The Pythagorean Theorem

In this video we're going
to get introduced to the

Pythagorean theorem,
which is fun on its own.

But you'll see as you learn
more and more mathematics it's

one of those cornerstone
theorems of really all of math.

It's useful in geometry,
it's kind of the backbone

of trigonometry.

You're also going to use
it to calculate distances

between points.

So it's a good thing to really
make sure we know well.

So enough talk on my end.

Let me tell you what the
Pythagorean theorem is.

So if we have a triangle, and
the triangle has to be a right

triangle, which means that one
of the three angles in the

triangle have to be 90 degrees.

And you specify that it's
90 degrees by drawing that

little box right there.

So that right there is-- let
me do this in a different

color-- a 90 degree angle.

Or, we could call
it a right angle.

And a triangle that has
a right angle in it is

called a right triangle.

So this is called
a right triangle.

Now, with the Pythagorean
theorem, if we know two sides

of a right triangle we can
always figure out

the third side.

And before I show you how to
do that, let me give you one

more piece of terminology.

The longest side of a right
triangle is the side opposite

the 90 degree angle-- or
opposite the right angle.

So in this case it is
this side right here.

This is the longest side.

And the way to figure out where
that right triangle is, and

kind of it opens into
that longest side.

That longest side is
called the hypotenuse.

And it's good to know, because
we'll keep referring to it.

And just so we always are good
at identifying the hypotenuse,

let me draw a couple of
more right triangles.

So let's say I have a triangle
that looks like that.

Let me draw it a
little bit nicer.

So let's say I have a triangle
that looks like that.

And I were to tell you
that this angle right

here is 90 degrees.

In this situation this is the
hypotenuse, because it is

opposite the 90 degree angle.

It is the longest side.

Let me do one more, just
so that we're good at

recognizing the hypotenuse.

So let's say that that is my
triangle, and this is the 90

degree angle right there.

And I think you know how
to do this already.

You go right what
it opens into.

That is the hypotenuse.

That is the longest side.

So once you have identified the
hypotenuse-- and let's say

that that has length C.

And now we're going to
learn what the Pythagorean

theorem tells us.

So let's say that C is equal to
the length of the hypotenuse.

So let's call this
C-- that side is C.

Let's call this side
right over here A.

And let's call this
side over here B.

So the Pythagorean theorem
tells us that A squared-- so

the length of one of the
shorter sides squared-- plus

the length of the other shorter
side squared is going to

be equal to the length of
the hypotenuse squared.

Now let's do that with an
actual problem, and you'll see

that it's actually not so bad.

So let's say that I have a
triangle that looks like this.

Let me draw it.

Let's say this is my triangle.

It looks something like this.

And let's say that they tell us
that this is the right angle.

That this length right here--
let me do this in different

colors-- this length right
here is 3, and that this

length right here is 4.

And they want us to figure
out that length right there.

Now the first thing you want to
do, before you even apply the

Pythagorean theorem, is to
make sure you have your

hypotenuse straight.

You make sure you know
what you're solving for.

And in this circumstance we're
solving for the hypotenuse.

And we know that because this
side over here, it is the side

opposite the right angle.

If we look at the Pythagorean
theorem, this is C.

This is the longest side.

So now we're ready to apply
the Pythagorean theorem.

It tells us that 4 squared--
one of the shorter sides-- plus

3 squared-- the square of
another of the shorter sides--

is going to be equal to this
longer side squared-- the

hypotenuse squared-- is going
to be equal to C squared.

And then you just solve for C.

So 4 squared is the same
thing as 4 times 4.

That is 16.

And 3 squared is the same
thing as 3 times 3.

So that is 9.

And that is going to be
equal to C squared.

Now what is 16 plus 9?

It's 25.

So 25 is equal to C squared.

And we could take the positive
square root of both sides.

I guess, just if you look at
it mathematically, it could

be negative 5 as well.

But we're dealing with
distances, so we only care

about the positive roots.

So you take the principal
root of both sides and

you get 5 is equal to C.

Or, the length of the
longest side is equal to 5.

Now, you can use the
Pythagorean theorem, if we give

you two of the sides, to figure
out the third side no matter

what the third side is.

So let's do another
one right over here.

Let's say that our
triangle looks like this.

And that is our right angle.

Let's say this side over here
has length 12, and let's say

that this side over
here has length 6.

And we want to figure out this
length right over there.

Now, like I said, the first
thing you want to do is

identify the hypotenuse.

And that's going to be the side
opposite the right angle.

We have the right angle here.

You go opposite
the right angle.

The longest side, the
hypotenuse, is right there.

So if we think about the
Pythagorean theorem-- that A

squared plus B squared is
equal to C squared-- 12

you could view as C.

This is the hypotenuse.

The C squared is the
hypotenuse squared.

So you could say
12 is equal to C.

And then we could say that
these sides, it doesn't matter

whether you call one of
them A or one of them B.

So let's just call
this side right here.

Let's say A is equal to 6.

And then we say B-- this
colored B-- is equal

to question mark.

And now we can apply the
Pythagorean theorem.

A squared, which is 6 squared,
plus the unknown B squared is

equal to the hypotenuse
squared-- is equal

to C squared.

Is equal to 12 squared.

And now we can solve for B.

And notice the difference here.

Now we're not solving
for the hypotenuse.

We're solving for one
of the shorter sides.

In the last example we
solved for the hypotenuse.

We solved for C.

So that's why it's always
important to recognize that A

squared plus B squared plus C
squared, C is the length

of the hypotenuse.

So let's just solve for B here.

So we get 6 squared is 36,
plus B squared, is equal

to 12 squared-- this
12 times 12-- is 144.

Now we can subtract 36 from
both sides of this equation.

Those cancel out.

On the left-hand side we're
left with just a B squared

is equal to-- now 144
minus 36 is what?

144 minus 30 is 114.

And then you
subtract 6, is 108.

So this is going to be 108.

So that's what B squared is,
and now we want to take the

principal root, or the
positive root, of both sides.

And you get B is equal
to the square root, the

principal root, of 108.

Now let's see if we can
simplify this a little bit.

The square root of 108.

And what we could do is
we could take the prime

factorization of 108
and see how we can

simplify this radical.

So 108 is the same thing as 2
times 54, which is the same

thing as 2 times 27, which is
the same thing as 3 times 9.

So we have the square root of
108 is the same thing as the

square root of 2 times 2
times-- well actually,

I'm not done.

9 can be factorized
into 3 times 3.

So it's 2 times 2 times
3 times 3 times 3.

And so, we have a couple of
perfect squares in here.

Let me rewrite it a
little bit neater.

And this is all an exercise in
simplifying radicals that you

will bump into a lot while
doing the Pythagorean theorem,

so it doesn't hurt to
do it right here.

So this is the same thing as
the square root of 2 times 2

times 3 times 3 times the
square root of that last

3 right over there.

And this is the same thing.

And, you know, you wouldn't
have to do all of

this on paper.

You could do it in your head.

What is this?

2 times 2 is 4.

4 times 9, this is 36.

So this is the square root of
36 times the square root of 3.

The principal root of 36 is 6.

So this simplifies to
6 square roots of 3.

So the length of B, you could
write it as the square root of

108, or you could say it's
equal to 6 times the

square root of 3.

This is 12, this is 6.

And the square root of 3,
well this is going to be a 1

point something something.

So it's going to be a
little bit larger than 6.