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Math Help - Centroid of a triangle in three space

  1. #1
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    Exclamation Centroid of a triangle in three space

    Find the distance from A to the centroid of the triangle. A(2,3/2,-4), B(3,-4,2), C(1,3,-7).
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  2. #2
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    Quote Originally Posted by johntuan View Post
    Find the distance from A to the centroid of the triangle. A(2,3/2,-4), B(3,-4,2), C(1,3,-7).
    Hi,

    the centroid has the coordinates \left(\frac{x_A+x_B+x_C}{3} , \frac{y_A+y_B+y_C}{3}, \frac{z_A+z_B+z_C}{3}\right)

    With your values the centroid is at \left(2, \frac16 , -3\right)

    Now use the distance formula:

    d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2} that means:

    d=\sqrt{0+\frac{16}{9} + 1}=\sqrt{\frac{25}{9}}=\frac53

    EDIT: I finally found my typos and I believe that now everything is correct.
    Last edited by earboth; October 8th 2007 at 10:24 PM.
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    if you don't mind, could you please tell me the derrivation of the first equation?. Thanks
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  4. #4
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    Hello, johntuan!

    Another approach . . .


    Find the distance from A to the centroid of the triangle
    with vertices: . A\left(2,\,\frac{3}{2},\,\text{-}4\right),\;B(3,\,\text{-}4,\,2),\;C(1,\,3,\,\text{-}7)

    You are expected to know that the centroid (G) is the common intersection of the medians.

    Also, the centroid is \frac{2}{3} the distance from a vertex to the midpoint of the opposite side.

    The midpoint of side BC is: . M\left(\frac{3+1}{2},\:\frac{\text{-}4+3}{2},\:\frac{2-7}{2}\right) \:=\:\left(2,\:\text{-}\frac{1}{2},\:\text{-}\frac{5}{2}\right)

    Distance AM is: . \overline{AM} \;=\;\sqrt{(2-2)^2 + \left(\frac{3}{2}+\frac{1}{2}\right)^2 + \left(-4+\frac{5}{2}\right)^2}  \;=\;\sqrt{0^2 + 2^2 + \left(-\frac{3}{2}\right)^2}

    . . \overline{AM}\;=\;\sqrt{4 + \frac{9}{4}} \;=\;\sqrt{\frac{25}{4}}\quad\Rightarrow\quad \overline{AM} \;=\;\frac{5}{2}


    Therefore: . \overline{AG} \;=\;\frac{2}{3}\cdot\overline{AM} \;=\;\frac{2}{3}\cdot\frac{5}{2} \;=\;\boxed{\frac{5}{3}}

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  5. #5
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    Quote Originally Posted by johntuan View Post
    if you don't mind, could you please tell me the derrivation of the first equation?. Thanks
    Hi,

    I've done this problem recently here:http://www.mathhelpforum.com/math-help/72933-post6.html
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