 And welcome back. Today, we're going to talk about the Pythagorean Inequalities theorem. So this is kind of a video where we're going to use the Pythagorean theorem for something other than solving for a hypotenuse or solving for the side, one of the legs of a triangle. Now I do have a previous video that I'll flash up on the screen now about how we use the Pythagorean theorem to see if a triangle was a right triangle or not. Basically, the idea is if the Pythagorean theorem works, then it's a right triangle. If it doesn't work, then it's not a right triangle. What this video goes over is, well, if it's not a right triangle, well, then what kind of triangle is it? Well, we have two other triangles that it could be. And you can kind of see right here in the words. It can either be an obtuse triangle or it could be an acute triangle. So here we go. If c squared is larger than a squared plus b squared, then the triangle is an obtuse triangle. And on the other side here, if c squared is smaller, is less than a squared plus b squared, then the triangle is an acute triangle. So now basically what we're doing is we're taking the Pythagorean theorem, using it a little bit differently, to see what kind of triangles we're going to come up with. So now here's a triangle. Now I do have the shapes kind of obviously. I drew these so that the math is going to fit with the picture. So you can tell right off the bat there's probably going to be an obtuse triangle, because it goes with this here. But what we're going to do is we're going to go through the math to actually prove that. So what we're going to do is we're going to find the legs, the hypotenuse. We're going to plug them into this little formula here. And then we're going to see why this is an obtuse triangle. So 15 is my hypotenuse. That's going to go in for my c squared. 15 is my hypotenuse. Now I'm going to do a little question when I come back to this here in a minute. 15 squared is that going to be equal to, let's do 12 squared plus 5 squared. They're my legs. Now I used a question mark there, because when you're doing these type of problems on a normal assignment, you won't know which side is going to be bigger. So you're going to have to have a question mark here. You don't know if there's a left side going to be bigger. Is the right side going to be bigger? You really don't know. So I'm just doing a question mark there for now, and I'll fill in one of these inequality symbols here in a moment. 15 squared, that's going to be 225. And then 12 squared is 144. 5 squared is 25. Add these together, you're going to get 169. Well, in fact, and you kind of saw this coming, in fact, 225 is much bigger than 169. So that means that this c squared side, now remember back up here, this is the c squared. That means the hypotenuse is way too big. 225 is bigger than 169. 225 is way too big. This 15 is way too big. Since that side, since the hypotenuse is way too big, that creates a big angle down here, and it creates an obtuse triangle. So again, this is again one of those problems where we're taking the numbers that we get, and we are interpreting what they mean. This isn't one of those problems where you just solve an equation, you find a number, and you circle it. That's not what we're doing. We're taking these numbers that we have, we're plugging them in, and we're basically solving, simplifying here. And now after we get those numbers, now we're figuring out what those numbers actually mean. Well, it means that this side with the 15 is too big. That means that this side's too big. It creates a big angle over here. It creates an obtuse angle. So we have an obtuse triangle. All right, now with this other side, change my colors here a little bit. This other side, 9, 8, and 7 are my three sides. I'm just going to jump into this. So this is an example. Obviously we'll have an acute angle this time. So I'm just going to jump right in. 9 squared, is that going to be equal? Don't forget your parentheses. Is that going to be equal to 7 squared plus 8 squared? So 9 is 81. If you watched my previous video, I did this exact example again. So I'll steamroll through this. 7 squared is 49. 8 squared is 64. Add those together. You get 113. This is 81 still over here, which means 113 way bigger than 81 over there. So basically what this means is that this hypotenuse is too small. That's why we keep talking about the hypotenuse here. That's why we have it first, look at both of these formulas. That's why the c squared is first. We're talking about the hypotenuse here. This hypotenuse is too small. Since it is too small, it creates a small angle over here, which means this is going to be an acute triangle. It's going to be an acute triangle, because that side is so small. OK, all right, that's it. That's basically using the Pythagorean inequality theorem. So again, the Pythagorean theorem can be used for much, much more than just finding a hypotenuse or finding a missing leg of a triangle. You can use it to determine if a triangle is a right triangle. And if it's not a right triangle, then you can use the Pythagorean theorem to figure out if it's an obtuse triangle or an acute triangle. So there's many, many uses for the Pythagorean theorem. Anyway, thank you for watching the video, and we'll see you next time.