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Similar triangles

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    Hello.
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    I will now introduce you to the
    concept of similar triangles.
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    Let me write that down.
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    6
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    So in everyday life what
    does similar mean?
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    8
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    Well, if two things are similar
    they're kind of the same but
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    they're not the same thing or
    they're not identical, right?
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    That's the same thing
    for triangles.
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    So similar triangles are
    two triangles that have
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    all the same angles.
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    14
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    For example, let me draw
    two similar triangles.
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    I'll try to make them look kind
    of the same because they're
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    supposed to look kind of the
    same, but just maybe
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    be different sizes.
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    So that's one, and I'll draw
    another one that's right here.
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    I'm going to draw it a little
    smaller to show you that
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    they're not necessarily the
    same size, they just are
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    same shape essentially.
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    One way I like to think about
    similar triangles are they're
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    just triangles that could be
    kind of scaled up or down in
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    size or flipped around or
    rotated, but they all have
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    the same angles so they're
    essentially the same shape.
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    For example, these two
    triangles, if I were tell you
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    that this angle -- and this is
    how they do it in class.
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    29
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    If I were to tell you this
    angle is equal to this angle
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    and I told you that this angle
    here is equal to this angle.
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    32
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    Well, a couple of things.
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    You already know that this
    angle's going to be equal to
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    this angle, and why is that?
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    Well because if two angles
    are the same, then the third
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    has to be the same, right?
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    Because all three
    angles add up to 180.
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    For example, if this is x,
    this is y, this one has to be
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    180 minus x minus y, right?
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    That's probably too
    small for you to see.
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    But that's the same thing here.
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    If this is x and this is
    y, then this angle right
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    here is going to be 180
    minus x minus y, right?
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    So if we know that two angles
    are the same in two triangles,
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    so we know that the third one's
    also going to be to same.
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    So we could also say this angle
    is identical to this angle.
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    And if all the angles are the
    same, then we know that we are
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    dealing with similar triangles.
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    What useful thing can we
    now do once we know that
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    a triangle is similar?
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    Well, we can use that
    information to kind of figure
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    out some of the sides.
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    So, even though they don't have
    the same sides, the ratio
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    of corresponding side
    lengths is the same.
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    I know I've just confused you.
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    Let me give you an example.
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    For example, let's say that
    this side is -- this side is 5.
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    Let's say that this side is,
    I don't know, I'm just going
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    to make up some number, 6.
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    And let's say that this
    side is 7, right?
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    And let's say we know that, I
    don't know, let's say we know
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    that this side here is 2.
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    64
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    So we know the ratio
    of corresponding
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    sides is the same.
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    So, if we look at these two
    triangles, they have completely
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    different sizes but they
    have corresponding sides.
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    For example, this side
    corresponds to this side.
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    How do we know that?
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    Well, in this case, they
    just happen to have
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    the same orientation.
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    But we know that because
    these sides are opposite
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    the same angle, right?
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    This is opposite angle y,
    and then this side is
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    opposite angle y again.
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    This whole triangle might be
    too small for you to see, but
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    hopefully you're getting
    what I'm saying.
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    So these are
    corresponding sides.
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    Similarly, this side, this
    blue side, and this blue side
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    are corresponding sides.
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    Why?
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    Not because they're kind of on
    the top left because we could
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    have rotated this and flipped
    it and whatever else.
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    It's because it's
    opposite the same angle.
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    86
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    That's the way I always
    think about triangles.
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    It's a good way to think about
    it, especially when you
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    start doing trigonometry.
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    So what does that us?
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    Well, the ratio between
    corresponding sides
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    is always the same.
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    So let's say we want to figure
    out how long this side of
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    the small triangle is.
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    Well there's a bunch of
    ways we could do it.
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    We could say that the ratio of
    this side to this side, so x to
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    7 is going to be equal to the
    ratio of this side to this side
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    -- is equal to the
    ratio of 2 to 5.
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    And then we could solve it.
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    And the only reason why we can
    do this -- you can't do this
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    with just random triangles, you
    can only do this with
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    similar triangles.
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    So we could then solve for x,
    multiply both sides but 7 and
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    you get x is equal
    to 14 over 5.
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    So it's a little
    bit less than 3.
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    So 14 over 5, so 2.8 or
    something like that,
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    that equals x.
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    And we could do the same thing
    to figure out this yellow side.
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    So if you know two triangles
    are similar, you know all the
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    sides of one of the triangles,
    you know one of the sides of
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    the other triangle, you can
    figure out all the sides.
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    I think I just confused
    you with that comment.
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    So, this side, so
    let's call this y.
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    you're doing one triangle's
    going to be the denominator
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    here, then that same
    triangle has to be the
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    denominator on the--.
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    If one triangle is the
    numerator on the left hand side
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    of the equal sign, right, so
    the smaller one's
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    the numerator.
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    Then it's also going to be the
    numerator on the right hand
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    side of the equal sign.
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    I just want to make sure
    you're consistent that way.
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    If you flip it then you're
    going to mess everything up.
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    And then we can just solve for,
    so y is equal to 12 over 5.
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    127
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    So, let's use this information
    about similar triangles
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    just to do some problems.
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    130
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    So let's use some of the
    geometry we've already learned.
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    I have two parallel lines, then
    I have a line like that, then
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    I have a line like this.
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    What did I say, I said that the
    lines are parallel, so this
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    line is parallel to this line.
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    And I want to know if this side
    is length 5, what is -- well,
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    let's say this length is length
    5, let's say that this length
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    is -- let me draw
    another color.
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    This length is, I
    don't know, 8.
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    140
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    I want to know what
    this side is.
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    Actually no, let me give you
    one more side just to make sure
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    you know all of one triangle.
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    Let's say that this side is 6,
    and what I want to do is I want
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    to figure out what this side is
    right here, this purple side.
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    So how do we do this?
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    So before we start using any of
    that ratio stuff, we have to
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    prove to ourselves and prove in
    general, that these are
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    similar triangles.
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    So how can we do that?
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    Let's see if we can figure
    out which angles are
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    equal to other angles.
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    So we have this angle here.
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    Is this angle equal to any
    of these three angles
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    in this triangle?
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    Well, yeah sure.
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    It's opposite this angle right
    here, so this is going to be
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    equal to this angle
    right here, right?
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    So we know that its opposite
    side is it's corresponding
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    side, so we know that it
    corresponds to -- we don't know
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    its length, but we know it
    corresponds to this
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    8 length, right?
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    I forgot to give you
    some information.
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    I forgot to tell you that
    this side is -- let me
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    give it a neutral color.
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    Let's say that this side is 4.
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    Let's go back to the problem.
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    So we just figured out these
    two angles are the same, and
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    that this is that angle's
    corresponding side.
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    Can we figure out any other
    angles are the same?
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    Let's say we know
    what this angle is.
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    172
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    I'm going to do kind of a
    double angle measure here.
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    So what angle in this triangle
    -- does any angle here
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    equal that angle?
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    Sure.
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    We know that these are parallel
    lines, so we can use alternate
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    interior angles to figure out
    which of these angles
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    equals that one.
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    But I just saw the time
    and I realize I'm
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    running out of time.
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    So I will continue this
    in the next video.
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Title:
Similar triangles
Video Language:
Polish
Duration:
09:34

English subtitles

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