Radius of a Circle Inscribed in a Triangle - Two Secret & EASY Formulas

00:05:33
https://www.youtube.com/watch?v=S9c4AFxdtAY

概要

TLDRIn this video, the presenter shares formulas for finding the radius of circles inscribed in triangles, applicable to both right and non-right triangles. For right triangles, the radius is calculated using the formula (a + b - c) / 2, while for non-right triangles, the formula is (2 * Area) / Perimeter. The presenter explains the geometric principles behind these formulas, including the relationship between the radius and the sides of the triangle. The video emphasizes the importance of continuous learning in mathematics and encourages viewers to engage with the content and explore additional resources for teaching math.

収穫

  • 📐 The radius of an inscribed circle in a right triangle is (a + b - c) / 2.
  • 🔺 For non-right triangles, the radius is (2 * Area) / Perimeter.
  • 📏 The radius is perpendicular to the sides at the point of tangency.
  • 📊 The area of the triangle relates to the inscribed circle's radius.
  • 🧮 Continuous learning in math is essential, even for experienced teachers.
  • 👩‍🏫 The presenter has extensive teaching experience in math education.
  • 📅 Weekly math problems are posted for engagement.
  • 🌐 Resources for math teachers are available online.

タイムライン

  • 00:00:00 - 00:05:33

    In this video, the speaker introduces formulas to find the radius of circles inscribed in triangles, applicable to both right and non-right triangles. The speaker emphasizes the continuous learning aspect of mathematics, even after years of teaching. For right triangles, the radius (R) is derived using the sides (a, b, c) with the formula R = (a + b - c) / 2, highlighting the relationship between the radius and the triangle's sides. For non-right triangles, the area is expressed in terms of the radius and the triangle's perimeter, leading to the formula R = (2 * Area) / Perimeter. The speaker encourages engagement with the content and offers resources for math teachers.

マインドマップ

ビデオQ&A

  • What is the formula for the radius of a circle inscribed in a right triangle?

    The formula is (a + b - c) / 2, where a and b are the legs and c is the hypotenuse.

  • How do you find the radius for a non-right triangle?

    The radius can be found using the formula (2 * Area) / Perimeter.

  • What is the significance of the radius being perpendicular to the sides at the point of tangency?

    It helps in establishing relationships between the sides of the triangle and the radius.

  • Can these formulas be used for any triangle?

    Yes, they apply to both right and non-right triangles.

  • What is the area of a triangle in relation to the inscribed circle?

    The area can be expressed as the sum of the areas of the smaller triangles formed with the radius and the sides.

  • What resources does the presenter offer for math teachers?

    The presenter has a website and teacher page with resources for engaging students in math.

  • How often does the presenter post math problems?

    The presenter posts a math problem once a week.

  • What is the presenter's background in teaching?

    The presenter has taught high school math for nearly two decades and community college math for ten years.

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  • 00:00:00
    well hello and welcome everybody hey in
  • 00:00:02
    this video what I'm going to do is share
  • 00:00:03
    with you a couple of formulas that well
  • 00:00:06
    they're Secret at least they were to me
  • 00:00:08
    a secret until here very recently what
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    the formulas will do is they'll help you
  • 00:00:12
    to find the radius of any Circle that's
  • 00:00:15
    inscribed in a triangle so that includes
  • 00:00:16
    right triangles and of course non-right
  • 00:00:18
    triangles and I think this is super cool
  • 00:00:20
    because I mean I've been teaching high
  • 00:00:22
    school math and well for almost two
  • 00:00:25
    decades and I did uh Community College
  • 00:00:27
    math for 10 years
  • 00:00:30
    I'm still learning stuff about math all
  • 00:00:32
    the time even at this level I think
  • 00:00:34
    that's really cool there's always
  • 00:00:35
    something to discover and that's pretty
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    fun to me anyway let's dive into the
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    first case right here with the right
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    triangles let's goe and name the sides a
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    b and c and the first fact we have to
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    draw upon is that uh the radius is
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    perpendicular to the side at the point
  • 00:00:49
    of tangency so I can construct the
  • 00:00:52
    radius here and here to side A and B as
  • 00:00:55
    well as C and they make right angles and
  • 00:00:57
    let's go ahead and label that radius R
  • 00:01:00
    so let's see if we can reexpress these
  • 00:01:03
    sides in terms of R and maybe come up
  • 00:01:06
    with some expression that's easy to use
  • 00:01:08
    so let's talk about that uh vertical
  • 00:01:10
    distance right there if we slide that
  • 00:01:12
    across that distance right there is also
  • 00:01:14
    the radius because we have a square well
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    that makes the side A minus r and r
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    because a minus r plus r would just be a
  • 00:01:22
    there we go that's good to go so now
  • 00:01:24
    let's do the horizontal distance and if
  • 00:01:26
    we slide that down drop it down we see
  • 00:01:28
    that this distance right here here is R
  • 00:01:30
    so the side would be r + B minus r right
  • 00:01:34
    so this whole side right there all right
  • 00:01:36
    so now the next fact that we're going to
  • 00:01:38
    have to use is that well when tangent
  • 00:01:41
    lines intersect those segments from the
  • 00:01:43
    point of tangency to the intersection
  • 00:01:45
    well they're
  • 00:01:46
    congruent so for example right here um
  • 00:01:51
    we have a minus r on the left well this
  • 00:01:53
    side would also be a minus r both of
  • 00:01:56
    these are tangent to the same Circle so
  • 00:01:58
    their segments are congruent and the
  • 00:02:01
    same is the case for B minus r so now
  • 00:02:04
    what I've got is that side C is also
  • 00:02:07
    equal to A- r + B minus r so since
  • 00:02:11
    that's true I could go ahead and write
  • 00:02:12
    an equation and simplify it right and
  • 00:02:15
    let's go ahead and solve for R so we're
  • 00:02:17
    going to move the 2 R to the left and
  • 00:02:19
    the C to the right side just so that we
  • 00:02:21
    end up with the well signs will work out
  • 00:02:23
    a little bit prettier and we can find
  • 00:02:26
    the radius by just the short legs a plus
  • 00:02:30
    B minus C / two that's the radius that
  • 00:02:33
    is a really cool slick formula I don't
  • 00:02:36
    know who says slick anymore but you know
  • 00:02:39
    it's pretty neat right that's a clean
  • 00:02:41
    formula I think that is super super cool
  • 00:02:44
    so now let's dive into the case for the
  • 00:02:46
    non-right triangle shall we um same fact
  • 00:02:50
    as before the perpendicular uh the
  • 00:02:52
    radius is perpendicular to the sides at
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    the point of tangency so we're going to
  • 00:02:55
    construct our radi like that and they're
  • 00:02:58
    all perpendicular now let's go ahead and
  • 00:03:00
    name our sides we've got uh vertices a b
  • 00:03:03
    and c and the opposite sides a b and c
  • 00:03:05
    like that and we're going to say the
  • 00:03:06
    center of the circle is O as in origin
  • 00:03:10
    all right now what we're going to do is
  • 00:03:12
    we're going to have to um find some
  • 00:03:14
    expression that relates the radius to
  • 00:03:17
    parts of this triangle right so it's not
  • 00:03:19
    a right triangle makes a little trickier
  • 00:03:21
    but we're going to use area so check
  • 00:03:23
    this out see this triangle right here
  • 00:03:26
    triangle B well I know it's area would
  • 00:03:30
    be 1/2 base time height so that's 1/2 of
  • 00:03:33
    the height which is R time the base
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    which is side A all right so angle
  • 00:03:40
    triangle B its area is 12 R * a let's do
  • 00:03:44
    the same thing with two other triangles
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    right here so I've got triangle boa
  • 00:03:49
    right boa is once again 1/2 base times
  • 00:03:52
    height the height is the perpendicular
  • 00:03:54
    distance to the base so the height here
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    is R the base is AB and let's do it once
  • 00:04:00
    again for angle AOC so AOC is 12 R *
  • 00:04:05
    side B all three of those triangles add
  • 00:04:08
    up together to make the area of the
  • 00:04:10
    entire triangle ABC as you can see all
  • 00:04:13
    right so let's go ahead and say uh write
  • 00:04:16
    our equation so the area of triangle ABC
  • 00:04:19
    the big triangle is equal to the three
  • 00:04:21
    smaller triangles right so that'd be
  • 00:04:23
    one2 RB plus 1 12 RC plus 1 12 R A we
  • 00:04:27
    can simplify that we can factor out a 12
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    R I'm going to put the r first time A +
  • 00:04:34
    B + C so the area is equal to R * 1 12
  • 00:04:40
    of the perimeter do you see a plus B+ C
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    would be the perimeter so R if we solve
  • 00:04:45
    for R multiply both sides by two divide
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    by perimeter twice the area divided by
  • 00:04:51
    the perimeter is the radius twice the
  • 00:04:53
    area divided by the perimeter is the
  • 00:04:55
    radius for a non-right triangle for a
  • 00:04:57
    right triangle a plus b so the leg minus
  • 00:05:00
    a hypotenuse divided by two is the
  • 00:05:01
    radius I think those are super super
  • 00:05:04
    cool I hope you really did enjoy this I
  • 00:05:06
    had a really good time putting it
  • 00:05:07
    together if you did like comment share
  • 00:05:09
    subscribe all that kind of stuff the
  • 00:05:11
    cool kids are doing check us out once a
  • 00:05:13
    week I post a math problem just for fun
  • 00:05:15
    kind of like this and if you're a math
  • 00:05:17
    teacher looking for some things you can
  • 00:05:18
    use in your classroom to engage students
  • 00:05:20
    save you a ton of time really get them
  • 00:05:23
    having a good time learning math I've
  • 00:05:24
    got a website and a teacher page teacher
  • 00:05:26
    store all the links are in the
  • 00:05:28
    description and until next next time I
  • 00:05:30
    hope you have a great day
タグ
  • inscribed circle
  • triangle
  • radius
  • right triangle
  • non-right triangle
  • math formulas
  • geometry
  • area
  • perimeter
  • math education