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# Pythagorean theorem proof using similarity

CCSS.Math:

## Video transcript

this triangle that we have right over here is a right triangle and it's a right triangle because it has a ninety degree angle or has a right angle in it now we call the longest side of a right triangle we call that side and you could either view it as the longest side of the right triangle or the side opposite the 90 degree angle it is called the hypotenuse it's a very fancy word for a fairly simple idea just the longest side of a right triangle or the side opposite the 90 degree angle and it's just good to know that because someone might say hypotenuse so like other just talking about this side right here the side longest the side opposite the 90-degree angle now what I want to do in this video 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 that 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 upper cases for points lower cases for lengths and let's call the length of the hypotenuse the length of a B let's call that C and let's see if we can come up with a relationship between a B and C and to do that I'm first going to construct another line or another segment I should say between C and the hypotenuse and I'm going to construct it so that they intersect at a right angle and you can always do that and we'll call this point right over here we'll call this point capital D and if you're wondering how can i how can you always do that you can imagine rotating this entire triangle like this and this isn't a rigorous proof but it just kind of gives you the general idea of how you can always construct a point like this so if I've rotated it around so now our hypotenuse we're now sitting on our 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 2 C you can imagine just dropping a rock from Point C we've made with a string attached and it would hit the hypotenuse at a right angle so that's all we did here to establish segment CD and to 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 three triangles here we have triangle ad see 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 have a right angle ADC has a right angle right over here clearly if this angle is 90 degrees and this angle is going to 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 so the smaller one has a right angle the larger one clearly has a right angle that's where we started from and they also both share they also both share this angle right over here angle DAC or BAC however you want to refer to it so we can actually write down that triangle I'm going to start with the smaller one ADC and maybe I'll shade it in right over here so this is a triangle we're talking about triangle ADC and I went from the blue angle to the right angle to the unlabeled angle from the point of view of triangle ADC this right angle isn't isn't applying to that right over there it's applying to the larger triangle so we could say triangle ADC triangle a DC is similar is similar to triangle once again you want to start at the blue angle a then we went to the right angle so we want to go to the right angle again to triangle so it's a CB a C B and because they're similar we can set up a relationship between the ratios of their sides for example we know the ratio of corresponding sides are going to in in general for a similar triangle we know that the ratio of the corresponding sides are going to be a constant so we could take the ratio of the hypotenuse of this side right of the smaller triangle so the hypotenuse is AC so AC over the hypotenuse over the larger one which is a B AC over a B is going to be the same thing as as ad as one of this of one of the legs ad ad then just to show that I'm just taking corresponding points on both similar triangles this is ad ad over AC over AC you can look at these triangles yourself and show look ad point ad is between the blue angle and the right angle and point sorry side ad is between the blue angle and the right angle side ac is between the blue angle and the right angle on the larger triangle so both of these are from the larger triangle these are the corresponding sides on the smaller triangle and if that is confusing looking at them visually you can act as long as we wrote our similarity statement correctly you can just find the corresponding points ac corresponds to a B 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 4ad or four for a b or ice art we do have a label for a B that is C right over here we don't have a label 4ad 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 then we'll call D B let's call that length e that'll just make things a little bit 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 have a times a which is a squared is equal to C times D which is CD so that's a little bit of an 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 a 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 could say triangle BDC we went from pink to right to not labeled so triangle B B DC triangle BDC is similar to triangle now we're gonna look at the larger triangle we're gonna start at the pink angle B and then we go to the right angle c a b b c a from right it from pink angle to right angle to non labelled angle at least from the point of view here we labeled it before with that blue so now let's set up some type of relation here we can say that the ratio on the smaller triangle b/c side b c over b a b c over b a once we're again we're taking the the hypotenuse of both of them so BC over B a is going to be equal to BD let me just in another color BD so this is one of the legs BD the way I drew is the shorter legs BD over BC I'm just taking corresponding vertices over BC and once again we know BC is the same thing as lowercase B BC is lowercase B B a is lowercase C D a is lowercase C and then BD we defined as lowercase e so this is lowercase e we can cross multiply here and we get B times B which and I mentioned this in many videos cross multiplying is really the same thing as multiplying both sides by both denominators B times B is B squared is equal to c e c e and now we can do something kind of interesting we can add these two statements let me rewrite the statement down here so B squared is equal to C e so if we add the left hand sides we get a squared plus B squared a squared plus B squared is equal to C D is equal to C D plus C e + c e and then we have a C in both of these terms so we could factor it out so this is going to be equal to we can factor out this C it's going to be equal to C times D plus e C times D plus E Plus E and close the parentheses now what is d + e d is this length a is this length so d + e is actually going to be C as well so this is going to be C so you have C times C which is just 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 in an arbitrary new color actually let me we 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 was just an arbitrary right triangle this is true 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 easily one of the most famous theorems and mathematics named for Pythagoras not clear if he is the first person to establish this but it's called the Pythagorean theorem Pythagorean Pythagorean theorem theorem and it's really the basis of well not all of geometry but a lot of the geometry that we're going to do and it forms the basis of a lot of trigonometry we're going to do and it's a really useful way if you know two of the sides of a right triangle you can always find the third