Garfield's proof of the Pythagorean theorem

what we're going to do with this video is study a proof of the Pythagorean theorem that was first discovered first discovered as far as we know first discovered by James Garfield in 1876 and what's exciting about this is he was not a professional mathematician you might know James Garfield as the 20th president of the United States he was elected president he was elected four years after 18 in 1880 and then he became president in 1881 and he did this proof while he was a sitting member of the United States House of Representatives and what's exciting about that it shows that Abraham Lincoln was not the only u.s. politician or not the only u.s. president who was really into geometry and what Garfield realized is if you construct a right triangle so I'm going to do my best attempt to construct one so let me construct one right here so let's say this side right over here is length B let's say this side is length a and let's say that this side the hypotenuse of my right triangle has length C so I'm just constructed and on a right triangle and let me make it clear it is a right triangle he essentially flipped and rotated this right triangle to construct another one that is congruent to the first one so let me construct that so we're going to have length B and it's collinear with length a it's along the same line I should say they don't overlap with each other so this is length side of length B and then you have a side of length let me draw it so it should be a little bit taller side of length of B then you have your side of length a at a right angle your side of length a comes in at a right angle and then you have your side of length C you have your side of length length C so the first thing we need to think about is what's the angle between these two sides what's this mystery angle what's that mystery angle going to be well it looks like something but let's see if we can prove to ourselves it really is what we think it looks like if we look at this original this original triangle and we call this angle theta what's this angle over here the angle that's between sides a of length a and length C what's this the measure of this angle going to be well theta plus this angle have to add up to 90 because you add those two together they add up to 90 and then you have another 90 you is going to get 180 degrees for the truth for the interior angles of this triangle so these two have to add up to 90 this angle is going to be 90 minus theta well if this triangle up here is congruent and we've constructed it so it is congruent the corresponding angle to this one is this angle right over here so this is also going to be theta and this right over here is going to be 90 minus theta so given that this is Theta this is 90 minus theta what is our angle going to be well they all collectively kind of go 180 degrees so you have theta plus 90 minus theta plus our mystery angle is going to be equal to 180 degrees the the Thetas cancel out theta minus theta and you have 90 plus our mystery angle is 180 degrees subtract 90 from both sides and you are left with your mystery angle equaling 90 degrees so that all worked out well so let me make that clear let me make that clear and that's going to be useful for us that's going to be useful for us in a second it's going to be useful so we can now say definitively that this is 90 degrees this is a right angle now what we are going to do is we are going to construct a trapezoid this side a is parallel to side B down here the way it's been constructed this is just one side right over here this goes straight up and now let's just connect these two sides right over there connect to these two sides right over there so there's a couple of ways to think about the area of this trapezoid one is we could just think of it as a trapezoid and come up with an A with with its area and then we could think about it as the sum of the areas of its components so let's just think first think of it as a trapezoid so what do we know about the area of trapezoid well the area of a trapezoid is going to be the height of the trapezoid which is a plus B a plus B that's the height of the trapezoid times the way I think of it the mean of the top and the bottom or the average of the top and the bottom so plus plower since that's this x times one half times a plus a plus B a a plus B a plus B and the intuition there you're taking the height times the average of the bait of this bottom and the top the average of the bottom and the top gives you the area of the trapezoid now how can we also figure out the area with its component parts so this should be regardless of how we calculate the area as long as we do correct things we should come up with the same result so how else can we come up with this area well we could say it's the area of the two the two right triangles the area of each of them is one-half a times B one half a times B but there's two of them let me do that B in that same blue color but there's two of these right triangles so let's multiply by 2 so this 2 times 1/2 a/b that takes into consideration this bottom right triangle and this top one and what's the area of this large one that I will color in in green what's the area of this large one well that's pretty straightforward it's just one half C times C so plus one half C times C which is one half C squared now let's simplify this thing and see what we come up with and you might guess where all of this is going so let's see what we get so we can rearrange this so this is one half let me rearrange this so it's one half times a plus B squared a plus B a plus B squared is going to be equal to is going to be equal to 2 times 1/2 well that's just going to be 1 so it's going to be equal to a times B a times B plus 1/2 C squared plus 1/2 C squared well I don't like these 1/2 sleighing around so let's multiply both sides of this equation by 2 I'm just going to multiply both sides of this equation by 2 on the left hand side I'm just left with a plus B squared so let me write that a plus B squared plus B squared and on the right hand side I am left with 2 a B 2 a B trying to keep the color coding right and then 2 times 1/2 C squared that's just going to be C squared plus C squared well what happens if you multiply out a plus B times a plus B what is a plus B squared well it's going to be a squared plus plus 2 a B plus 2 a B a B plus B squared and then our right hand side right hand side is still is still it's going to be equal to all of this business and changing all of the colors is difficult for me so let me copy and let me paste it so it's still going to be equal to the right hand side well this is interesting how can we simplify this is there anything that we can subtract from both sides well sure there is you have a 2 a B on the left hand side you have a 2 a B on the right hand side let's subtract 2 a B from both sides if you subtract 2 a B from both sides what are you left with you are left with the Pythagorean theorem so you're left with a squared plus B squared is equal to is equal to C squared very very exciting and for that we have to thank the 20th President of the United States James Garfield and I really you know this this is really exciting because this the Pythagorean theorem it was around for thousands of years before James Garfield and he was able to contribute just kind of fiddling around while he was a member of the US House of Representatives