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Sorry for starting the
presentation with a cough.
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I think I still have a little
bit of a bug going around.
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But now I want to continue
with the 45-45-90 triangles.
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So in the last presentation we
learned that either side of a
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45-45-90 triangle that isn't
the hypotenuse is equal to the
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square route of 2 over 2
times the hypotenuse.
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Let's do a couple
of more problems.
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So if I were to tell you that
the hypotenuse of this
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triangle-- once again,
this only works for
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45-45-90 triangles.
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And if I just draw one 45
you know the other angle's
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got to be 45 as well.
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If I told you that the
hypotenuse here is,
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let me say, 10.
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We know this is a hypotenuse
because it's opposite
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the right angle.
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And then I would ask you
what this side is, x.
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Well we know that x is equal
to the square root of 2 over
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2 times the hypotenuse.
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So that's square root
of 2 over 2 times 10.
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Or, x is equal to 5
square roots of 2.
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Right?
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10 divided by 2.
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So x is equal to 5
square roots of 2.
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And we know that this side
and this side are equal.
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Right?
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I guess we know this is an
isosceles triangle because
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these two angles are the same.
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So we also that this
side is 5 over 2.
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And if you're not
sure, try it out.
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Let's try the
Pythagorean theorem.
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We know from the Pythagorean
theorem that 5 root 2 squared,
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plus 5 root 2 squared is equal
to the hypotenuse squared,
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where the hypotenuse is 10.
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Is equal to 100.
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Or this is just 25 times 2.
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So that's 50.
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But this is 100 up here.
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Is equal to 100.
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And we know, of course,
that this is true.
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So it worked.
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We proved it using the
Pythagorean theorem, and
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that's actually how we
came up with this formula
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in the first place.
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Maybe you want to go back to
one of those presentations
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if you forget how we
came up with this.
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I'm actually now going
to introduce another
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type of triangle.
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And I'm going to do it the same
way, by just posing a problem
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to you and then using the
Pythagorean theorem
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to figure it out.
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This is another type
of triangle called a
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30-60-90 triangle.
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And if I don't have time
for this I will do
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another presentation.
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Let's say I have a
right triangle.
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That's not a pretty one,
but we use what we have.
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That's a right angle.
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And if I were to tell you that
this is a 30 degree angle.
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Well we know that the
angles in a triangle
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have to add up to 180.
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So if this is 30, this is 90,
and let's say that this is x.
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x plus 30 plus 90 is equal to
180, because the angles in
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a triangle add up to 180.
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We know that x is equal to 60.
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Right?
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So this angle is 60.
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And this is why it's called a
30-60-90 triangle-- because
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that's the names of the three
angles in the triangle.
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And if I were to tell you that
the hypotenuse is-- instead of
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calling it c, like we always
do, let's call it h-- and I
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want to figure out the other
sides, how do we do that?
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Well we can do that
using pretty much the
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Pythagorean theorem.
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And here I'm going to
do a little trick.
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Let's draw another copy of this
triangle, but flip it over
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draw it the other side.
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And this is the same triangle,
it's just facing the
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other direction.
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Right?
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If this is 90 degrees
we know that these two
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angles are supplementary.
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You might want to review the
angles module if you forget
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that two angles that share kind
of this common line would
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add up to 180 degrees.
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So this is 90, this
will also be 90.
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And you can eyeball it.
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It makes sense.
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And since we flip it, this
triangle is the exact
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same triangle as this.
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It's just flipped
over the other side.
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We also know that this
angle is 30 degrees.
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And we also know that this
angle is 60 degrees.
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Right?
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Well if this angle is 30
degrees and this angle is 30
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degrees, we also know that this
larger angle-- goes all the way
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from here to here--
is 60 degrees.
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Right?
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Well if this angle is 60
degrees, this top angle is 60
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degrees, and this angle on the
right is 60 degrees, then we
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know from the theorem that we
learned when we did 45-45-90
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triangles that if these two
angles are the same then the
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sides that they don't share
have to be the same as well.
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So what are the sides
they don't share?
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Well, it's this side
and this side.
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So if this side is h
then this side is h.
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Right?
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But this angle is
also 60 degrees.
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So if we look at this 60
degrees and this 60 degrees, we
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know that the sides that they
don't share are also equal.
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Well they share this side, so
the sides that they don't share
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are this side and this side.
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So this side is h, we also
know that this side is h.
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Right?
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So it turns out that if you
have 60 degrees, 60 degrees,
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and 60 degrees that all the
sides have the same lengths, or
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it's an equilateral triangle.
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And that's something
to keep in mind.
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And that makes sense too,
because an equilateral triangle
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is symmetric no matter
how you look at it.
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So it makes sense that all of
the angles would be the same
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and all of the sides would
have the same length.
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But, hm.
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When we originally did this
problem we only used half of
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this equilateral triangle.
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So we know this whole side
right here is of length h.
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But if that whole side is of
length h, well then this side
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right here, just the base of
our original triangle-- and I'm
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trying to be messy on purpose.
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We tried another color.
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This is going to be
half of that side.
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Right?
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Because that's h over 2,
and this is also h over 2.
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Right over here.
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So if we go back to our
original triangle, and we said
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that this is 30 degrees and
that this is the hypotenuse,
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because it's opposite the right
angle, we know that the side
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opposite the 30 degree side
is 1/2 of the hypotenuse.
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And just a reminder,
how did we do that?
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Well we doubled the triangle.
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Turned it into an
equilateral triangle.
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Figured out this whole
side has to be the same
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as the hypotenuse.
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And this is 1/2 of
that whole side.
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So it's 1/2 of the hypotenuse.
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So let's remember that.
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The side opposite the 30 degree
side is 1/2 of the hypotenuse.
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Let me redraw that on another
page, because I think
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this is getting messy.
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So going back to what
I had originally.
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This is a right angle.
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This is the hypotenuse--
this side right here.
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If this is 30 degrees, we just
derived that the side opposite
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the 30 degrees-- it's like what
the angle is opening into--
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that this is equal to
1/2 the hypotenuse.
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If this is equal to 1/2
the hypotenuse then what
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is this side equal to?
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Well, here we can use the
Pythagorean theorem again.
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We know that this side squared
plus this side squared-- let's
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call this side A-- is
equal to h squared.
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So we have 1/2 h squared plus A
squared is equal to h squared.
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This is equal to h squared
over 4 plus A squared,
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is equal to h squared.
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Well, we subtract h
squared from both sides.
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We get A squared is equal to h
squared minus h squared over 4.
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So this equals h squared
times 1 minus 1/4.
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This is equal to 3/4 h squared.
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And once going that's
equal to A squared.
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I'm running out of space,
so I'm going to go all
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the way over here.
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So take the square root of both
sides, and we get A is equal
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to-- the square root of 3/4
is the same thing as the
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square root of 3 over 2.
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And then the square root
of h squared is just h.
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And this A-- remember,
this isn't an area.
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This is what decides the
length of the side.
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I probably shouldn't
have used A.
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But this is equal to the square
root of 3 over 2, times h.
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So there.
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We've derived what all the
sides relative to the
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hypotenuse are of a
30-60-90 triangle.
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So if this is a 60 degree side.
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So if we know the hypotenuse
and we know this is a 30-60-90
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triangle, we know the side
opposite the 30 degree side
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is 1/2 the hypotenuse.
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And we know the side opposite
the 60 degree side is the
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square root of 3 over 2,
times the hypotenuse.
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In the next module I'll show
you how using this information,
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which you may or may not want
to memorize-- it's probably
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good to memorize and practice
with, because it'll make you
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very fast on standardized
tests-- how we can use this
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information to solve the sides
of a 30-60-90 triangle
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very quickly.
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See you in the next
presentation.
