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The Pythagorean Theorem

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    In this video we're going
    to get introduced to the
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    Pythagorean theorem,
    which is fun on its own.
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    But you'll see as you learn
    more and more mathematics it's
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    one of those cornerstone
    theorems of really all of math.
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    It's useful in geometry,
    it's kind of the backbone
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    of trigonometry.
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    You're also going to use
    it to calculate distances
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    between points.
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    So it's a good thing to really
    make sure we know well.
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    So enough talk on my end.
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    Let me tell you what the
    Pythagorean theorem is.
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    So if we have a triangle, and
    the triangle has to be a right
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    triangle, which means that one
    of the three angles in the
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    triangle have to be 90 degrees.
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    And you specify that it's
    90 degrees by drawing that
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    little box right there.
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    So that right there is-- let
    me do this in a different
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    color-- a 90 degree angle.
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    Or, we could call
    it a right angle.
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    And a triangle that has
    a right angle in it is
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    called a right triangle.
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    So this is called
    a right triangle.
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    Now, with the Pythagorean
    theorem, if we know two sides
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    of a right triangle we can
    always figure out
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    the third side.
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    And before I show you how to
    do that, let me give you one
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    more piece of terminology.
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    The longest side of a right
    triangle is the side opposite
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    the 90 degree angle-- or
    opposite the right angle.
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    So in this case it is
    this side right here.
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    This is the longest side.
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    And the way to figure out where
    that right triangle is, and
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    kind of it opens into
    that longest side.
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    That longest side is
    called the hypotenuse.
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    And it's good to know, because
    we'll keep referring to it.
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    And just so we always are good
    at identifying the hypotenuse,
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    let me draw a couple of
    more right triangles.
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    So let's say I have a triangle
    that looks like that.
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    Let me draw it a
    little bit nicer.
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    So let's say I have a triangle
    that looks like that.
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    And I were to tell you
    that this angle right
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    here is 90 degrees.
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    In this situation this is the
    hypotenuse, because it is
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    opposite the 90 degree angle.
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    It is the longest side.
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    Let me do one more, just
    so that we're good at
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    recognizing the hypotenuse.
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    So let's say that that is my
    triangle, and this is the 90
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    degree angle right there.
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    And I think you know how
    to do this already.
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    You go right what
    it opens into.
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    That is the hypotenuse.
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    That is the longest side.
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    So once you have identified the
    hypotenuse-- and let's say
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    that that has length C.
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    And now we're going to
    learn what the Pythagorean
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    theorem tells us.
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    So let's say that C is equal to
    the length of the hypotenuse.
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    So let's call this
    C-- that side is C.
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    Let's call this side
    right over here A.
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    And let's call this
    side over here B.
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    So the Pythagorean theorem
    tells us that A squared-- so
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    the length of one of the
    shorter sides squared-- plus
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    the length of the other shorter
    side squared is going to
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    be equal to the length of
    the hypotenuse squared.
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    Now let's do that with an
    actual problem, and you'll see
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    that it's actually not so bad.
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    So let's say that I have a
    triangle that looks like this.
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    Let me draw it.
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    Let's say this is my triangle.
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    It looks something like this.
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    And let's say that they tell us
    that this is the right angle.
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    That this length right here--
    let me do this in different
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    colors-- this length right
    here is 3, and that this
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    length right here is 4.
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    And they want us to figure
    out that length right there.
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    Now the first thing you want to
    do, before you even apply the
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    Pythagorean theorem, is to
    make sure you have your
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    hypotenuse straight.
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    You make sure you know
    what you're solving for.
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    And in this circumstance we're
    solving for the hypotenuse.
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    And we know that because this
    side over here, it is the side
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    opposite the right angle.
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    If we look at the Pythagorean
    theorem, this is C.
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    This is the longest side.
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    So now we're ready to apply
    the Pythagorean theorem.
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    It tells us that 4 squared--
    one of the shorter sides-- plus
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    3 squared-- the square of
    another of the shorter sides--
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    is going to be equal to this
    longer side squared-- the
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    hypotenuse squared-- is going
    to be equal to C squared.
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    And then you just solve for C.
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    So 4 squared is the same
    thing as 4 times 4.
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    That is 16.
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    And 3 squared is the same
    thing as 3 times 3.
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    So that is 9.
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    And that is going to be
    equal to C squared.
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    Now what is 16 plus 9?
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    It's 25.
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    So 25 is equal to C squared.
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    And we could take the positive
    square root of both sides.
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    I guess, just if you look at
    it mathematically, it could
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    be negative 5 as well.
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    But we're dealing with
    distances, so we only care
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    about the positive roots.
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    So you take the principal
    root of both sides and
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    you get 5 is equal to C.
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    Or, the length of the
    longest side is equal to 5.
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    Now, you can use the
    Pythagorean theorem, if we give
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    you two of the sides, to figure
    out the third side no matter
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    what the third side is.
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    So let's do another
    one right over here.
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    Let's say that our
    triangle looks like this.
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    And that is our right angle.
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    Let's say this side over here
    has length 12, and let's say
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    that this side over
    here has length 6.
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    And we want to figure out this
    length right over there.
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    Now, like I said, the first
    thing you want to do is
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    identify the hypotenuse.
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    And that's going to be the side
    opposite the right angle.
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    We have the right angle here.
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    You go opposite
    the right angle.
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    The longest side, the
    hypotenuse, is right there.
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    So if we think about the
    Pythagorean theorem-- that A
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    squared plus B squared is
    equal to C squared-- 12
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    you could view as C.
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    This is the hypotenuse.
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    The C squared is the
    hypotenuse squared.
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    So you could say
    12 is equal to C.
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    And then we could say that
    these sides, it doesn't matter
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    whether you call one of
    them A or one of them B.
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    So let's just call
    this side right here.
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    Let's say A is equal to 6.
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    And then we say B-- this
    colored B-- is equal
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    to question mark.
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    And now we can apply the
    Pythagorean theorem.
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    A squared, which is 6 squared,
    plus the unknown B squared is
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    equal to the hypotenuse
    squared-- is equal
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    to C squared.
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    Is equal to 12 squared.
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    And now we can solve for B.
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    And notice the difference here.
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    Now we're not solving
    for the hypotenuse.
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    We're solving for one
    of the shorter sides.
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    In the last example we
    solved for the hypotenuse.
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    We solved for C.
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    So that's why it's always
    important to recognize that A
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    squared plus B squared plus C
    squared, C is the length
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    of the hypotenuse.
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    So let's just solve for B here.
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    So we get 6 squared is 36,
    plus B squared, is equal
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    to 12 squared-- this
    12 times 12-- is 144.
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    Now we can subtract 36 from
    both sides of this equation.
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    Those cancel out.
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    On the left-hand side we're
    left with just a B squared
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    is equal to-- now 144
    minus 36 is what?
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    144 minus 30 is 114.
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    And then you
    subtract 6, is 108.
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    So this is going to be 108.
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    So that's what B squared is,
    and now we want to take the
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    principal root, or the
    positive root, of both sides.
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    And you get B is equal
    to the square root, the
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    principal root, of 108.
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    Now let's see if we can
    simplify this a little bit.
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    The square root of 108.
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    And what we could do is
    we could take the prime
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    factorization of 108
    and see how we can
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    simplify this radical.
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    So 108 is the same thing as 2
    times 54, which is the same
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    thing as 2 times 27, which is
    the same thing as 3 times 9.
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    So we have the square root of
    108 is the same thing as the
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    square root of 2 times 2
    times-- well actually,
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    I'm not done.
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    9 can be factorized
    into 3 times 3.
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    So it's 2 times 2 times
    3 times 3 times 3.
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    And so, we have a couple of
    perfect squares in here.
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    Let me rewrite it a
    little bit neater.
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    And this is all an exercise in
    simplifying radicals that you
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    will bump into a lot while
    doing the Pythagorean theorem,
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    so it doesn't hurt to
    do it right here.
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    So this is the same thing as
    the square root of 2 times 2
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    times 3 times 3 times the
    square root of that last
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    3 right over there.
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    And this is the same thing.
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    And, you know, you wouldn't
    have to do all of
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    this on paper.
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    You could do it in your head.
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    What is this?
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    2 times 2 is 4.
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    4 times 9, this is 36.
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    So this is the square root of
    36 times the square root of 3.
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    The principal root of 36 is 6.
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    So this simplifies to
    6 square roots of 3.
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    So the length of B, you could
    write it as the square root of
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    108, or you could say it's
    equal to 6 times the
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    square root of 3.
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    This is 12, this is 6.
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    And the square root of 3,
    well this is going to be a 1
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    point something something.
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    So it's going to be a
    little bit larger than 6.
Title:
The Pythagorean Theorem
Video Language:
English
Duration:
10:46
dhbot edited English subtitles for The Pythagorean Theorem Jun 10, 2020, 4:58 PM
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Amara Bot edited English subtitles for The Pythagorean Theorem Jan 9, 2011, 9:54 PM
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