hi everyone and welcome back to another national phone maths lesson and today we will be looking at pythagoras we will first look at right angle triangles once again and describe them in a little bit more detail than we've done before in previous lessons and then move on how we can use this thing called pythagoras in relation to right angle triangles and finally we will look at some example problems and how this thing called pythagoras can be used in the real world to solve problems so right angle triangles we know that a right angle triangle is simply a triangle with a 90 degree angle like this one here and just like any triangle a right angle triangle has three sides as we can see here but one other unique thing about a right angle triangle is that it has a special side called the hypotenuse and although the name of this side seems quite complicated all we really need to know is that the hypotenuse and the right angle triangle always lies opposite the right angle so as we can see in this diagram we have the right angle down here and then the hypotenuse lies directly opposite of the right angle and the other unique thing about this side called the hypotenuse that is it's always the biggest side within a right angle triangle so the hypotenuse is only unique to 90 degree or right angled triangles and always lies opposite the right angle and it's always the biggest side and why is this the case the hypotenuse is the biggest side always within a right angle triangle well we know that all angles in a triangle add up to 180 degrees and we know we have a 90 degree angle here so this always must be the biggest angle within a right angle triangle and if that's the biggest angle and that means that the two lines coming out from the angle are the most spread apart they are the most far apart and that would then mean that the opposite side is going to be the biggest because if it's the biggest angle then the opposite side is going to be the biggest and that's where we come up with the hypotenuse so how does right angle triangles and this side called the hypotenuse relate to this thing called pythagoras ethereum well all this really is pythagoras as a way of calculating the length of one side and a right angle triangle when the other two sides are known so that basically just means is if we know the length of one side and also another side then we can use this thing called pythagoras to calculate the length of the other side and when we talk about pythagoras theorem theorem just relates to the fact that the way of calculating the other side within a right angle triangle has been proven and can be used for any right angle triangle there is so this equation right here is pythagoras theorem and it can be used to relate each of the sides to one another and can be used to find out what the length is of an unknown side so we can see in the diagram here we have the hypotenuse is leveled as a and the two other sides b and c and because we already know that the hypotenuse is always the biggest side within a triangle we know that a is going to be bigger than sides b and c so here's an example if we have a right angle triangle and we know the length of two sides so here we have a length of seven meters and five meters and these two lengths can be represented in pythagoras theorem by b and c and then we can use this equation to then find out the length of hypotenuse a and it doesn't matter which of the two sides we know on which one side we don't know so in this case we could be given the hypotenuse which is that five meters and then the other side which is four meters and then be asked to calculate what this side c is in this case by rearranging this equation and the one thing that's worth noting is that within this equation right here a is always the hypotenuse as we can see here but b and c it doesn't matter which side is b and which side is c the important thing is that the hypotenuse is always a in this form of the equation so we can say that the hypotenuse a squared is equal to b squared plus c squared and that there is pythagoras's theorem okay so the best way to properly understand pythagoras is to go over a few example problems which we'll do just now so in this one we've been asked to calculate the length of side x as we can see over here and we've been given the length of two other sides we can see this length here is five meters and this one is four meters so the first thing we have to ask ourselves is that is this triangle a right angle triangle and the answer is of course yes because right here we can see that the right angle has been symbolized therefore we can say yes now since we know it's a right angle triangle that means we know that we can now use pythagoras's theorem so the next thing we have to do is find where the hypotenuse is in the triangle which side is hypotenuse and we know that the hypotenuse always lies opposite the right angle so that must mean that this side here of 5 meters is the hypotenuse and because that's the case we now know that 5 meters is equal to a in our little equation then we can also say that the two other sides are b and c and as i said it doesn't matter which one is which so we're labeling this side of full metals b and the side x as c so if we replace the values we can then say that 5 squared is equal to 4 squared plus x squared and because we want to calculate the length of side x we can rearrange this to say x squared plus 4 squared is equal to 5 squared and to solve for x what we want to do is move this four squared over to the other side so if it's plus four squared on this side it's going to go over to the other side and become minus four squared so we can say that x squared is equal to five squared take away four squared and we know that five squared is the same as five times 5 which is equal to 25 and 4 squared is the same as 4 times 4 which is 16. so we can say then 25 minus 16 is equal to 9 and to get x what we have to do is take the square root of both of these because if we have x squared is equal to 9 that would mean that x multiplied by x is equal to nine so we have to take this square root of nine so we can say that then x is equal to the square root of nine and we know that the square root of nine is the same as saying what what times by itself is equal to nine and we know that is three so our answer x is equal to three meters because we can say that three times three is equal to nine therefore three is equal to the square root of nine and then we can replace x in our diagram with the three meters that we've just calculated and there you go that's how you use pythagoras to solve the length of one side and a right angled triangle so here we have another example where we've been asked to calculate a length x and this time we have a length three meters and eight meters either side so as a right angle triangle yes we can see that's indicating right here and now we want to find the hypotenuse and we know that this always lies opposite the right angle and that means this time x is the hypotenuse so if we say that using pythagoras theorem a squared is equal to b squared plus c squared and we know that a is always hypotenuse so in this case a is in fact x and the other two sides eight meters and three meters are b and c so we can say x squared is equal to three squared plus eight squared so x squared is equal to three squared well three squared is equal to nine and 8 squared is 8 times 8 which is equal to 64. and if we add 9 and 64 that is equal to 73 and the same as last time in order to get x we have to take the square root so we can say that x is equal to the square root of 73 and this is one we have to do by calculator so if you put the square root of 73 in your calculator we get an answer of around 8.5 meters so again using pythagoras we can replace the x in our diagram with 8.5 meters to calculate the length of the missing side and it doesn't matter which of the three sides and the right angle triangle we've been asked to calculate they can always be solved using pythagoras the important thing is again is just remembering which one is a b and c and we've always said that a and this form of the equation is always equal to the hypotenuse if we get that right then it's an easy job the rest of the time so let's look at pythagoras in a practical form where you can use it to solve in real world problems so you can see by this magnificent drawing that we have a building a cable attached to the building and that leads onto the ground and we can see that the cable is 15 meters high from where it attaches to the building and it goes out seven meters and the question asks us to calculate the length of this cable so we can see that in here we can represent this as a right angle triangle as shown below with side seven meters and 15 meters and this side here represents the cable and now we know that because the cable side lies opposite of the right angle triangle then that is going to be a all the hypotenuse so let's start this question off by writing down pythagoras theorem so we can say that a squared the hypotenuse squared is equal to b squared plus c squared and we know that the cable side is a and that's hypotenuse so we're just going to keep that the same so we can say then a squared is equal to 7 squared plus 15 squared so a squared is equal to 49 which is 7 squared plus 15 squared is equal to 2 to 5. adding both of them together 49 plus two to five equals 274 and then we can then say that a taking the square root of 274 is equal to approximately 16.5 meters and again to calculate this you can just put this in your calculator and because a is 16.5 meters we can then say the length of this cable is also 16.5 meters so that just shows you how we can use pythagoras and real world problems and how it can be useful to solve them so let's recap what we've covered in this lesson about right angle triangles and pythagoras we said the side opposite the 90 degree angle within a right angle triangle is called the hypotenuse and this is always the biggest side within the right angle triangle and we know though that if two sides within a right angle triangle if their length unknown we can calculate the other length by using pythagoras theorem and i've described that as the way of calculating the length of one side in a right angle triangle when the other two sides are known and we do this by this equation a squared is equal to b squared plus c squared where a b and c are the different lengths of the triangle and a in this equation a is always the hypotenuse i hope this video has been helpful to you and if you have any questions don't hesitate to ask until then i'll see you in the next video thanks again for watching