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# Introduction to the Pythagorean theorem

Right triangles and the Pythagorean Theorem. Created by Sal Khan.

Video transcript

Welcome to the presentation of the Pythagorean Theorem I apologize if my voice sounds a little hoarse-y, a little hoarse not hoarse-y I was singing a little bit too much last night So, please forgive me Well anyway we will now teach you about the Pythagorean Theorem And you might have heard of this before As far as I know it is the only mathematical theorem named after the founder of a religion, Pythagoras actually his whole religion was based on Mathematics But I'm not historian here, so I'll leave that to the historians Anyway let's get started with what the Pythagorean Theorem is all about If I were to give you a triangle, let me give you a triangle and I were to tell you that it's not a normal triangle, it is a right triangle And all a right triangle is, is a triangle that has one side equal to 90 degrees And I'll leave you think about whether it's ever possible for a triangle to have more that one side that has 90 degrees Anyway just granted that a right triangle is a side, that has at least, well let me just say, a right triangle is a triangle that has only one side that is 90 degrees And if you have a right triangle with the Pythagorean Theorem allows you to do is, if I give you a right triangle and I give you two of the sides, we can figure out the third side So, before I throw the theorem at you let me actually, give you more definitions, actually, just one more So, if this is the right angle and a right triangle it's at 90 degrees And we symbolize that by drawing the angles like this, kinda like a box, instead of drawing a curve like that How bout not messing up the drawing too much? The angle opposite the right angle or I mean, the side opposite the right angle is called the hypotenuse And I really should look up where this word comes from Because I think it's a large and unwieldy word and it's a little daunting at first I have a sister told me she had a Math teacher once who made people memorize, it's a "high pot that is in use " So, I don't know if that helps you or not But overtime you'll use hypotenuse so much it'll seem just like a normal word Although when you look at it, it does look strange Anyway, going back to definitions, the hypotenuse is the side opposite the 90 degree angle And if you look at any right triangle you'll also quickly realize that the hypotenuse is the longest side of the right triangle So, I think we're done with definitions So, what does the Pythagorean Theorem tells us? Well let's call C is equal to the length of the hypotenuse, length of hypotenuse And let A be the length of this side and let B equal the length of this side What the Pythagorean Theorem tells us that A squared plus B squared is equal to C squared Now that's very simple formula, might be one of the most powerful formula in Mathematics >From this you go into Euclidian Geometry, you go into Trigonometry, you can do anything with this formula Well do that in future lectures Anyway, let's actually test this formula or not test it, let's use the formula Maybe in another presentation I'll ask you to prove the minimum of visual portfolio I apologize ahead of time, that I'm a bit scattered brain today It's a been a while since I directed a video and once again I told you I sang a little bit too much last night, so my throat is sore Okay, so we have a triangle and remember it has to be a right triangle So, let's say this is a right triangle here, its 90 degrees And if I were to tell you that this side is of length 4 and actually, let me change that This side is of length 3, this side is of length 4 And we wanna figure out the side of this length The first thing I do when I look at a right triangle is I figure out which side I the hypotenuse is Which side is the hypotenuse? Well there's two ways to do it, there's actually one way You look at where the right angle is and the side opposite to that So, this is the hypotenuse This would be C in our formula in the Pythagorean Theorem, I mean we can call it whatever we want, but just for simplicity remember A squared plus B squared equals C squared So, in this case we see that the other 2 sides, each other squared when added together will equal c squared So, we get 3 squared plus 4 squared is equal to C squared Where C is our hypotenuse So, 3 squared is 9 plus 16 is equal to C squared 25 is equal to C squared And it seems to be plus or minus 5 But we know that you can't have a minus 5 length in Geometry So, we know that C is equal to 5 So, using the Pythagorean Theorem we just figured out that if we know the sides of, one side is 3, the other side is 4 then we can use Pythagorean Theorem for that Hypotenuse of this triangle it has the length of 5 Let's do another example Let's say, once again this is a right angle This side is of length 12, this side is of length 6 and I wanna figure out what this side is So, let's write down the Pythagorean Theorem A squared plus B squared is equal to C squared Where C is the length of the hypotenuse So, the first thing that I wanna do when I look at our triangle that I just drew Is which side is the hypotenuse? Well this right here, is the right angle So, the hypotenuse is this side right here And we can also eyeball it and say "oh that's definitely the longest side of this triangle " So, we know that A squared plus B squared is equal to 12 squared, which is 144 Now, we know that we have one side but we don't have the other side So, I'm gonna ask you as question Does it matter, which side we substitute for A or B? Well no This is because A or B kinda do the same thing in this formula So, we can pick any side to be A other than the hypotenuse And we'll pick the other side to be B So, let's just say that this side is B and let's say this side is A So, we know what A is, so we get 6 squared plus B squared is equal to 144 So, we get 36 plus B squared is equal to 144 B squared is equal to 144 minus 36, B squared is equal to 112 Now we got to simplify what the square root of 112 is What we did on those radical modules was probably helpful here So, B is equal to the square root of 112 Let's think about it, how many times does 4 go into a 112? 4 goes into a 125 times, it will go into it 28 times And 4 goes into 28, 7 times I actually think that this is equal to 16 times 7, am I right? 7 times 10 is 70 plus 42 is a 112 Right So, B equals the square root of 16 times 7 You see, I just factored that as a product of a perfect and a prime number Or actually it doesn't have to be prime number just a non-perfect square And then I get B is equal to 4 square root of 7 So, there we go And this is 12, this is 6, this is 4 square root of 7 I think that's all the time I have, for this presentation Right after this, I'll do one more presentation where I give a couple of more Pythagorean Theorem problems See you soon