Pythagoras Theorem video

In this video we're going to look at Pythagoras theorem. Now Pythagoras theorem is used to find missing sides in a right angle triangle and it can only be used in right angle triangles. Now Pythagoras theorem states that a squared plus b squared equals c squared and this is where c is a hypotenuse. Now the hypotenuse is the longest side in a right angle triangle so highlight this word hypotenuse, it is the longest side in a right angle triangle. Very easy to spot as the longest side, the hypotenuse is always opposite the right angle. Okay, so to find the missing side I always start with a hypotenuse, hypotenuse squared is equal to and then we go from here. So we're going to look at a couple examples on how you can use this. So it says find the length of the missing side of the triangles. So I'm just going to write down my formula. C squared equals a squared plus b squared and then we'll go from here. So my hypotenuse is x squared, so x squared equals four squared plus three squared and then this is a very nice one for us, we'll just work through this, that's going to be 16 plus 9. So x squared is equal to 25 and then that means x itself is the square root of 25. So x works out to be 5 and give it the units 5 centimeters. Okay we have sort of stumbled across a very important right angle triangle, probably the most important right angle triangle. This is a 3, what's called a 3-4-5 triangle and this is what's called a Pythagorean triple. So a Pythagorean triple is just three whole numbers basically that work in Pythagoras is there. And once you find one Pythagorean triple you can find out lots of other ones. So for example if I multiply these all by 2 and that was 10, 8 and 6, that would be a Pythagorean triple. If I multiply them all by 3 and got 15, 12 and 9, that would also be a Pythagorean triple. So what I mean by that is if I had 10 squared that would work out the same as 8 squared plus 6 squared, likewise 15 squared would be the same as 12, 12 squared plus 9 squared. Now there are lots of other Pythagorean triples, let's see if I can think of one off the top of my head. Another Pythagorean triple would be 5, 12 and 13. So 13 squared is equal to 5 squared plus 12 squared. So then any multiples of these three numbers would also form a Pythagorean triple. So once you find one Pythagorean triple and you can get to other ones. There's a very good wee tip if you want to make up a few questions for yourself or for your friends, just work with this, work with your original Pythagorean triple and either maybe multiply them all by 10 or maybe divide them all by 10, so 0.5, 0.4, 0.3 and that will become a Pythagorean triple. Okay the next one I'm going to just say c squared again. So c squared is equal to a squared plus b squared and then in this case what is your hypotenuse? Let's have a look. Okay it's the 3.2. So 3.2 squared is equal to x squared plus 1.4 squared. Okay my numbers aren't going to work out just as nice but that is absolutely fine. So 3.2 squared is 10.24 x squared and then 1.4 squared is 1.96 and then 10.24 minus 1.96 is equal to x squared and then if we just work out whatever that is, x squared is going to be equal to 8.28 then x is just going to be equal to the square root of your 8.28 so x works out to be and it's not a nice number here so I've had to do this to two decimal places, 2.8 and the units are centimeters and that's the two decimal places. Okay next example we have got, we'll write down a formula, c squared equals a squared plus b squared so your hypotenuse squared is equal to the sum of the other two sides squared so in this case what's my hypotenuse? It is 8.2 feet. I'll just say 8.2 squared is equal to x squared plus 3.7 squared. 8.2 squared is going to work out to be 67.24, I've done that in my calculator don't worry and x squared plus 3.7 squared is 13.69, bring that across to the other side to give your x squared and then we'll tidy it up that means your x squared is equal to what was it 67.24 minus 13.69, sorry 53.55 which means your x is a square root of 53.55 which means your x is going to be equal to 7.32 feet and again that was the two decimal places. Okay a couple of problems for us now, some different types of questions. It says in this question find the length of AC. Now let's help just draw on what AC is, AC is your long diagonal which you can see forms a right angle triangle so I'm going to call it x and then I'll just say c squared equals a squared plus b squared that's my Pythagoras theorem so x squared is equal to 4 squared plus 8 squared, nicer numbers this time so x squared equals 16 plus 64 x squared is equal to 80 which means x is equal to the square root of 80 which means x is equal to and if you do the square root of 80 you're going to get and I've gone to 1 dp 8.9 centimeters to 1 dp because they've asked you for 1 dp in this particular question. Okay in this question we've got an isosceles triangle it says find the height of this triangle of 2 dps so if we draw if we drop a perpendicular line try and make it look down and what we're trying to find is a height I'm going to call that h so I'm going to redraw that triangle very roughly but there we are that's my height I know that's 9 and this length here along the bottom is going to be it's going to be this 6 split into 2 so it is going to be just 3 and then I'm good to go. So write down your Pythagoras theorem c squared equals a squared plus b squared which means 9 squared is equal to 8 squared plus 3 squared which means 81 is equal to 8 squared plus 9 then 81 minus your 9 is equal to 8 squared 72 is equal to 8 squared 8 sin is equal to the square root of 72 so h is equal to 8 point something what's going to be 8.49 meters and that's to two decimal places. Okay it says in this question find the value of x to one decimal place bit of a tricky question this right we've got a right angle triangle here and then we've got a second smaller looking right angle triangle here what I'm going to do is use the left hand right angle triangle to work out this wee height and then once I've got this height as a number I can then work use this right angle triangle to find my x so I'm going to call this height y so I'm going to call that height y first of all and I'm going to set out use Pythagoras to find in this left triangle to find my value of y so we're going to say the left triangle this is I'm looking at let's just say c squared equals a squared plus b squared so 13 squared is equal to y squared plus 5 squared which is 169 is equal to y squared plus 25 and then 169 minus 25 is equal to y squared which means 144 is equal to y squared which means y is equal to the square root of 144 okay I'm just going to stop here a minute and talk about this this is an exact perfect answer now it happens to be that this one is a square number so you can actually find the square number square number here and square over here I should say if you if you couldn't I would use this as your value of y for the next part of the question it's very very easy to do so it keeps it really really accurate instead of me having to round the whole pile of decimal places but in this particular example it worked out very nicely for us anyway our y was equal to 12. This is my the right triangle then and all I'm going to do I think I'll just do a diagram here just to keep me right I've now worked this out to be 12 that's my x that I'm trying to find and that's my 3 so x squared is equal to 12 squared plus 3 squared sorry I didn't say what it was because remember my Pythagoras theorem is c squared equals a squared plus b squared so my hypotenuse is x this time so x squared equals 12 squared plus 3 squared and go on with this x squared is equal to 144 plus 9 which means x squared is equal to 153 which means x is equal to square root of 153 which means x is equal to 12.4 kilometers to one decimal place okay our last example says in triangle ABC angle B is equal to 90 sorry angle B is equal to 90 AB is equal to 10 BC is equal to 14 find the length of AC so let's do a new diagram doesn't have to be drawn brilliantly the only thing I'd really worry about is putting the B in the right place so here's my B it's a 90 degrees AB is equal to 10 so it's the shorter of the two so there's AB and then BC is equal to 14 and it says find the length of AC so whatever you're trying to find mark it as x or y I prefer to use x in these questions so c squared equals a squared plus b squared there's your Pythagoras theorem x squared in this case is equal to 14 squared plus 10 squared x squared is equal to 196 plus 100 x squared is equal to 296 xn is equal to the square root of 296 and then x works out to be to one decimal place 17.2 centimeters oh sorry that's the one decimal place and that is us done okay right folks we've had a wee look at Pythagoras theorem always write down your formula c squared equals a squared plus b squared and always start with your hypotenuse squared is equal to whatever it is so here was an example where you're you're on no one the thing you're trying to find was a hypotenuse over here's an example where in this triangle your hypotenuse was 13 so you had to start with 13 squared equals and then you're on no one thing squared plus 5 squared but if you always start with a hypotenuse squared equals 3 it makes life very very easy for you