Pythagorean theorem proofs
Garfield's proof of the Pythagorean Theorem James Garfield's proof of the Pythagorean Theorem.
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- What we're doing in this video is study a proof of the Pythagorean theorem,
- that was first discovered,as far as we know by James Garfield in 1876.
- What's exciting about this is that he was not a professional mathematician.
- You might know James Garfield as the twentieth president of the United States.
- He was elected president, four years after 1880, and then he became president in 1881.
- He did this proof while he was a sitting member of the United States house of representatives.
- What's exciting about this is that Abraham Lincoln was not the only US politician
- or the only US president who was into geometry. And what Garfield realised is that we can construct a right triangle-
- Let's say this side over here is length 'b'(blue) and this side is length 'a'(red),
- and let's say this side, the hypotenuse of my right triangle has length 'c'.
- I'll make it clear it is a right triangle. He essentially flipped and rotated this triangle,
- to construct another one that is congruent to the first one.
- Let me construct that. So we're gonna have length 'b'and
- it's collinear with length 'a', it's along the same line as length 'a'.
- they don't overlap with each other.
- So this is side of length 'b' and then you have your side of length 'a'
- at a right angle and then you have your side of 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 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 triangle, and we call this angle 'theta',
- what's this angle over here, the angle that's between the sides of length a and c.
- what's the measure of this angle going to be?
- Well, theta plus this angle have to add up to 90, because the other angle is 90.
- So 90 and 90 you get 180 degrees for the interior angles of this triangle.
- So if these two angles together is 90, then this angle is '90 minus theta'.
- We've constructed this triangle congruent to the original one, so the angle corresponding to theta is also going to be theta.
- and this angle right over here is gonna be 90 - theta.
- So given that this is theta and this is 90 - theta, what is our angle going to be?
- Well they all collectively kind of go 180 degrees.
- So you've theta + (90-theta)+ our mystery angle is gonna be equal to a 180 degrees.
- The thetas cancel out(theta - theta), 90 + our mystery angle is a 180 degrees,
- We subtract 90 from both sides and you are left with
- your mystery angle equalling 90 degrees.
- so that all worked out well.
- so let me make that clear and that's gonna be useful for us.
- So now we can 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 its been constructed
- and this is just one side right over here, this goes straight up
- and now let's just connect these two sides right over there.
- So there's a couple of ways to think about the area of this trapezoid.
- One is we can just think of it as a trapezoid and come up with its area,
- And then we could think about it as the sum of the areas of its components.
- So let's just first think of it as trapezoid.
- So, what do we know about the area of the trapezoid?
- The area of a trapezoid,is gonna be the height of the trapezoid, which is (a+b) times, the way I think of it, the mean or average of the top and the bottom.
- So, ar(trapezoid) = (a+b) x 1/2(a+b)
- In the intuition there you are taking the height times 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 as far as we do the 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 right triangles.
- The area of each of them is one half of a times b.
- But there's two of them, Let me do that say in blue colour,
- But there's two of these right triangles, so let's multiply them by two.
- So 2 times half ab, that takes into consideration this bottom right triangle, and this top one.
- and what's the area of this large one, that I'll colour in green
- 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 square.
- Now , let's simplify this thing and see what we come up with and you might guess where all of this is going.
- So, we can rearrange this. So this one half times (a+b) squared is going to be equal to two times one half,
- well that's just going to be one, so its gonna be equal to a times b plus one half c squared.
- I don't like these one halves lying around, so let's multiply both sides, this equation, by 2.
- I'm just gonna multiply both sides by two.
- So, on the left hand side, I'm just left with (a+b) squared,
- and on the right hand side, I'm left with 2ab and then two times one half c squared, i.e., plus c squared.
- What happens if you multiply out (a+b) times (a+b)? We get (a+b) squared.
- That is a sq. + 2ab+ b sq. = 2ab + c sq.
- Subtracting 2ab from both sides, we are left with, a sq. + b sq. = c sq. i.e. the Pythagorean theorem.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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