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Let's continue with the
30, 60, 90 triangles.
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So just review what we just
learned, or hopefully learned--
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at minimum what we just saw,
--is if we have a 30, 60, 90 --
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and once again, remember: this
is only applies to 30, 60, 90
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triangles --and if I were to
say the hypotenuse is of length
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h, we learned that the side
opposite the 30-degree angle,
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and this is the shortest side
of the triangle, is going to be
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h over 2, or 1/2 times
the hypotenuse.
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And we also learned that the
longer side, or the side
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opposite the 60-degree side,
is equal to the square
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root of 3 over 2 times h.
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So let's do a problem where
we use this information.
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Let's say I had this
triangle right here.
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It's a 90-degree triangle;
let's say that this
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is 30 degrees.
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And we could also figure out
obviously if that's 30, this
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is 90, that this is
also 60 degrees.
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And let's say that the
hypotenuse is 12.
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The length is 12 and we know
that this is the hypotenuse
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because it's opposite
the right angle.
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What is the side right here?
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Well, is the side opposite the
60-degree angle, or is it
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opposite the 30-degree angle?
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It's the 30-degree angle
that opens into it, right?
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I drew this triangle a little
bit different on purpose.
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The 30-degree angle opens up
into this side, and it's
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also the shortest side.
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We learned that the side
opposite the 30-degree angle is
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half the hypotenuse, so the
hypotenuse is 12;
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this would be 6.
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And this side, which is
opposite the 60-degree side, is
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equal to the square root of 3
over 2 times the hypotenuse.
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So it's the square root of 3
over 2 times 12, or it's just
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equal to 6 square roots of 3.
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Another interesting thing is,
of course, the longer
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non-hypotenuse side is square
root of 3 times longer
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than the short side.
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I don't confuse you too much.
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Let's do another one.
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Let's say this is 30 degrees--
it's our right triangle --and I
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were to tell you that this side
right here is 5, what is
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the length of this side?
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Well first of all let's
figure out what we have.
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5 is which side?
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So if this is 30 degrees,
we know that this is
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going to be 60 degrees.
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So 5 is opposite the 60-degree
side, and x is the hypotenuse.
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Since x is opposite the
90-degree side, it's also
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the longest side of
the right triangle.
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So we know from our formula
that 5 is equal to the square
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root of 3 over 2 times the
hypotenuse, which in
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this example is x.
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And now we just solve for x.
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We can multiply both
sides by the inverse
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of this coefficient.
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So if you just multiply 2 times
the square root of 3-- can
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ignore this --we get 10 over
the square root of three here.
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And, of course, this 2
cancels out with this 2.
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This square root of 3 cancels
out this square root
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of 3 is equal to x.
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And now if you watched the last
couple of presentations, you
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realize that this could be the
right answer, but we have a
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square root of 3 in the
denominator, which people don't
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like because it's an irrational
number in the denominator.
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And I guess we could
have a debate as to
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why that might be bad.
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So let's rationalize
this denominator.
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We say x is equal to 10 over
the square to 3; to rationalize
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this denominator we can
multiply the numerator and the
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denominator by the
square root of 3.
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Because as long as we multiply
the numerator and the
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denominator by the same thing,
it's like multiplying by 1.
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So this is equal to 10 square
roots of 3 over square root of
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3 times square of 3;
well that's just 3.
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So x equals 10 square
roots of 3 over 3.
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That's the hypotenuse.
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I know I confused you.
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And, of course, if this is 10
square root of 3 over 3--
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that's the hypotenuse --we know
that the 30-degree side-- this
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is 30 degrees --we know the
30-degree side is half of
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that, so it's 5 square
root of 3 over 3.
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Anyway, I think that might
give you a sense of the
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30, 60, 90 triangles.
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I think you might be ready now
to try some of the level two
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Pythagorean Theorem problems.
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Have fun.
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