Let's continue with the
30, 60, 90 triangles.

So just review what we just
learned, or hopefully learned--

at minimum what we just saw,
--is if we have a 30, 60, 90 --

and once again, remember: this
is only applies to 30, 60, 90

triangles --and if I were to
say the hypotenuse is of length

h, we learned that the side
opposite the 30-degree angle,

and this is the shortest side
of the triangle, is going to be

h over 2, or 1/2 times
the hypotenuse.

And we also learned that the
longer side, or the side

opposite the 60-degree side,
is equal to the square

root of 3 over 2 times h.

So let's do a problem where
we use this information.

Let's say I had this
triangle right here.

It's a 90-degree triangle;
let's say that this

is 30 degrees.

And we could also figure out
obviously if that's 30, this

is 90, that this is
also 60 degrees.

And let's say that the
hypotenuse is 12.

The length is 12 and we know
that this is the hypotenuse

because it's opposite
the right angle.

What is the side right here?

Well, is the side opposite the
60-degree angle, or is it

opposite the 30-degree angle?

It's the 30-degree angle
that opens into it, right?

I drew this triangle a little
bit different on purpose.

The 30-degree angle opens up
into this side, and it's

also the shortest side.

We learned that the side
opposite the 30-degree angle is

half the hypotenuse, so the
hypotenuse is 12;

this would be 6.

And this side, which is
opposite the 60-degree side, is

equal to the square root of 3
over 2 times the hypotenuse.

So it's the square root of 3
over 2 times 12, or it's just

equal to 6 square roots of 3.

Another interesting thing is,
of course, the longer

non-hypotenuse side is square
root of 3 times longer

than the short side.

I don't confuse you too much.

Let's do another one.

Let's say this is 30 degrees--
it's our right triangle --and I

were to tell you that this side
right here is 5, what is

the length of this side?

Well first of all let's
figure out what we have.

5 is which side?

So if this is 30 degrees,
we know that this is

going to be 60 degrees.

So 5 is opposite the 60-degree
side, and x is the hypotenuse.

Since x is opposite the
90-degree side, it's also

the longest side of
the right triangle.

So we know from our formula
that 5 is equal to the square

root of 3 over 2 times the
hypotenuse, which in

this example is x.

And now we just solve for x.

We can multiply both
sides by the inverse

of this coefficient.

So if you just multiply 2 times
the square root of 3-- can

ignore this --we get 10 over
the square root of three here.

And, of course, this 2
cancels out with this 2.

This square root of 3 cancels
out this square root

of 3 is equal to x.

And now if you watched the last
couple of presentations, you

realize that this could be the
right answer, but we have a

square root of 3 in the
denominator, which people don't

like because it's an irrational
number in the denominator.

And I guess we could
have a debate as to

why that might be bad.

So let's rationalize
this denominator.

We say x is equal to 10 over
the square to 3; to rationalize

this denominator we can
multiply the numerator and the

denominator by the
square root of 3.

Because as long as we multiply
the numerator and the

denominator by the same thing,
it's like multiplying by 1.

So this is equal to 10 square
roots of 3 over square root of

3 times square of 3;
well that's just 3.

So x equals 10 square
roots of 3 over 3.

That's the hypotenuse.

I know I confused you.

And, of course, if this is 10
square root of 3 over 3--

that's the hypotenuse --we know
that the 30-degree side-- this

is 30 degrees --we know the
30-degree side is half of

that, so it's 5 square
root of 3 over 3.

Anyway, I think that might
give you a sense of the

30, 60, 90 triangles.

I think you might be ready now
to try some of the level two

Pythagorean Theorem problems.

Have fun.