. Today what we're going to do is we're going to talk about the Pythagorean Theorem. We're going to go through just some basic examples of using the Pythagorean Theorem. The first example I'll do is to use to find the hypotenuse. And then the second example I'll do is going to use to find the leg of a triangle. Now, the Pythagorean Theorem is usually written a couple of different ways. Well, you can't write it different ways, but it's usually looked at it as a couple of different ways. The first way is a normal right triangle, whether you're legs of a, leg of b, and then you're hypotenuse of c. And then so the equation is written as a squared plus b squared is equal to c squared. There's your very basic Pythagorean Theorem that you're used to seeing. Now, another way that I have seen it written is you can also look at it more visually like this, where the legs, you take the legs of the triangle, you square them, add them together. That should be equal to the hypotenuse squared. That's just kind of a different way to look at the Pythagorean Theorem that you may not have seen before. So a little something a little bit different. It's the same equation, it's solved the same, that kind of stuff. Just looks a little bit different. OK, now on to the first example. The first example we're going to do is we're going to find the hypotenuse of a triangle. These are going to be some just very basic examples, but what I'll also go over is how to simplify some of your, some of the answers, especially on this first one. So what I also have written up here is the Pythagorean Theorem. Actually, you know what? I'm going to move that around a little bit because I can. So bring that down here, where I'm going to use it to work with. So this Pythagorean Theorem, what I need to do first is I need to identify the legs and then also identify the hypotenuse. So as I look at my right triangle, the legs are always going to be right next to this right angle symbol. Now make sure that you are always, always make sure that you're, with the Pythagorean Theorem, you can only use it on right triangles. Well, actually, that's kind of a fib. Later on in another video I'll be doing, you can actually use the Pythagorean Theorem on other triangles, but we'll work on that later. For right now, the Pythagorean Theorem is pretty much always used on right triangles. So right here's my right angle. The legs of my right triangle are always going to be right next to the right angle symbol. So this 2 and this 6 are, in fact, my right angle or my legs of my triangle, which makes x over here. This is going to be my hypotenuse. So what I want to do is I'm going to take these numbers. I'm going to plug them in. So a is a leg. So we'll make that 2. So here we go. 2 squared plus 6 squared is equal to x squared. Now notice one thing that I did is I put parentheses around everything that I plugged in. You got to make sure and do that every time. It's not so important for this example, but for your more complicated examples, if you've got numbers in there, if you have some other symbols or you've got negatives or something else like that, you need to have parentheses around the numbers that you plug in. The basic rule is whenever you use substitution, whenever you take something out and plug it back in, always use parentheses. Always, always, always. OK, so now on to solving this, 2 squared is 4. 6 squared is 36 is equal to, well, x when you square it is just going to give you x squared. And then now, again, now what we have is we have an equation. And just like with any other equation you've ever solved, what we're going to do is we're going to solve for x. So the first thing I got to do is I've got to add these together here on this side. 40 is equal to x squared. Now I need to get x by itself, which means I need to get rid of this squared right here. To get rid of squared, what I need to do is I need to square root the problem. So what I'm going to do is I'm going to square root this side. Squaring and square rooting, they cancel each other out. So that means I can get rid of that, which means I'll be left with just the x. And on this other side, and again, just what you know from solving equations, whatever I do to one side, I must also do to the other side of the equation. So that also means I need to square root that 40. Now at this point, what I'm going to get into is simplifying a radical. This square root symbol right here, it's also called a radical. That's the actual name of the symbol itself. So we're going to simplify the radical here. Basically what we're going to do is the square root of 40 is way too big of a number to have underneath the radical. So I'm going to take this number, and I'm going to reduce it down. I'm going to show you a couple different ways to do this. I'm going to show you actually a wrong way, and then the right way to do this. It'll make sense here in just a moment. So what I do here is I want to try to make this square root of 40 smaller. So what I'm going to do is I'm going to split this up. Now this kind of looks like factor trees, if you remember those from your previous math years. What I'm going to do is I'm going to split this up. You all know that 8 times 5 multiplies back to get 40. So what I've done there is I've split it up into 8 and 5. Now the idea behind this is that when we split up, if we have a large number underneath the radical, when we split it up, we want to get a number that will reduce, that we can actually take the square root of. Well, in this case, we don't have a number to take the square root of. We don't know what the square root of 8 is. We don't know what the square root of 5 is. So it's actually pretty useless for us to split it up this way. So actually, I'm going to go backwards a little bit. So instead of the square root of 8 and the square root of 5, what I'm going to do is I'm going to split this up actually into the square root of 4 and the square root of 10. 10 times 4 gives me back to 40. Now the reason I would split it up that way is because some of you already have seen this. Right here, right here, the square root of 4, I actually know what that is. So what I can do is I can reduce this to 2. The square root of 10, I really don't know what that is. So I'm just going to leave that as the square root of 10. And that's how I simplify a radical. So notice what I've done here. The square root of 40, we split it up a little bit to make it smaller. So notice the number underneath the radical is now smaller. And that's why we call it simplifying the radical. We made it a simpler number. So that's kind of the whole process of finding the hypotenuse. Now one other thing that you can do is you can write this as a decimal. And that's what students usually do. And again, it depends on what the problem asks you to do. But what you can do is you can just plug the square root of 40 into your calculator. Or you can plug in 2 times the square root of 10 into your calculator and you get something to the effect of 6.3, I think it's 6.3, 2, 4, 5. I think that's what it is, hopefully I'm not wrong. Okay, is equal to x. So this is the decimal version. You can either plug the square root of 40 in or you can actually plug 2 times the square root of 10 and you will get that same decimal. It's the same answer, but it just kind of depends on what your problem asks for. Sometimes the problems will ask you to write it in simplest, write your answer in simplest radical form. And sometimes the problem will ask you to round to the nearest 10, the word to the nearest hundred, or to some decimal place. So it just kind of depends. But anyway, what we want to concentrate on is the Pythagorean theorem. So we'll get back into that. Anyway, so our third side there, the hypotenuse, we can either say is 2 square root of 10 or we can say it is 6.32, something to that effect. All right, so that is one example of finding the hypotenuse of a triangle. Now let's find the side. Can we go a little bit faster through this example? And this example is actually a little bit easier because I'll tell you beforehand it comes out as whole numbers, but I can also go over some vocabulary that one will need to know. Okay, so actually I'm gonna move this around just like last time so I can actually use this. First thing I want to do when I'm using the Pythagorean theorem is I want to find my sides. Or actually I want to find my legs. Here's the right angle symbol, which means my legs are going to be right next to it. So my legs in this case are 6 and x are my legs. Which means my hypotenuse is going to be 10. Now remember, your hypotenuse is always gonna be the longest side of your triangle. So in this case, whenever I solve for x, I should get a number that is smaller than 10. So it kind of gives me a hint of what my answer is going to be. All right, so I'm gonna start solving 6 squared is 36 x and when you square it you get x squared and 10 when you square it you get 100. All right, and then now we're just like solving a normal equation we gotta get x by itself. Which means I need to take this positive 36, subtract it, take the positive 36, subtract it over to the other side. So now I have x squared is equal to 64. And then just like last time to get rid of squares I need to square root both sides. I need to square root this side squaring and square rooting cancel. I'm left with just x. Square root to 64, now coincidentally we actually know what the square root of 64 is. That actually comes out evenly, so we get eight. So this isn't a problem like last time where we had to simplify the radical or do any of that kind of jazz. What we did this time, we got a whole number for one of our legs so we know that this side right here is actually just eight. Now I'm gonna redraw this triangle to get a little bit of different color in here. I'm gonna redraw this triangle real quick. Cause what I wanna do is I wanna go over a vocab word that you're going to see in my little right angle symbol in here. You're gonna see a vocab word in here. It's called a Pythagorean triple. Pythagorean triple. Okay, now what a Pythagorean triple is is simply just a, it's a right triangle that has whole numbers for all three of its sides. So notice this side has a six, we have an eight and we have a 10. Everything's a whole number, no decimals or fractions or any of that kind of jazz. So that's what we call a Pythagorean triple. There's a lot of common Pythagorean triples out there and as you do more and more problems you'll start to refine them. Three, four, five is one of them. Six, eight, 10 is another, what's another? Five, what is it? Five, five, 12, 13 I think is another one. Seven, 24, 25 I believe is another one. I'm trying to do these off the top of my head. But yeah, there's a lot of Pythagorean triples out there and again as you do more and more problems you'll find more of them. Anyway, that's it for today. Log off for now. So that was Pythagorean theorem finding the hypotenuse and also finding the side of a right triangle. And again remember with the Pythagorean theorem you can only use it when you have a right triangle. Alrighty, and that's it for now. Thank you for watching this video and we'll see you next time.