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High school geometry
Garfield's proof of the Pythagorean theorem
Former U.S. President James Garfield wrote a proof of the Pythagorean theorem. He used a trapezoid made of two identical right triangles and half of a square to show that the sum of the squares of the two shorter sides equals the square of the longest side of a right triangle. Created by Sal Khan.
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what is theta(27 votes)
It is a greek letter which is used to mark an unknown angle in a triangle when you are going to use trigonometry to work it out,(20 votes)
At7:32, what do you mean by "multiply out" (a+b) square and get a square + 2ab + b square?(33 votes)
Yeah, thats why (a+b)^2 always equals to a^2 + 2ab + b^2, it is an algebraic exponent property.
For Example: When a = 2 and b = 3,
(2+3)^2 = 2^2 + 2(2 x 3) + 3^2
5^2 = 4 + 2 x 6 + 9
25 = 13 + 12
25 = 25
As L.H.S. = R.H.S. : (a+b)^2 = a^2 + 2ab + b^2(5 votes)
I'm reading through the slightly different answers people all gave to the different people who asked the same question- "what is theta?". Well, my question is what is the most commonly used definition for theta?(18 votes)
Theta is usually used to express an unknown angle.(53 votes)
- How is it posable for someone to come up with a theorem that is right? how was he able to make this up suddenly? A squrd x B squrd = C squrd, how can someone come up with something like this?(9 votes)
- Theorems aren't right they are just scaffolding which is used to build up more theorems. For this they only need to be consistent, which is different from being right or true or fact or real or any of these other categories. As for how thoughts become theorems, it's a process of de-simplification. Which isn't ever just made up, but is built upon other ideas then articulated. Flying squirrels and ants perform trigonometry, but humans talk about it and put it in language. Humans 40,000 years ago started articulating and thinking and then 4,000 years ago, which is when written history starts, we see they were turning thought into theory, and since then it's just been a matter of building up scaffolding, in different languages. One being algebra which started being used to express this ancient relationship of the subtend to the arms (of a triangle but also of a square and a rectangle as the ancient Chinese, Indians, and Babylonian theories went) after the Middle Ages when Italian and German and others discovered ancient Greek and current Arabic/Hindu math, formed groups and schools of method and adopted symbols and abstractions instead of writing everything out in Latin. It was also around that time, coinciding with the rise of professional mathematicians, when simply 'learning for learnings sake' or 'contributing to the stores of wisdom in God's House', sadly gave way to the modern idea of 'originality' and 'research' which has a completely different ethic about it. But it seems Mr. Garfield here wasn't hindered and took the time to light upon an ancient idea even though it had been looked at by countless eyes before.(18 votes)
Was James Garfield the most recent to prove the Pythagorean Theorem in a new way?(13 votes)
no because in0:01Sal stated "a proof" not the only one. Also it is around 1876 someone probably already found one.(1 vote)
why do we need to equate the area of trapezoid to the are of the three triangles??(7 votes)
Mr. Garfield realized that there are two ways of calculating the area, and it is understood that, since we are calculating the area of the same figure, both those methods SHOULD give the same result. So he just tried to calculate using both those ways, and saw what it gave him. This is a common way of proof in mathematics - U know something through method A, and you know it using method B also. And since you know that both these, should be giving the same result, by equating them, you might stumble upon wonderful new things :). That is the magic of mathematics.
Beware that sometimes, you may end up with something obvious like a.b = a.b, but sometimes, you might end up proving Pythagoras theorem or that e^i(pi) + 1= 0. :D(17 votes)
At7:38how does he get the 2ab by multiplying (a+b) squared out? I don't understand. Shouldn't it just be a squared plus b squared?(3 votes)
Work it out with real numbers to see how it comes out.
Let a = 3
Let b = 5
So we start with (a + b)^2
Plugging in the numbers, we'd have (3 + 5)^2 or 8-squared which is equal to 64.
But if we presume that (a + b)^2 is equal to a-squared + b-squared, and if we plug in the numbers, we'd have 3-squared ( which is 9) plus 5-squared (which is 25) and added together they make only 34. Not even close to 64.
On the other hand, we see that a^2 + 2ab + b^2 when substituted would work out to:
3-squared + 2(3 times 5) + 5-squared, or 9 + 2(15) + 25, which is equal to 9 + 30 + 25 = 64.(9 votes)
Is Ө just a symbol, or does it have mathematical meaning?(5 votes)
θ is a greek letter like π. We use θ (theta) as a variable mostly for angle measures in trigonometry.(9 votes)
Did anyone understand this? I am very confuzzled.(6 votes)- Yes I did indeed understand this.(2 votes)
Why (A+B) squared is equal to A squared + 2AB + B squared?(2 votes)
First, try to change the expression (A+B)^2 into something manageable and understandable. (A+B)^2 = (A+B) * (A+B). Using the Distributive Property, (A+B) * (A+B) = (A+B) * A + (A+B) * B. To further simplify the expression, (A+B) * A + (A+B) * B = A^2 + AB + AB + B^2, which is equal to A^2 + 2AB + B^2.
'^' is the same as raised to the nth power. '*' is the same as multiplied with.
Hope this helps!(8 votes)
7:32
Video transcript
What we're going
to do in this video is study a proof of
the Pythagorean theorem that was first
discovered, or 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 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 is that it shows that Abraham
Lincoln was not the only US politician or not the only
US 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've just constructed
enough 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 side of
length b, and then you have a side of
length-- let me draw a it so this will be a little
bit taller-- side of length b. And 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. 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 that 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 sides of
length a and length c? What's 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're going to get 180
degrees 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
appears 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
go 180 degrees. So you have theta,
plus 90 minus theta, plus our mystery angle is going
to be equal to 180 degrees. 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, and 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,
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 could 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 a trapezoid so what do we know about
the area of a trapezoid? Well, the area of a
trapezoid is going to be the height of the
trapezoid, which is 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. Since that's this times one
half times a plus a plus b. And the intuition there,
you're taking the height times the average of
this bottom and the top. The average of the
bottom and the top gives you the area
of the trapezoid. Now, how could we
also figure out the area with its
component parts? 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 right triangles. The area of each of them
is 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 triangle. So let's multiply by two. So two times one 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 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. Let me rearrange this. So one half times
a plus b squared is going to be equal
to 2 times one half. Well, that's just
going to be one. So it's going to be equal
to a times b, plus one half c squared. Well, I don't like these
one halfs laying 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. And on the right-hand
side, I am left with 2ab. Trying to keep the
color coding right. And then, 2 times
one half 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 2ab plus 2ab plus b squared. And then, our
right-hand side it's going to be equal to
all of this business. And changing all 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 2ab on
the left-hand side. You have a 2ab on
the right-hand side. Let's subtract 2ab
from both sides. If you subtract 2ab 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 c squared. Very, very exciting. And for that, we have to
thank the 20th president of the United States,
James Garfield. This is really exciting. 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.