hello and welcome back to stem with Steve today we're going to do an introductory lesson to Pythagoras where we're going to look at his theorem which was on right angle triangles and what he found was that basically there's a relationship that exists between the longest side and the other two sides now this was over 2 and a half thousand years ago that he was able to find this without computers without modern day technology and it's kind of cool with the relationship that he found so let's Jump On In so Pythagoras's Theorem is pretty simple it's basically the hypotenuse or the biggest side squared is equal to the other two Sid squared added together now the trick here is that area or that space on this on these re on these squares are similar or they're exactly the same they're exactly the same measurements I'm going to go through it and unpackage what that meant so most people think of Pythagoras as the Greek scholar had a big school and people would come and have chats to him about what he found in uh in this sort of relationship so would spend days talking looking at drawing right angle triangles and then trying to see if they're proportional or not now most people know this um version of Pythagoras but there was a bit of a darker secret to him he actually was running a cult now if people didn't believe in Pythagoras's Theorem um and there's one guy in particular they B he basically take them out on a boat and kick them off if they didn't believe in his theorem and there's a particular reason why which we'll unpackage later um but yeah that's the basic general gist so the leading question I have is if had a ladder that was on a 4 M high wall and the ladder was 3 m away at the base what would be the measurement of the ladder or how long would it be and we're going to do that first with this little um diagram which we're about to do so the learning objectives that I have is that we're going to learn how to label a triangle with the sides identify where the hypotenuse is and then introduce Pythagoras's Theorem and then we're going to learn how to rewrite the theorem using different side notation so the first step I want you guys to do in your books is to draw a right angle triangle with the measurements 5 4 and three now you need to use a ruler here and measure of 3 cm 4 cm and hopefully you'll find when you draw that last one if it's a perpendicular it'll be 5 cm long afterwards then draw a grid so the bottom will have 3 cm x 3 so we'll have nine little boxes this one will have four boxes by four so 16 boxes and then you'll find the big one will have 5x5 or 25 boxes if you add sorry 16 + 9 together you will find that it actually equals 25 but this is the fundamentals of what Pythagoras found so draw it with the three squares so just pause the video and have a go with that and if you're not sure there's a video I'm about to show that goes through exactly how it looks cool so here is the video right angle triangle hypot is C if you square all the sides find the three becomes nine the four become 16 cool music as well and the five will have 25 now if you restack these boxes together this is where something cool happens they're exactly the same oh so when we look at a triangle we need to have a bit of common language between it and we need to understand what vertices and edges are so if we have a right angle triangle like this labeled a b and c those letters represent the vertices or the points of the shape the edges then are on the the edges on the sides of basically the sides of the shape and the way that we describe them is with um little letters so little letters represent the side because it's opposite this angle so little B because it's opposite the capital B this one over here would be little a and then this one over here would be little C to make that work so then to describe the length very similar we would point at that and that would be the length a because it goes from verticy a all the way through to C this length here another way of describing it would be length BC and then this length here would be a so if just pause the video and see if you can label that one awesome so you should have found AC was equal to 10 BC was equal to six sorry AB was equal to 8 and last one BC was equal to six cool so um couple ways of remembering this some people use the word hippopotamus so hypotenuse sounds like a hippo hippos are really big so that's the bit why it's linked to the biggest side the hypotenuse is always opposite that right angle that's really important to Le so in your books you'll need to write this so the hypotenuse is always the longest side in a right angle triangle it's always opposite the right angle and we call the other sides the legs so the smaller sides we call those the legs and then here's a pretty cookie cutter video um image of what the hypotenuse and what a right angle triangle looks like so the sides are there hypotenuse opposite the right angle as you can see there now the way reason why it's opposite the right angle if you think about the hypotenuse as it gets bigger and bigger it increases the length of the triangle side doesn't it so if you have a small angle see it makes a small side but if you make a big angle makes a big side if you have a right angle makes a really big sign so hypotenuse if we look at that try and identify should have found that it was this guy here so that's the AC in this one we should see that's AB this one AC again this one it's BC cool so Pythagoras's Theorem he actually did not know what um Al was so he didn't use the rule that we saw at the very beginning he had a slightly different way of describing it he described it in words or probably in Greek not English but it would look something like this the hypotenuse squared or the biggest side squ is always equal to the sum of the other two sides squared so let's say that again the hypotenuse squar is equal to the other two sides squared added together and that's basically Pythagoras's Theorem now because we have some understanding of algebra there's a better way of describing that which is what this rule is so h^2 is = A2 + B2 so the area that you see in the H is equal to the area of a 2 + b s going in so if we had a triangle like this with a 34 triangle which we've done before just to show you graphically how this looks if you go out four blocks that way and then you get 16 and if you go three blocks down you then get nine cuz three * 3 is 9 then if we go out this way we would get the five by adding those together to get 25 this is what we call the 345 triangle it's Pythagoras's fundamental triangle it's where everything sort of comes from systemic of this so then we could put into the rule just to see if this actually sort of balances out as well so H we can put in is five um we can then put 4 S + 3 S and see what happens so 16 + 9 we get 25 and then to work out H we need to then do the square root of it because that's the opposite of squaring and we're left with five cool the other cool way of looking at this graphically if I take those nine blocks at the bottom and put it around the corners on this side over here the the blue um Square you can see they fit perfectly and then the other side the 16 fits then perfectly if we rotate it in and jam it into that place there so the area of those two we should see now see the area of the two small sides is equal to the area of that big side ohol another thing that you might see depending on the school system that you're in they have different ways of writing Pythagoras's Theorem so sometimes you might see it like this instead of H they say c^2 is equal A2 + B2 now personally I don't like it because it's just a random letter I like saying h^ squ is equal a s + b s simply because H represents the hypotenuse and then I have that link in my mind when I do more complex things um you can then also change it around so say you had a triangle with the label e EFG you could then say it's e^2 is = F2 + G2 similarly if we just use symbols doesn't really matter we could say in this case smiley face square is equal to frowny face square plus P sign squ it doesn't really matter what the symbols are it's just so that we know the position of where they are so if we wanted to write down now Pythagoras's theorem for these just pause the video and see if you can get them cool so what you should have found with the first one the hypotenuse is C so therefore it would be c^2 = A2 + B2 the next one with B was F2 is = d^2 + e 2 with C you should have found z^2 = y^2 + X2 D you should have seen V ^2 = S2 + t^2 e you should have seen E2 is = G2 + F2 and then the last one X2 is = A2 + B2 cool so quick little review you should know now how to label the triangle sides the hypotenuse is always the longest side and we're using use this rule H2 is a b s sorry I should say a s + b s but that works as well now the bookwork that you guys need to do then is 3K questions 1 3 4 6 and 8 and if you want a really big challenge don't spend more than 20 minutes on this question number 10 that's it for me hopefully you've picked up a bit on Pythagoras then if you do have any questions please put them in the video down below um and yeah don't forget to like And subscribe see you next time on Steam with Steve adios