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.