We've already seen several videos where we, one, learned
what the Pythagorean theorem is and how it can be applied
to find the sides of right triangles, and even to find
the distances between points on the Cartesian plane, so
this is the Pythagorean theorem.
And just as a bit of review, it tells us that if I have a
right triangle, and if c is the side opposite the right
angle, right?
That's my right angle. c is that side.
It's the longest side.
We call that the hypotenuse.
Then the square of the other sides, the sum of the squares
of the other sides, will be equal to c squared.
So if this side is a and this side is b, the Pythagorean
theorem tells us that a squared plus b squared is
going to be equal to c squared.
And we did several problems where we used this to solve
different triangles.
But what I want to show you in this video is really a
visual-- you can view it as a visual proof of the
Pythagorean theorem to see why this actually makes sense.
And to understand that, I'm going to draw a little bit of
a diagram here.
So let me draw a little right triangle here.
So let's say that that is my right triangle.
And let's say that this is a, this is b,
that is my right angle.
Then that right there, this right here, that length right
there is c.
Now, what I'm going to do is draw this exact same triangle,
but I'm going to rotate it a little bit.
You're going to see what I'm talking about.
This right here is the exact same triangle that I just
drew, except I rotated it.
So this is a, this is b, and now this is c.
This is the exact-- you could do this, you know.
You could take a ruler out and actually do it.
Pick an a, b, and c, and you can actually rotate this
triangle around to the right like this.
You could just rotate it like that, and you would get this
triangle, just like that.
Now, I'm going to rotate it again.
And I'm just going to imagine copy and pasting it.
And then I'll get this triangle, just like that.
And once again, this is a, this is b, and this is c.
These triangles are all the exact same triangle, the exact
same measure.
Now let me rotate this again, and I'll get this triangle.
I will get this triangle right here.
And once again, this is a.
This length right here is a.
This length right here is b.
And this is the hypotenuse of this right angle, this is c.
Now what do we have here?
We have all of these right angles, and inside, I have
this other square that is inscribed
into this right angle.
And I can prove to you, if you like, that these are also
right angles.
And you know, why don't I just show it to you.
You may or may not already be familiar that the angles in a
triangle add up to 180.
So if you call this angle x, you call this angle y, we know
that x plus y plus 90 degrees have to be equal to 180.
So x plus y plus 90 have to be equal to 180.
Or you could say that if you subtract 90 from both sides of
this equation, x plus y are going to be equal to 90.
I just subtracted 90 from both sides.
This disappears.
180 minus 90 is 90.
And this x and y, they're all over.
If this angle is x right here, so is this angle because this
is the exact same triangle.
So is this angle.
If this angle is y, so is that angle, so is that angle, and
so is that angle.
Now, you may or may not already know that if you have
three angles that form a line-- so let me draw it.
If you have three angles that form a line like this, so
let's say we have angle a, angle b-- let me draw b a
little bit different-- and angle c, just like that, and
together they kind of form you could imagine a semi-circle or
they go from one line all the way back, that a, b plus c are
going to be equal to 180.
a plus b plus c is going to be equal to 180 degrees.
They complete a 180-degree arc.
Now, if you look at, let's say, this point right here on
our diagram, you can see that angle x plus this angle right
here, which I'll call angle z, x plus z plus y is going to be
equal to 180 degrees.
They form a line.
So x plus z plus y, or maybe I should write it as x plus y
plus z is equal to 180.
Now, we already know x plus y is equal to 90, so this is
equal to 90.
So 90 plus z is equal to 180.
z minus-- well, both sides of this equation minus 90
degrees, you get 180 minus 90.
I just subtract 90 from both sides, so z
is equal to 90 degrees.
So these are all 90-degree angles.
Their sides are all the same length of c, so this is a
square in the middle of this diagram.
Now, let's think a little bit about what the area
of the square is.
And there's two ways we can come up with the
area of that square.
So let me draw something here.
So the first easy way to think of the area of the square,
well, it's a square.
This length is c, that length is c.
To figure out the area, I just multiply one side
by the other side.
It's c times c, or c squared.
So if I want the area of that right there, it is c
squared, c times c.
Area is equal to c squared.
But how else could we figure out that area?
Well, we could say that would be the area of this larger
square, of this whole square over here minus the area of
these triangles.
That also would give us the area of that inside square.
So what is the area of this outside, bigger square?
Well, each of its sides are a plus b.
And so you take a plus b.
If you want the area, you could say, a plus b times a
plus b, or a plus b squared, right?
That's the area of the outer square, but we still have to
get the area of our inner square.
Area of small square.
We still have to subtract out the area of these triangles.
And what are the areas of these triangles?
Well, we have to subtract out.
The area of each of them is 1/2 times base times height,
or base times height, ab, 1/2 ab.
That's the area of each of these triangles.
Now we have 1, 2, 3, 4 of these triangles.
So it's 4 times 1/2 ab.
This is another way of getting the area of this inside
smaller square.
Now let's just do a little bit of algebra-- we're pretty
familiar with this-- to essentially simplify the
right-hand side of this equation right here.
So a plus b times times a plus b, a times a is a squared,
plus 2 ab, plus b squared.
And then we have minus-- what do we have right here?
Minus 2ab, right?
4 times 1/2 is 2, so minus 2 ab.
Well, these cancel out.
2ab minus 2ab, we're just left with a squared and b squared.
So the area of this inside circle, which is c squared,
it's also equal to a squared plus b squared.
And I've just to shown you a visual proof for the
Pythagorean theorem, because remember, a and b were the
smaller sides of a right triangle having c as its
hypotenuse.
And from there, we constructed this bigger diagram.
Hopefully, you found that fun.