If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:10:23

Video transcript

in case you haven't noticed I've gotten somewhat obsessed with doing as many proofs of the Pythagorean theorem as I can do so let's do one more and like how all of these proofs start let's construct ourselves a right triangle so let's going to I'm going to construct it so this hypotenuse sits on the bottom so that's the hypotenuse of my right triangle try to draw it as big as possible so that we have space to work with so that's going to be my hypotenuse and then let's say that this is my the longer side that's not the hypotenuse it doesn't we could have two sides that are equal but I'll just draw it so it looks a little bit longer let's call that side length a and then let's draw this side right over here it has to be a right triangle so maybe it goes right over there that's side of length B let me extend the length a a little bit so it definitely looks like a right triangle and this is our 90 degree angle so the first thing that I'm going to do is take this triangle and then rotate it counterclockwise by 90 degrees so if I rotate it counterclockwise by 90 degrees I'm literally just going to rotate it like that and draw another completely congruent version of this one so I'm going to rotate it by 90 degrees and if I did that this the hypotenuse would then sit straight up so I'm going to drew my best attempt to draw it almost to scale as much as I can eyeball it this side of length a will now come out will I look something like this well now look something like this it'll actually be parallel to this over here so let's see let me see how well I could draw it all I could draw it so this is a side of length a and if we care this would be ninety degrees the rotation between the corresponding sides are just going to be 90 degrees in every case that's going to be 90 degrees that's going to be 90 degrees now let me draw side B so it's going to look something like something like that or the length the side that's length B and this and the right angle is now here so all I did is I rotated this by 90 degrees counterclockwise now what I want to do is construct a parallelogram I'm going to construct a parallelogram by essentially and let me label this so this is height C right over here let me do it in that white color this is height C now what I want to do is go from this point and go up C as well go up C go up C as well now so this is height C as well and what is this length what is the length over here from this point to this point going to be what is this length going to be well a little clue is is this is a parallelogram I had this line right over here is going to be parallel to this line it's maintained the same distance and since it's traveling the same distance in the X direction or in the horizontal direction and the vertical direction this is going to be the same length so this is going to be of length a now the next question I have for you is what is the area of this parallelogram that I have just constructed well to think about that let's redraw this part of the diagram so that the parallelogram is kind of sitting on the ground so this is length a this is this is length a this is length this is length C this is length C and if you look at this part right over here it gives you a clue the height of the parallelogram there's this green color the height of the parallelogram is given right over here this side is perpendicular is perpendicular to the base so the height of the parallelogram the height of the parallelogram is a as well so what's the area well the area of a parallelogram is just the base times the height so the area of this parallelogram right over here is going to be a squared a squared now let's do the same thing but let's rotate our original right triangle let's rotate it the other way so let's rotate it 90 Gries clockwise and this time instead of pivoting on this point we're going to pivot on that point right over there so what are we going to get so the side of length see if we rotate it if we rotate it like that it's going to end up right over here try to draw it as close to scale as possible so that side has length C now the side of length B is going to pop out look something like this look something like this it's going to be parallel to that this is going to be a right angle so let me draw it like that that looks pretty good and then the side of length a the side of length a is going to be out here the side of length a is going to be right over here so that's a and then this right over here is B and I want to do that B in blue do the B in blue and then this right angle once we've rotated is just sitting right over here now let's do the same exercise let's construct a parallelogram right over here so this is height C this is height C as well so by the same logic we used over here if this length is B this length is B as well these are parallel lines we're going the same distance in the horizontal direction we're going the same where rising the same in the vertical direction we know that because they're parallel so this is length B down here this is length B up there now what is the area of this parallelogram right over there what is the area of that parallelogram going to be well once again to help us visualize it we can draw it kind of sitting sitting on flat so this is that side then you have another side right over here they both have length B and you have the sides of length C so that's C that's C what is its height well you see it right over here its height its height has length B as well it gives right there we know that this is that this is 90 degrees this is 90 degrees we did a 90 degree rotation so this is how we constructed the thing so given that the area of a parallelogram is just the base times the height base times the height the area of this parallelogram is B squared so now things now things are starting to get interesting now what I'm going to do is I'm going to copy and paste this this part right over here because this is in my in my mind the most interesting part of our diagram let me see how well I can how well I can select it so let me select this part right over here so let me copy and then I am going to scroll down and then let me paste it let me paste it so this diagram that we've constructed right over here it's pretty clear what the area of it is the combined diagram now let me let me delete a little few parts of it just so that whoops I want to do that in black so that it cleans it up so let me clean this thing up so we really get the part that we want to focus on so cleaning that up cleaning and cleaning this up cleaning this up all right over there so it what is and actually let me let me delete this right down here as well let me delete this right over here although we know that this length was C and actually I'll draw it right over here this is from our original construction we know that this length is C we know this height is C we know this down here is C but my question for you is what is the area of this combined what is the area of this combined shape well it's just a squared plus B squared let me write that down the area it's just a squared plus B squared the area of those two parallelograms now how can we express how can we maybe rearrange pieces of this shape so that we can express it in terms of C well it might have jumped out at you when I drew this or when I drew this line right over here we know that this is a length I want to do that in white we know we know that this part right over here is of length C this comes from our original construction whoops lost my diagram this is of length C that's of length C and then this right over here is of length C and so what we could do is take this top right triangle which is completely congruent to our original right triangle and shift it down so remember the entire including this top right triangle is a squared plus B squared and what we're cluding this part down here which is our original triangle but what happens if we or take that so let me cut let me actually cut and then let me paste it and all I'm doing is I'm moving that triangle down here so now it looks like this so I've just rearranged the area that was a squared B squared so this entire area of this entire square is still a squared but plus B squared a squared is this entire area right over here it was before a parallelogram I just shifted that top part of the parallelogram down B squared is this entire area is this entire area right over here well what's this going to be in terms of C well we know that this entire thing is a C by C square so the area in terms of C is just C squared so a squared plus B squared is equal to C squared and we have once again proven the Pythagorean theorem