Introduction to the Pythagorean Theorem

Right triangles and the Pythagorean Theorem

Welcome to the presentation
on the Pythagorean theorem.

I apologize if my voice
sounds a little horsie.

A little hoarse, not horsie.

I was singing a little
bit too much last night,

so please forgive me.

Well, anyway, we
will now teach you

about the Pythagorean theorem.

And you might have
heard of this before.

As far as I know, it is the
only mathematical theorem

named after the
founder of a religion.

Pythagoras, actually, I
think his whole religion

was based on mathematics.

But I'm no historian here.

So I'll leave that
to the historians.

But anyway, let's get started on
what the Pythagorean theorem is

all about.

If I were to give
you a triangle--

let me give you a
triangle-- and I

were to tell you that it's
not a normal triangle.

It is a right triangle.

And all a right triangle
is is a triangle

that has one side
equal to 90 degrees.

And I'll leave
you to think about

whether it's ever
possible for a triangle

to have more than one
side that's 90 degrees.

But anyway, just granted
that a right triangle

is a side that has
at least-- well,

let me say a right
triangle is a triangle that

has only one side
that's at 90 degrees.

And if you have a right
triangle, what the Pythagorean

theorem allows you to do is
if I give you a right triangle

and I give you two
of the sides, we

can figure out the third side.

So before I throw
the theorem at you,

let me actually give you a
couple of more definitions.

Actually, just one more.

So if this is the right
angle in a right triangle--

it's at 90 degrees.

And we symbolize that by
drawing the angles like this,

kind of like a box instead
of drawing it like a curve,

like that.

I hope I'm not messing
up the drawing too much.

The side opposite the right
angle is called the hypotenuse.

And I really should look up
where this word comes from.

Because I think it's a
large and unwieldy word,

and it's a little
daunting at first.

My sister told me that she
had a math teacher once

who made people memorize it's
a high pot that is in use.

So I don't know if
that helps you or not.

But over time, you'll
use the term hypotenuse

so much that it'll seem
just like a normal word.

Although when you look
at it, it really does

look kind of strange.

Anyway, going back
to definitions,

the hypotenuse is the side
opposite the 90-degree angle.

And if you look at
any right triangle,

you'll also quickly realize that
the hypotenuse is the longest

side of the right triangle.

So I think we're done
now with definitions.

So what does the
Pythagorean theorem tell us?

Well, let's call C is equal to
the length of the hypotenuse.

And let A be the
length of this side.

And let B equal the
length of this side.

What the Pythagorean
theorem tells

us is that A squared plus B
squared is equal to C squared.

Now, that very
simple formula might

be one of the most powerful
formulas in mathematics.

From this, you go into
Euclidean geometry.

You go into trigonometry.

You can do anything
with this formula,

but we'll leave that
to future lectures.

Let's actually
test this formula.

Or not test it--
let's use the formula.

Maybe in another
presentation, I'll

actually do a proof or, at
minimum, a visual proof of it.

I apologize ahead of time that
I'm a bit scatterbrained today.

It's been a while since
I last did a video.

And once again, I told you
I sang a little bit too much

last night.

So my throat is sore.

OK, so we have a triangle.

And remember, it has
to be a right triangle.

So let's say that this is
a right angle right here.

It's 90 degrees.

And if I were to tell you
that this side is of length 4.

And actually, let
me change that.

This side is of length 3.

This side is of length 4.

And we want to figure out
the side of this length.

So the first thing I do when
I look at a right triangle

is I figure out what
the hypotenuse is.

Which side is the hypotenuse?

Well, there's two ways to do it.

There's actually one way.

You look at where
the right angle is.

And it's the side
opposite to that.

So this is the hypotenuse.

This would be C in our formula,
the Pythagorean theorem.

We could call it
whatever we want.

But just for simplicity,
remember A squared plus B

squared is equal to C squared.

So in this case, we see that the
other two sides, each of them

squared, when added together
will equal C squared.

So we get 3 squared
plus 4 squared

is equal to C squared,
where C is our hypotenuse.

So 3 squared is 9, plus
16 is equal to C squared.

25 is equal to C squared.

And C could be plus or minus 5.

But we know that you can't have
a minus 5 length in geometry.

So we know that C is equal to 5.

So using the
Pythagorean theorem,

we just figured out that if we
know the sides-- if one side is

3, the other side
is 4, then we can

use Pythagorean
theorem to figure out

that the hypotenuse of this
triangle has the length 5.

Let's do another example.

Let's say, once again,
this is the right angle.

This side is of length 12.

This slide is of length 6.

And I want to figure
out what this side is.

So let's write down the
Pythagorean theorem.

A squared plus B squared is
equal to C squared, where

C is that length
of the hypotenuse.

So the first thing I want to
do when I look at our triangle

that I just drew is which
side is the hypotenuse.

Well, this right here
is the right angle.

So the hypotenuse is
this side right here.

And we can also
eyeball it and say, oh,

that's definitely the longest
side of this triangle.

So we know that A
squared plus B squared

is equal to 12
squared, which is 144.

Now we know we have
one side, but we

don't have the other side.

So I've got to ask
you a question.

Does it matter which side
we substitute for A or B?

Well, no, just
because A or B-- they

kind of do the same
thing in this formula.

So we could pick any side to
be A other than the hypotenuse.

And we'll pick the
other side to be B.

So let's just say
that this side is B,

and let's say that
this side is A.

So we know what A is.

So we get 6 squared plus
B squared is equal to 144.

So we get 36 plus B
squared is equal to 144.

B squared is equal
to 144 minus 36.

B squared is equal to 112.

Now we've got to simplify what
the square root of 112 is.

And what we did in
those radical modules

probably is helpful here.

So let's see.

B is equal to the
square root of 112.

Let's think about it.

How many times
does 4 go into 112?

4 goes into 120 five times,
so it'll go into it 28 times.

And then 4 goes
into 28 seven times.

So I actually think that
this is equal to 16 times 7.

Am I right?

7 times 10 is 70,
plus 42 is 112.

Right.

So B equals the square
root of 16 times 7.

See, I just factored
that as a product

of a perfect square
and a prime number.

Or actually, it doesn't have
to be a prime number, just

a non-perfect square.

And then I get B is equal
to 4 square roots of 7.

So there we go.

If this is 12, this is 6,
this is 4 square roots of 7.

I think that's all the time I
have now for this presentation.

Right after this, I'll
do one more presentation

where I give a couple of more
Pythagorean theorem problems.

See you soon.