Review: Pythagoras' Theorem & its Converse

Okay, now before you actually put any pen to paper underneath the subheading, what is Pythagoras' theorem? What is Pythagoras' theorem? If I gave you a right angled triangle, because that's what we're looking at today, that's a terrible right angle I know, okay, what does Pythagoras' theorem tell us about the right angled triangle? Any takers? Okay, the longest side, which we give a special name, the hypotenuse, okay, what do we do with the hypotenuse? What does it relate? It relates the hypotenuse to the other two sides, right? We tend to call them A and B, right? How does it relate them? What is the relationship between the three sides of a right angled triangle? Anyone want to help me out? We've been doing this for years now. What's the relationship between the three sides of a right angled triangle according to this old dead Greek guy? What's the relationship? Yep. We're so close. Got the basic idea, yeah. Okay, thank you. It was very close. Okay, so in case you haven't already, you should draw that little diagram underneath. Now, in words, what does it mean? In words, C is the longest side, the hypotenuse. So we say that the square of the hypotenuse, the square of the hypotenuse, okay, it equals two. Now, how would you explain what the right hand side is? That's the left hand side. How would you word the right hand side? Okay, now hold on. I've got this form written down. I'll talk about the other form in a second, right? So everything's squared. I have no square root signs anywhere yet, okay? So I think what I've got on the right hand side, it's a sum, right? It's a sum. Equals to the sum of, well, what is it the sum of? It's the squares of the two shorter sides, isn't it? Sum of the squares of the two shorter sides. You could just as easily have said the two other sides, because the shorter sides have to be the other sides. The hypotenuse is the longest, okay? Now, sometimes you will see this, occasionally, right? Which is taking the square root of both sides and then ignoring the negative value, okay? Now, that's a necessary result. If you, you know, we usually don't want the square of the hypotenuse. We want the hypotenuse, okay? And you can see where it comes from, okay? However, I'm going to recommend that you come back to this. I think this is what you should write to begin with and not this. In the same way, if you wanted to find one of these other two sides, right? If you do want the hypotenuse as the subject, I think rather than writing this, for instance, A star, okay? Both of these lines, both of these results come from Pythagoras' theorem, but they're not Pythagoras' theorem, okay? So, if you can distinguish between those two, these are kinds of like conclusions that you draw or something, but they're not the theorem itself, okay? So, I reckon this is sort of the line you should always start with, okay? Now, Pythagoras' theorem, let's just really, really quickly use it for an example. Okay, how can we go ahead and find the hypotenuse? What should be the very first thing that we write? Maybe I should give this a name here. Hold you right there. Good. Perfect, okay. Now, by the way, you know, in geometry, what we try and emphasize is we explain, justify, why you say different lines, right? Provide a reason, okay? So, generally, if you say here, Pythagoras' theorem, that should be sufficient. H squared equals 12 squared plus 5 squared, so we can actually do this. We don't need a calculator, do we? 144, 25. So, what's H? 13, isn't it? And, in fact, some of you might have recognized 5, 12, 13. Those three numbers, we give them a name. 3, 4, 5 and 6, 8, 10. What do we call those? Yeah, that's right. It comes from this guy's name again, right? So, I should write down H equals 13. And 5, 12, 13 is what we call a Pythagorean triad. And these triple sets of numbers is an infinite set of numbers. Great. Now, before we move off Pythagoras' theorem, I want to talk about the converse. Now, apart from being the shoe brand, what does it mean for something to have a converse? What is the converse of something? Actually, I'm going to put you under less pressure. I won't ask you to say it, but who reckons in their head they have an idea of what a converse is? Hands up, anyone? I'm just thinking of shoes. Okay, fine. Do you reckon you have one? Okay, so opposite. Opposite, I think, is a very helpful word. It is different, but I'm going to go after it because I think it's helpful. Have a look at this. This is Pythagoras' theorem anyway. Now, the opposite of that is to kind of turn it around, do it in reverse. I should have probably said right at the beginning, in right triangles. As in, triangles with a right angle. So if I'm to turn this around, I'm going to say if Now, I'm going to do all this. If c² equals a² plus b² in a triangle. Any triangle you want. If that happens to be true, then the triangle is right angled. So you can see that opposite is on the right track, I think. It's not quite the opposite. That's why we have these two English words, converse and opposite. But it is taking that idea and then flipping it around. That's what we mean by the converse. Now, Pythagoras' theorem is true. The converse of Pythagoras' theorem is also true. The reason why I'm making a big deal about it is that it's not always the case. You need to If you have a square, Don't write this down, I'm just trying to get real. If you have a square, all the angles are They're all equal You can tell me more than that, right? What are they all equal to? Because they're quadrilateral. They're all equal to 90, aren't they? 90 degrees. So that's a true statement. But the converse of the statement is not true. Right? Think about it. If I turn it around, what would happen if I turned it around? If all the angles in a four-sided figure are 90 degrees, it's not necessarily a square, is it? What else could it be? It could be a rectangle, right? So the fact that Pythagoras' theorem is true and the converse is true is actually kind of a big deal. It's important to us. So how do you use this in your exam? They'll give you a triangle. And they'll give you numbers like 1, 2, 9, 13, 14, 15, 16, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 25 plus 16 of course is? 41, last I checked. And they say, uh-oh, hold on, 41 is not equal to 49. And then they'll say, okay, therefore, is it right angled or not? It's not, because it doesn't fit in the relationship, right? Now can anyone tell me what's the problem with this when I'm setting it up, even though I've got the right answer? Why did I say this? Why did I, why did I get to here and then change an equal sign into a not equal sign? Like seriously, remember 30 seconds ago, what was the thought process going through my brain, okay? It was here, I don't know, here, I don't know. Ah, now I know, okay, so that's why I said it. But all of these are the same guy, right? Surely if this one, this line isn't equal, then none of them are equal, right? Please tell me that's true, right? Okay, so therefore, please, please do not start with this line, okay? The whole thing is bogus, okay? Because you started off assuming something which is wrong from the beginning, okay? Instead, let's take each of these two sides and we're working out separately, okay? The numbers are going to be the same, we draw the same conclusion at the end, but you're not going to be lying to yourself, okay? So let me just rehearse that for you and this is the way I'd like you to set it up, so you can follow along with me. a squared plus b squared, that's still going to be 5 squared plus 4 squared, still 25 and 16, still 41, okay? c squared, still 7 squared, still 39. So now, at this point, I can conclude, aha, the sum of those two squares is not equal to the square root of r squared, therefore, it's not right. Okay, now can you see all the numbers, all the maths of it, it all works out the same, okay? But the way you present it is totally different. Okay, let's move on.