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Pythagorean theorem proof using similarity

Video transcript
This triangle that we have right over here is a right triangle And it's a right triangle because it has a 90 degree angle or has a right angel in it Now we call the longest side of a right triangle, we call that side and you can either view the longest side of the right triangle or the side opposite of 90 degree angle it is called the hypotenuse It is a very fancy word for a fairly simple idea just the longest side of a right triangle or the side opposite of the 90 degree angle And it's good to know that because some might say hypotenuse "oh, they're just talking about this side here, the side longest, the side opposite of the 90 degree angle" Now what I wanna do is prove a relationship, a very famous relationship and you might see where this is going, a very famous relationship between the lengths of the sides of a right triangle So, let's say the length of AC, so uppercase "A," uppercase "C " Let's call that length lowercase "a " Let's call the length of BC, lowercase "b" right over here I'll use uppercases for points and lowercases for lengths And let's call the length of the hypotenuse, so length of AB, let's call that C And let's see if we can come with a relationship between A, B and C And to do that I'm first gonna construct another line or a another segment, I should say, between C and the hypotenuse And I'm gonna construct it so that they intersect at a right angle And you can always do that, we'll call this point right over here, we'll call this point capital "D " And if you're worrying, how can you always do that? You can imagine rotating this entire triangle like this And this is proof but it just gives you the general idea of how you can construct a point like this So, if I've rotated it around, so now our hypotenuse we're now sitting on a hypotenuse This is point, this is now point B, this is point A, so we've rotated the whole thing all the way around This is point C, you can imagine just dropping a rock from point C with maybe a string attached, and it would hit the hypotenuse at a right angle So, that's all we did here to establish segment CD where we put our point D right over there And the reason why I did that, is now we can do all sorts of interesting relationships between similar triangles Because we have 3 triangles here: we have triangle ADC, we have triangle DBC and then we have the larger, original triangle We can, hopefully, establish similarity between those triangles And first I'll show you that ADC is similar to the larger one Because both of them a have a right angle ADC has right angle right over here So, if this angle is 90 degrees, this angle is gonna be 90 degrees as well they're supplementary they have to add up to 180 And so they both have a right angle in them The smaller one has a right angle, the larger clearly has a right angle that's where we started from And they both share, they also both this angle right over here Angle DAC or BAC however you wanna refer to it We can actually write down that triangle, I'm gonna start with the smaller one: ADC, maybe I'll shade it in, right over here So, this is the triangle we're talking about, triangle ADC and I went to the blue angle to the right angle to the unlabelled angle from the point of view of ADC This right angle isn't applying to that right over there It's applying to the larger triangle So, we can say that triangle ADC, triangle ADC is similar to, is similar to triangle, once again you wanna start at the blue angle A then we went to the right angle So, we won't' have to go to the right angle again To triangle- this was ACB ACB And because they're similar we can setup a relationship between the ratios of their sides For example we know the ratio of corresponding sides we're gonna- well in general for similar triangles we know that the ratios of the corresponding sides are gonna be constant So, we can take the ratios, the hypotenuse of this smaller triangle So, the hypotenuse is AC or the hypotenuse of the larger one which is AB AC over AB is going to be the same thing as AD as one of this, as one of the legs AD, AD, just to show that I'm just taking corresponding points on both similar triangles This is AD over AC, over AC You can look at these triangles yourself and show "look, AD, point AD is between the blue angle and red angle, and point- sorry Side AD is between the blue angle and the red angle " Side AC is between the blue angle and the red angle of the larger triangle So both of these are from the larger triangles These are the corresponding sides of the smaller triangle and if that is confusing, looking at them visually, you can- as long as you wrote our similarity statement correctly you can find the corresponding points AC corresponds to AB on the larger triangle AD on the smaller triangle corresponds to AC on the larger triangle And we know that AC, we can rewrite that as lowercase "a," AC is lowercase "a " AC is lowercase "a " We don't have any label for AD or for AB, we do have a label for AB that is c over here We don't have a label for AB, so let's just call that, so AD, let's just call that lowercase "d " So, lowercase "d" applies to that part right over there, c applies to that entire part right over there And we'll call DB, let's call that length e, that'll just make things simpler for us So, AD we'll just call d And so we have A over C is equal to D over A If we cross multiply, you a times a which is a squared is equal to c times d, which is cd So, that's a little bit of interesting result Let's see what we can do with the other triangle right over here So, this triangle right over here So, once again it has right angle, the larger one has a right angle and they both share this angle right over here So, by angle, angle similarity the two triangles are going to be similar So, we can say triangle BDC, we went from pink to right, to not labeled So, triangle BDC, triangle BDC is similar to triangle, now we're gonna look at the larger triangle, now we're gonna start the pink angle B, now we go to the right angle CA BCA >From pink angle to right angle to non-labeled angle, at least from the point of view here before the blue Now we setup some type of relationship here We can say that the ratio on the smaller triangle BC, side BC over BA BC over BA Once again we're taking the hypotenuses of both of them So, BC over BA is going to be equal to BD Here's another color, BD, so this one of the legs BD, the way I drew it as a shorter legs BD over BC, I'm just taking the corresponding vertices, over BC And once again, we know, BC is the same as lowercase "b," BC is lowercase "b " BA is lower case "c " And then BD we defined as lower case "e " So, this is lowercase "e " We can cross multiply here and we b times b Which and I mentioned this in many videos cross multiplying both sides by both denominators b times b is equal to ce And now we can do something kind of interesting We can add these two statements down here Let me rewrite this statement down here So, b squared is equal to ce So, if we add the left hand sides, we get a squared plus b squared, a squared plus b squared is equal to cd, is eqaual to cd plus ce And then we have a ce in both of these terms so we can factor it out So, this is gonna be equal to, we can factor out the c, it's gonna be c times d plus e c times d plus e, and close the parenthesis Now what is d plus e? d is this length e is this length So, d plus e is actually gonna be c as well So, this is gonna be c So, if c times c is the same thing as c squared So, now we have an interesting relationship, we have that a squared plus b squared is equal to c squared Let me rewrite that a squared, I'll do that- well let me just arbitrary new color I deleted that by accident, so let me rewrite it So, we've just established that A squared plus B squared is equal to C squared And this is just an arbitrary right triangle this is for any two right triangles We've just established that the sum of the squares of each of the legs is equal to the square of the hypotenuse And this is probably one of the, what's easy one of the most famous theorems of Mathematics, named after Pythagoras Not clear if he was the first person to establish this But it's called the Pythagorean Theorem Pythagorean Theorem And it's the really the basis of, well not all of Geometry but a lot of the Geometry that we're gonna do And it forms the basis of all the Trigonometry that we're gonna do And it's a really useful way if you know of the 2 sides of a right triangle, you can always find the third