Return to Video

Right Triangles Inscribed in Circles (Proof)

  • 0:00 - 0:01
  • 0:01 - 0:04
    Let's say we have a circle,
    and then we have a
  • 0:04 - 0:05
    diameter of the circle.
  • 0:05 - 0:09
    Let me draw my best diameter.
  • 0:09 - 0:10
    That's pretty good.
  • 0:10 - 0:13
    This right here is the diameter
    of the circle or it's a
  • 0:13 - 0:15
    diameter of the circle.
  • 0:15 - 0:16
    That's a diameter.
  • 0:16 - 0:19
    Let's say I have a triangle
    where the diameter is one side
  • 0:19 - 0:26
    of the triangle, and the angle
    opposite that side, it's
  • 0:26 - 0:29
    vertex, sits some place
    on the circumference.
  • 0:29 - 0:34
    So, let's say, the angle or the
    angle opposite of this diameter
  • 0:34 - 0:35
    sits on that circumference.
  • 0:35 - 0:38
    So the triangle
    looks like this.
  • 0:38 - 0:44
    The triangle looks like that.
  • 0:44 - 0:47
    What I'm going to show you
    in this video is that
  • 0:47 - 0:51
    this triangle is going
    to be a right triangle.
  • 0:51 - 0:54
  • 0:54 - 0:57
    The 90 degree side is going
    to be the side that is
  • 0:57 - 0:59
    opposite this diameter.
  • 0:59 - 1:00
    I don't want to label it
    just yet because that would
  • 1:00 - 1:02
    ruin the fun of the proof.
  • 1:02 - 1:05
    Now let's see what we
    can do to show this.
  • 1:05 - 1:09
    Well, we have in our tool kit
    the notion of an inscribed
  • 1:09 - 1:13
    angle, it's relation to
    a central angle that
  • 1:13 - 1:15
    subtends the same arc.
  • 1:15 - 1:16
    So let's look at that.
  • 1:16 - 1:19
    So let's say that this is an
    inscribed angle right here.
  • 1:19 - 1:23
    Let's call this theta.
  • 1:23 - 1:25
    Now let's say that
    that's the center of
  • 1:25 - 1:27
    my circle right there.
  • 1:27 - 1:30
    Then this angle right here
    would be a central angle.
  • 1:30 - 1:33
    Let me draw another triangle
    right here, another
  • 1:33 - 1:33
    line right there.
  • 1:33 - 1:35
    This is a central
    angle right here.
  • 1:35 - 1:38
    This is a radius.
  • 1:38 - 1:40
    This is the same radius
    -- actually this
  • 1:40 - 1:41
    distance is the same.
  • 1:41 - 1:44
    But we've learned several
    videos ago that look, this
  • 1:44 - 1:49
    angle, this inscribed angle,
    it subtends this arc up here.
  • 1:49 - 1:52
  • 1:52 - 1:56
    The central angle that subtends
    that same arc is going
  • 1:56 - 1:57
    to be twice this angle.
  • 1:57 - 1:59
    We proved that
    several videos ago.
  • 1:59 - 2:02
    So this is going to be 2theta.
  • 2:02 - 2:05
    It's the central angle
    subtending the same arc.
  • 2:05 - 2:10
    Now, this triangle right here,
    this one right here, this
  • 2:10 - 2:12
    is an isosceles triangle.
  • 2:12 - 2:14
    I could rotate it and
    draw it like this.
  • 2:14 - 2:16
  • 2:16 - 2:22
    If I flipped it over it would
    look like that, that, and then
  • 2:22 - 2:25
    the green side would
    be down like that.
  • 2:25 - 2:29
    And both of these sides
    are of length r.
  • 2:29 - 2:31
    This top angle is 2theta.
  • 2:31 - 2:34
    So all I did is I took it
    and I rotated it around to
  • 2:34 - 2:35
    draw it for you this way.
  • 2:35 - 2:37
    This side is that
    side right there.
  • 2:37 - 2:42
    Since its two sides are equal,
    this is isosceles, so these to
  • 2:42 - 2:44
    base angles must be the same.
  • 2:44 - 2:48
  • 2:48 - 2:50
    That and that must be the same,
    or if I were to draw it up
  • 2:50 - 2:55
    here, that and that must be
    the exact same base angle.
  • 2:55 - 2:58
    Now let me see, I already
    used theta, maybe I'll
  • 2:58 - 3:00
    use x for these angles.
  • 3:00 - 3:05
    So this has to be x,
    and that has to be x.
  • 3:05 - 3:08
    So what is x going
    to be equal to?
  • 3:08 - 3:12
    Well, x plus x plus 2theta
    have to equal 180 degrees.
  • 3:12 - 3:14
    They're all in the
    same triangle.
  • 3:14 - 3:16
    So let me write that down.
  • 3:16 - 3:23
    We get x plus x plus 2theta,
    all have to be equal to 180
  • 3:23 - 3:31
    degrees, or we get 2x plus
    2theta is equal to 180 degrees,
  • 3:31 - 3:36
    or we get 2x is equal
    to 180 minus 2theta.
  • 3:36 - 3:43
    Divide both sides by 2, you get
    x is equal to 90 minus theta.
  • 3:43 - 3:51
    So x is equal to
    90 minus theta.
  • 3:51 - 3:53
    Now let's see what else
    we could do with this.
  • 3:53 - 3:55
    Well we could look at this
    triangle right here.
  • 3:55 - 3:59
    This triangle, this side over
    here also has this distance
  • 3:59 - 4:02
    right here is also a
    radius of the circle.
  • 4:02 - 4:04
    This distance over here we've
    already labeled it, is
  • 4:04 - 4:05
    a radius of a circle.
  • 4:05 - 4:09
    So once again, this is also
    an isosceles triangle.
  • 4:09 - 4:13
    These two sides are equal,
    so these two base angles
  • 4:13 - 4:14
    have to be equal.
  • 4:14 - 4:17
    So if this is theta,
    this is also going to
  • 4:17 - 4:18
    be equal to theta.
  • 4:18 - 4:21
    And actually, we use that
    information, we use to actually
  • 4:21 - 4:25
    show that first result about
    inscribed angles and the
  • 4:25 - 4:27
    relation between them and
    central angles subtending
  • 4:27 - 4:28
    the same arc.
  • 4:28 - 4:30
    So if this is theta, that's
    theta because this is
  • 4:30 - 4:31
    an isosceles triangle.
  • 4:31 - 4:36
    So what is this whole
    angle over here?
  • 4:36 - 4:40
    Well it's going to be theta
    plus 90 minus theta.
  • 4:40 - 4:42
    That angle right there's
    going to be theta
  • 4:42 - 4:45
    plus 90 minus theta.
  • 4:45 - 4:46
    Well, the thetas cancel out.
  • 4:46 - 4:50
    So no matter what, as long as
    one side of my triangle is the
  • 4:50 - 4:53
    diameter, and then the angle or
    the vertex of the angle
  • 4:53 - 4:57
    opposite sits opposite of
    that side, sits on the
  • 4:57 - 5:02
    circumference, then this angle
    right here is going to be a
  • 5:02 - 5:09
    right angle, and this is going
    to be a right triangle.
  • 5:09 - 5:12
    So if I just were to draw
    something random like this --
  • 5:12 - 5:16
    if I were to just take a point
    right there, like that, and
  • 5:16 - 5:20
    draw it just like that,
    this is a right angle.
  • 5:20 - 5:23
    If I were to draw something
    like that and go out like
  • 5:23 - 5:25
    that, this is a right angle.
  • 5:25 - 5:28
    For any of these I could
    do this exact same proof.
  • 5:28 - 5:30
    And in fact, the way I drew it
    right here, I kept it very
  • 5:30 - 5:34
    general so it would apply
    to any of these triangles.
  • 5:34 - 5:34
Title:
Right Triangles Inscribed in Circles (Proof)
Description:

Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle

more » « less
Video Language:
English
Duration:
05:35

English subtitles

Revisions Compare revisions