Pythagoras' Theorem Explained

Hi guys, in this example I'm very excited, we're going to start to see Pythagoras' theorem in picture form, in a very visual format, so you can start to see what all this square stuff is actually meaning. So let's take a look at this triangle we have here, we have a black triangle, it's surrounded by all these grids as you can see. Now the black triangle is a right angled triangle, we know it's right angled because it's got this little green thing here denoting and representing 90 degrees. So as we look around we have these grids, and we have the long hypotenuse here on that side and the right angle over here as mentioned earlier. So what we're going to do is we're going to play with Pythagoras' theorem in such a way that we can start to see if his theorem holds true or not. So first of all the best way to play with this so we can visually understand it is to work with the lengths of this black triangle. So let's take a closer look at the length up here in orange. The length up here is 1, 2, 3 units, so let's write that down as 3 units over there. And the length across here is 1, 2, 3, 4 units, let's write down 4 over here. And finally the hypotenuse or the longer length along here, that's actually 5 units, 1, 2, 3, 4, 5, let's write down 5 over here. Okay, let's see if we can do this the shortcut method and we come up with a theorem maybe a bit better than Pythagoras' theorem. Let's actually do this. Let's add the first two numbers, the way Pythagoras added the first two numbers, and let's see if it equals the hypotenuse, the last number on the right. Okay we have 3 plus 4 equals 5. Now does 3 plus 4 equal 5? No it doesn't, does it? 3 plus 4 is 7. 7 doesn't equal 5, well there goes that theory, that's not going to work too good. Well let's try and play now with Pythagoras' theorem. Now what he did is he squared each number. As you can see each number is squared, that's a square, that's a square, and the hypotenuse is also a square, so let's square the numbers. So let's take 3 and square it, so we've got 3 squared, and then take the 4 and it's 4 squared, and then take the 5, the 5 here, 5 by 5, that's 5 squared. Okay let's add the first two numbers, and let's see if it equals the third number. Let's have a look if that works. Now we have 3 squared here, which is right over here, 3 squared equals 9, and we have the 4 squared, that's 4 by 4, that equals 16 units squared, and finally we have the 5 squared here, 5 by 5, that actually equals 25. Let's see if this holds true, we have 9 plus 16 equals 25, let's see if that holds true right now. 9 plus 16, well that's 25, so we have here 25 equaling, of course, 25. So isn't that interesting? When we square each number, it actually holds true. So 25 does in fact equal 25. So Pythagoras' theorem actually holds true. Now let's see how it holds true visually up here displayed in the picture. So let's have a look here, we have 9 and 16 equaling 25. Well we have 9 here, 3 by 3 is 9, we have 4 by 4 which is 16 over here. Now if we take the 9 and we add it to the 16, 16 plus 9 is 25, it's the same number here, it's 25 units. So clearly, for a right angled triangle, these two numbers when they are squared, they equal the hypotenuse squared. So clearly Pythagoras' proof actually is true. Take a minute to check out our website at http://www.vividmaths.com You'll find additional resources, cheat sheets, transcripts and other maths videos that are not available on YouTube. Lastly, don't forget to subscribe to get access to all the written answers for your maths questions plus other special offers.