Proof of Pythagoras' Theorem

Here's what I want to try and understand. I see that there's a relationship going on here, right? You might notice, you know, the hypotenuse is always the longest, 5 is the longest side of this triangle, 13 is the longest here, 17 is the longest here, and so on. I want to know, is there a quick way, rather than me drawing every single triangle, right? Is there a faster way to work out this relationship? What's going on, okay? And that's what we're going to show you. I'm going to show you that, okay? So in your square, here's the construction I'd like you to put in a few lines with your rulers, okay? What we're going to do is, up in the top right-hand, left-hand corner, we're going to draw one of these right-angled triangles. Because it's a square, you take our right angle here, and here, and here. So any triangle you draw, which shares this corner, will be right-angled, okay? So draw one roughly, I'll give you a distance. I said 3 centimetres, didn't I? 3 centimetres. Okay, so this time measure that out, I hope yours is big enough to do that. If you measure that 3 centimetres, and put in a little mark there, okay? I want you to also measure out that same distance, 3 centimetres, from this opposite corner. And I'm going to go this way, 3 centimetres. Something like that. Sorry, I missed a step. Okay, and I've got another point there. Okay, that's the line I want you to join up. That's going to form a right-angled triangle. Sorry about that. Better to catch it earlier, rather than later on. Okay, so you've drawn that in. You've got that distance there, is the same as that distance there. Now in exactly the same way, I want you to take that same distance, and measure it from this corner over here. So I've got 3 centimetres happening there. And then there will be a little mark. And then I want you to join up this here, like so. So what you should find now is I've got two triangles that kind of look the same, but they've been rotated around. Think back to last year when you were looking at symmetry. What we're creating here is a shape with rotational symmetry, or point symmetry. We're almost there. You measured 3 centimetres here. We're going to do one more measurement, 3 centimetres. This time from the bottom left-hand corner. 3 centimetres. Okay, you're going to put in a little mark there. And now I'm going to join up the last two lines. Okay, so I'm going to join up this triangle, which you can see is opposite my very first triangle. It's opposite, directly facing, from that side to that side. And my last triangle will be there. Okay, gasp. Alright, give me a second. You should have that measured out. Okay. So, now that we've put this in, I'm going to make... Some of you will have different lengths here, so... This part's going to be tricky, so hold on to your hats, okay? Don't worry, though, I wouldn't pressure too much, you know, writing everything down, because everything I'm about to write down I will keep on the board. So it might be wise of you, actually, just put your pen down for a minute. Just watch. When I'm finished writing, everything will stay on the board, and you can copy it down. But for now, it'll be more important just to follow what's actually happening. Okay. So look at the very first triangle we drew up in the top left-hand corner. Okay, in fact, I'm going to get a... Very first triangle, okay? So here we go, I've got a triangle up here, and I'm trying to work out, what's the relationship between all of these sides? So to work out how they relate, I kind of need to label them so I can say, this one's this, this one's that, rather than talk about, you know, that side, and then the other side, and then the one which isn't the two sides I just talked about. Labels are much more helpful. So I'm going to call that A, B, and C. So I just want you to just mentally kind of block out the rest of the diagram for a second, so you can see you've got A, B, and C, and that's where I'm trying to work out a relationship. Okay? So you see that triangle? Now think, remember how we constructed this. That triangle over there is exactly the same as these three other triangles. They've just been sort of moved around like a tile, okay? So therefore, where else can I put A? If you look carefully on my diagram, right? William, do you want to make a suggestion? Oh, where the other edges are? Very good. That's how we measured out, like, the three centimeters, right, on your diagram. So I've got A up here, I've got A down here, and I've got A over here as well. I'll give you a second. Why don't you label your diagram just like mine, before it gets a little more busy, okay? Okay. What else do we have here? I've got this side C. I'm going to leave off B for a second, okay, because I think I've got enough B's on there. Where else can you see the letter C? Morgan? All the way to the side, the big square. Yeah, fantastic. So there's my big square in blue, and all of these ones on the inside, they're all C as well, because all of these triangles are the same, okay? All those triangles are the same. Now, I can go through and I can prove it properly for you, but just for the sake of time, I'm just going to state it, and you can have a think about it later. This shape in the middle here, it can be one of two kinds of shapes. It's got two choices, right? Before we say which one it is, what are the two possibilities? Rachel? Okay, it could be a square, or it could be a diamond, and the other name for that is, it starts with an R. A rhombus. A rhombus, very good. Same thing, different name. Rhombus is a more descriptive name, though, so I prefer that. So this thing could be a rhombus or a square. Now, just as it happens, it is actually a square. If you'd like me to show you the full proof for that, I will. It doesn't take that long. But for now, it's not that important. So now, what am I trying to do? I'm trying to get some relationships between all of these different shapes together. So, for instance, have a look at the big square, the big blue square that we started with. It's a square. How long are each of the sides? 15. Everyone's got a different number, but according to how I've labelled them, every side is A plus B. Can you see that? A plus B, every side is the same. So if I want to know the area of the whole square, how do you find the area of a square if you know its side length? You just, it's a square, so you square it, right? A plus B times A plus B. So I'm just going to write square. Are you happy with that? Yeah. Okay. Now, that's one way of looking at it. In maths, one of the most useful things to do is look at a problem, but look at it from two different angles, right? And you always discover something new. So I looked at it as one big square, but it's not just a big square. We cut it up into a whole bunch of slices, right? Have a look at the little different chunks. What have we got here? Okay, so for starters, I've got this square in the middle, and you all told me that the side length is C, so it's C squared. But I also have these triangles that I drew, right? How many triangles have I got? I've got four of them. Four times. Now, think back. This has been a little while now. What's the area of a triangle? Yeah, Tinder. Length times breadth. Very good. Let me say that again in case you didn't catch it. Length times breadth, and then you divide by two. You take half, okay? Now, the length and breadth of all my triangles is the same. The length is A, and the breadth is B. So it's four times length, breadth, divide by two. Okay, I'm going to pause for a second. Why don't you jot that down? Okay, that's an important idea. Before we carry on any further, the idea is A plus B all squared. You can even label it the way I have. It's the big square, right? What's C squared? That's the smaller square on the inside. Small square, okay? And then what's this thing trailing off on the end? It's four triangles, right? Four triangles. Okay, you'll find me doing this a lot, by the way, and I highly encourage you to do it. Whenever you write an equation, it's vital that you know what the equation means. It's not just random letters flying around, okay? They actually stand for different things. Fantastic habit to get into, particularly when you write an equation for the first time. You say, now, where did that come from? What's it actually represent? Okay, now, this is what we're going to work on here. It does look a little bit complicated, but we can work with it, right? We'll start with the right-hand side first because it's actually a bit easier. I've got a C squared. Now, have a look at this. We can simplify this, can't we? Think back to your fractions, right? That 4 is really on the numerator, right? So I've got a 4 over 2. How should I write that instead? 4 over 2. So that's 4 divided by 2, isn't it? And 4 divided by 2 is just 2. It's just 2. Okay? So can you see what I've done over here? I've cancelled here an equivalent fraction up here. 4 over 2 is equal to 2 over 1. We don't really need to write over 1, do we? So we've got 2AB. Now let's have a look at the left-hand side. This is tricky. A plus B, when you stick a square on it, right? That's shorthand. It's shorthand for this. Okay? Now, everyone got to different points in the topic of algebra. Whatever it was we wanted to talk about. And I told you algebra last year. So I'm not going to assume you know what this is right away. We're just going to work on it together. Okay? Think about the distributive law. I wonder if you remember this from last year when you were expanding brackets. Okay? So, for example, I'll give you a numerical example. Don't write this one. Just follow it. Oops. If I gave you this. Okay, so this is a fairly big number that we're going to have because I'm multiplying lots of numbers together. Okay? How can I unpack this? I've got a couple of choices. I can add these two. Gives me 4. I can add these two. Gives me 30. And then I multiply through. That's fine. Okay? But there's another way I can do it. Right? One plus three times that means I should have one of them and I should have three of them. Just look at that for a second. See if that sort of seems familiar to you. Multiplying this across here means I want one of them and then I want three of them. So I've written them now out separately. Okay? Now, then I can go ahead and I can expand. I'll get the same answer at the end. I'm going to do exactly the same thing here. Watch really carefully. A plus B times all of that. Will be A times that plus can anyone fill in for me what's going to be on the end? Yeah, Ethan. B dash thingy. Bracket. A plus B bracket. Cool. I'm not going to get all hung up on it because actually they're not even called brackets. They're called parentheses but no one calls them parentheses. Line thingy, curvy thing is fine for now. Okay. So there you go. I've expanded this out. Okay? I'm just going to write the right hand side again. I haven't done anything to that. Okay. Now this we can have a look at. Expanding brackets is something we've seen before. Okay? But we haven't had that much practice on it. So let's give this a go. A times A. We have a shorthand for that. It's A squared. A times B. We just have a shorthand for that which is AB. We've seen that, haven't we? AB. A times B. Okay? I've done the first lot. That's turned into that. What's going to happen here? I'm going to try and do it in order as best I can. That way I won't forget turns by accident. So I'll do these two first. B and A. When I multiply them together, you can multiply numbers in whatever order you like. Three times five. Five times three. Same thing. So I'm going to write it in this order. Okay? BA, AB. Same thing. And then the B squared that you mentioned before. Are we okay so far? Yep. Following? Okay, I'm just going to write the right-hand side another time. I haven't done anything to it. Okay, now on the left-hand side, this is my second last line. So if you followed this far, well done. See these two guys? We have a special name for these, right? They look the same. So we call them like terms. Do you remember this from last year? Yeah. We can collect like terms. I've got one here, and I've got one here. So how many do I have in total? Two. I've got A squared. Two of them, like you just told me. And B squared. So far so good? Here comes the last line. This is an equation, right? I always love to picture up with equations what you're saying is. You've got two things. Where's my other black one? I've got two things, and they're exactly equal to each other. You might be looking at them in slightly different ways, but they're the same. So anything you do to one side, you can do to the other. Well, have a look. I've got a 2AB over here, and a 2AB over here. They're the same, right? So I can get rid of them from both sides. It's like having a set of scales, and you take the same thing away from both sides. It's still equal, right? So I'm going to do that. I'm going to remove it from one side, and remove it from the other. You see that? And here's my final line. What's left on the left-hand side? A squared, and B squared. What's left on the right-hand side? C squared. That there. That, that, in big, bold letters, put a box around it, that is Pythagoras' theorem. He said, if you know what the two shorter sides are, you square them, and that'll give you the longer side, the hypotenuse squared. That's Pythagoras' theorem. You might notice, by the way, it's got an apostrophe on the end, because it's not Pythagoras' theorem. It's the theorem that belongs to Pythagoras, so there's an apostrophe.