"Converse of Pythagoras' Theorem (2 of 2: Counter-example)"

So here's another triangle. And just like before, I don't know whether it's right-angled or not. What I'm going to show you now is sort of a faster way of doing things. Maybe you're like, I don't want to label everything. It takes me a while. To be fair, if you do the working that I'm about to show you, it's pretty clear what's going on, but you still need to be very cautious in the way you set out your working. Okay? I'm not going to label any of the sides. I'm trying to do this slightly faster. So I'm going to go straight to talking about the squares of the two shorter sides. Which ones are those? Which are the shorter numbers? Yeah, go ahead. 8 and 15. You can just see them in order, right? So I'm going to say 8 squared plus 15 squared, and I'm going to work out what it is. Okay? 8 squared. Can we do 8 squared in our heads? 64. Thank you. Can we do 15 squared in our heads? Merrick, yeah. 155. Oh, so close. It is 200 and something and 5. It's 225. Now, maybe if you're like, oh, I wasn't sure. We do have calculators for this, right? This is why they are there. For numbers like this, it starts to get a bit big. But to be honest, you're going to see 15 squared a whole lot. So maybe get used to calculating it and working it out. 64 plus 225. Do we need to write this vertical or do you reckon we can handle it? Yeah, do you want to help me? 200 and the 10s add up to 8. The units add up to 9. And I gave you nice numbers. You didn't have to carry ones or anything, okay? So there we go. That's the sum of the squares of the shorter sides. We're going to have a look at the longest side now. That's 16 squared, okay? Hmm. This one is less common. But if you're a computer nerd like me, you know numbers like this because they're powers of 2. 16 squared. Does anyone know what 16 times 16 is? Yeah. Oh, is it 289? Now, here's an interesting thing, right? Even if you don't know exactly what 16 times 16 is, there's a way you can quickly tell, sadly for us in this triangle, it's not going to be 289. That last digit there, that units digit, right? How can you get 9? Well, you could have like 3 times 3. That would be 9. You could have 9 times 1, I guess. Those are your only two factors, right? But here I've got two 6s. What's 6 times 6? It's 36, right? So the only unit you're going to get out of this is a 6. That's, in fact, the last digit. Does anyone know what the rest of the digits are? Very good. 256. And by the way, you can see it is just a bit bigger than this one. This is 15 squared. This is the next one, 16 squared, just a little bit bigger, okay? Now, what's happened here? My a squared and my b squared, when I add them together, they do not match with my c squared, do they? So I can write this, a squared plus, sorry, 8 squared plus 15 squared is not equal to 16 squared, right? They don't line up. a squared plus b squared aren't equal to 16 squared. Therefore, and those three dots indicate, okay, I'm going to make a logical step from here. The triangle isn't right angled. That's a bit sad, but that's okay. By the way, 289, it actually happens to be a square number. Does anyone know what it's the square of? Yeah. It's the next one, 17 squared. So we've got 5, 12, 13 here. That's a Pythagorean triad. 8, 15, 17 is another Pythagorean triad. Some of you might have actually already identified that by doing your homework. We had asked you to find a few, but this one's a bit of an imposter. It's just a little bit off, okay? Now, what we've used over here on the right-hand side, you might have noticed, is neither Pythagoras' theorem nor its converse. It's actually something different. It's the opposite, right? It's when the triangle isn't right angled, and that's why you've got in your little table there, you've got a right-hand side over there, right? So you can see there's Pythagoras' theorem, and on the left, you've got the original version, a squared plus b squared equals 16 squared. Underneath, you've got your converse that looks like this. Let me show you how I would put the right-hand side of the table. And I apologize, it looks a bit cheesy, but it'll work. I'm going to do this first so you can actually read it. Stay. There we go. Okay. Let me make it a teeny bit smaller for you. There you go. So can you see what I've got here and what it means, right? If you know that your triangle is right angled, you can say, you can conclude, you've got this a squared plus b squared equals c squared thing. That's really nice, right? If, on the other hand, like our 5-12-13 triangle over here, if you test out the sides, even if you don't know whether it's right angled or not, even if the question doesn't tell you, if this works out, you can say, oh, yeah, it must be right angled. But what did we do just now with our 8-15-16 triangle? Which one did we use? Can you see it wasn't either of these two because these are where everything is nice and it works out. Which one is it? Yeah, Krisha. We used the opposite convex. Yeah, it's this bottom right-hand corner, right? We checked out a squared plus b squared. Then we checked out c squared. And we're like, uh-oh, they're not equal to each other. And that means, no, no, no, this can't possibly be a right angled triangle. So that's why I've got a red line through it. You've got this too, by the way, which we didn't really deal with. It doesn't come up very frequently. But I look at this and I'm kind of like, oh, it's missing a cell. This is to make it complete. If you've got a triangle and it's not right angled, then you can actually say Pythagoras will not work out. But you weren't thinking of that anyway because we use Pythagoras when it is right angled. So I'm going to leave this up here for a minute because I didn't want to print it in color because that would be a horrendous waste of color toner. But you might have some colors on you and you can probably draw it in a more vivid way if you've got them at your table. And I'm also going to put this up on canvas too so that you have a reference to it. But for me, I had a friend at school and he made really cool notes on all of his subjects, English and history and that kind of thing. And he was really generous. He'd share his notes with me. But I never did as well in my exams as he did. And the reason why is because the value of notes is not in reading them. The value of notes is in making them. If you construct them yourselves, that's where you actually start to put things to sink into your brain. All right, so I'll leave that up for another brief moment. I will put it back if some of you are like, no, wait, I haven't finished yet. But does anyone have any questions about that so far? Hmm. Okay, now, I've one last... Oh, yeah, sorry, you did have a question. Yeah, go ahead. Are you asking me if I write this sort of the other way around? Yeah? In a manner of speaking, it's still called Pythagoras' theorem. And the real reason why that is is because, you know this equal sign, right? The reason why, by the way, an equal sign is an equal sign is because it's literally two objects that are identical. Did you notice that? Some of the symbols are exactly what they look like, right? Like the division sign. Have you ever thought about the division sign? If you've got two things and then you want to divide them up, then you just sort of separate them. Ta-da! That's what the division sign means. Same deal with the equal sign. Now, if they are two things that are truly identical, then it kind of doesn't matter which one you say is which. So you could actually just interchange them and it'd still be Pythagoras' theorem. Okay.