Pythagorean Theorem

Pythagorean Theorem

Let's now talk about what is
easily one of the most famous

theorems in all of
mathematics.

And that's the Pythagorean
theorem.

And it deals with
right triangles.

So a right triangle is a
triangle that has a 90 degree

angle in it.

So the way I drew it
right here, this is

our 90 degree angle.

If you've never seen a 90 degree
angle before, the way

to think about it is, if this
side goes straight left to

right, this side goes straight
up and down.

These sides are perpendicular,
or the angle between them is

90 degrees, or it is
a right angle.

And the Pythagorean theorem
tells us that if we're dealing

with a right triangle-- let me
write that down-- if we're

dealing with a right triangle--
not a wrong

triangle-- if we're dealing with
a right triangle, which

is a triangle that has a right
angle, or a 90 degree angle in

it, then the relationship
between their sides is this.

So this side is a, this side
is b, and this side is c.

And remember, the c that we're
dealing with right here is the

side opposite the
90 degree angle.

It's important to keep track
of which side is which.

The Pythagorean theorem tells us
that if and only if this is

a right triangle, then a squared
plus b squared is

going to be equal
to c squared.

And we can use this
information.

If we know two of these, we
can then use this theorem,

this formula to solve
for the third.

And I'll give you one more piece
of terminology here.

This long side, the side that
is the longest side of our

right triangle, the side that
is opposite of our right

angle, this right here-- it's
c in this example-- this is

called a hypotenuse.

A very fancy word for
a very simple idea.

The longest side of a right
triangle, the side that is

opposite the 90 degree angle,
is called the hypotentuse.

Now that we know the Pythagorean
theorem, let's

actually use it.

Because it's one thing to know
something, but it's a lot more

fun to use it.

So let's say I have the
following right triangle.

Let me draw it a little
bit neater than that.

It's a right triangle.

This side over here
has length 9.

This side over here
has length 7.

And my question is, what
is this side over here?

Maybe we can call that--
we'll call that c.

Well, c, in this case, once
again, it is the hypotenuse.

It is the longest side.

So we know that the sum of the
squares of the other side is

going to be equal
to c squared.

So by the Pythagorean theorem,
9 squared plus 7 squared is

going to be equal
to c squared.

9 squared is 81, plus
7 squared is 49.

80 plus 40 is 120.

Then we're going to have the 1
plus the 9, that's another 10,

so this is going to
be equal to 130.

So let me write it this way.

The left-hand side is going to
be equal to 130, and that is

equal to c squared.

So what's c going
to be equal to?

Let me rewrite it over here.

c squared is equal to 130, or
we could say that c is equal

to the square root of 130.

And notice, I'm only taking
the principal root here,

because c has to be positive.

We're dealing with a distance,
so we can't take the negative

square root.

So we'll only take
the principal

square root right here.

And if we want to simplify this
a little bit, we know how

to simplify our radicals.

130 is 2 times 65, which
is 5 times 13.

Well, these are all prime
numbers, so that's about as

simple as I can get.
c is equal to the

square root of 130.

Let's do another one of these.

Maybe I want to keep this
Pythagorean theorem right

there, just so we always
remember what

we're referring to.

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

Let's see.

Let's say it looks like that.

And this is the right
angle, up here.

Let's say that this side,
I'm going to call it a.

The side, it's going
to have length 21.

And this side right here is
going to be of length 35.

So your instinct to solve for a,
might say, hey, 21 squared

plus 35 squared is going to
be equal to a squared.

But notice, in this situation,
35 is a hypotenuse.

35 is our c.

It's the longest side of
our right triangle.

So what the Pythagorean theorem
tells us is that a

squared plus the other
non-longest side-- the other

non-hypotenuse squared-- so a
squared plus 21 squared is

going to be equal
to 35 squared.

You always have to remember, the
c squared right here, the

c that we're talking about, is
always going to be the longest

side of your right triangle.

The side that is opposite
of our right angle.

This is the side that's opposite
of the right angle.

So a squared plus 21 squared
is equal to 35 squared.

And what do we have here?

So 21 squared-- I'm tempted to
use a calculator, but I won't.

So 21 times 21: 1 times 21
is 21, 2 times 21 is 42.

It is 441.

35 squared.

Once again, I'm tempted to use
a calculator, but I won't.

35 times 35: 5 times 5 is 25.

Carry the 2.

5 times 3 is 15, plus 2 is 17.

Put a 0 here, get rid
of that thing.

3 times 5 is 15.

3 times 3 is 9, plus 1 is 10.

So it is 11-- let me do it in
order-- 5 plus 0 is 5, 7 plus

5 is 12, 1 plus 1 is 2,
bring down the 1.

1225.

So this tells us that a squared
plus 441 is going to

be equal to 35 squared,
which is 1225.

Now, we could subtract
441 from both

sides of this equation.

The left-hand side just
becomes a squared.

The right-hand side,
what do we get?

We get 5 minus 1 is 4.

We want to-- let me write this
a little bit neater here.

Minus 441.

So the left-hand side, once
again, they cancel out. a

squared is equal to-- and then
on the right-hand side, what

do we have to do?

That's larger than that, but 2
is not larger than 4, so we're

going to have to borrow.

So that becomes a 12, or
regrouped, depending on how

you want to view it.

That becomes a 1.

1 is not greater than
4, so we're going to

have to borrow again.

Get rid of that.

And then this becomes an 11.

5 minus 1 is 4.

12 minus 4 is 8.

11 minus 4 is 7.

So a squared is equal to 784.

And we could write, then,
that a is equal to the

square root of 784.

And once again, I'm very tempted
to use a calculator,

but let's, well, let's not.

Let's not use it.

So this is 2 times, what?

392.

And then this-- 390 times
2 is 78, yeah.

And then this is
2 times, what?

This is 2 times 196.

That's right.

190 times 2 is-- yeah,
that's 2 times 196.

196 is 2 times-- I want to
make sure I don't make a

careless mistake.

196 is 2 times 98.

Let's keep going down here.

98 is 2 times 49.

And, of course, we know
what that is.

So notice, we have 2 times
2, times 2, times 2.

So this is 2 to the
fourth power.

So it's 16 times 49.

So a is equal to the square
root of 16 times 49.

I picked those numbers because
they're both perfect squares.

So this is equal to the square
root of 16 is 4, times the

square root of 49 is 7.

It's equal to 28.

So this side right here is going
to be equal to 28, by

the Pythagorean theorem.

Let's do one more of these.

Can never get enough practice.

So let's say I have
another triangle.

I'll draw this one big.

There you go.

That's my triangle.

That is the right angle.

This side is 24.

This side is 12.

We'll call this side
right here b.

Now, once again, always identify
the hypotenuse.

That's the longest side,
the side opposite

the 90 degree angle.

You might say, hey, I don't know
that's the longest side.

I don't know what b is yet.

How do I know this is longest?

And there, in that situation,
you say, well, it's the side

opposite the 90 degree angle.

So if that's the hypotenuse,
then this squared plus that

squared is going to be
equal to 24 squared.

So the Pythagorean theorem-- b
squared plus 12 squared is

equal to 24 squared.

Or we could subtract 12 squared
from both sides.

We say, b squared is equal to
24 squared minus 12 squared,

which we know is 144, and that
b is equal to the square root

of 24 squared minus
12 squared.

Now I'm tempted to use a
calculator, and I'll give into

the temptation.

So let's do it.

The last one was so painful,
I'm still recovering.

So 24 squared minus 12 squared
is equal to 24.78.

So this actually turns into--
let me do it without a-- well,

I'll do it halfway.

24 squared minus 12 squared
is equal to 432.

So b is equal to the
square root of 432.

And let's factor this again.

We saw what the answer is, but
maybe we can write it in kind

of a simplified radical form.

So this is 2 times 216.

216, I believe, is
a-- let me see.

I believe that's a
perfect square.

So let me take the square
root of 216.

Nope, not a perfect square.

So 216, let's just keep going.

216 is 2 times 108.

108 is, we could say,
4 times what?

25 plus another 2-- 4 times
27, which is 9 times 3.

So what do we have here?

We have 2 times 2, times 4, so
this right here is a 16.

16 times 9 times 3.

Is that right?

I'm using a different
calculator.

16 times 9 times 3
is equal to 432.

So this is going to be equal
to-- b is equal to the square

root of 16 times 9, times 3,
which is equal to the square

root of 16, which is 4 times the
square root of 9, which is

3, times the square root
of 3, which is equal

to 12 roots of 3.

So b is 12 times the
square root of 3.

Hopefully you found
that useful.