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Thread: The centroid of a triangle with coordinates

  1. #1
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    The centroid of a triangle with coordinates

    Triangle DEF has vertices D(1,3) and E(6,1), and centroid at C(3,4). Determine the coordinates of point F.

    I know that the medians in a triangle intersect at the centroid and that the centroid divides each median in a ration of 2:1. Also, I know that each median intersects a side at its midpoint with the shorter part of the median. What I did was find the midpoint between point E and F by taking half of the line D to C and adding onto C to give point G. The coordinates of point G was found to be (4, 4.5). Since I knew that the change in the coordinates between points E and G, (-2, 3.5), was only half of the line EF I doubled it and added it point E to give the answer (2, 8). I wanted to know if there was an easier way to do this.
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  2. #2
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    Quote Originally Posted by darksoulzero View Post
    Triangle DEF has vertices D(1,3) and E(6,1), and centroid at C(3,4). Determine the coordinates of point F.

    I know that the medians in a triangle intersect at the centroid and that the centroid divides each median in a ration of 2:1. Also, I know that each median intersects a side at its midpoint with the shorter part of the median. What I did was find the midpoint between point E and F by taking half of the line D to C and adding onto C to give point G. The coordinates of point G was found to be (4, 4.5). Since I knew that the change in the coordinates between points E and G, (-2, 3.5), was only half of the line EF I doubled it and added it point E to give the answer (2, 8). I wanted to know if there was an easier way to do this.
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  3. #3
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    centroid of triangle

    Hi darksoulzero,

    Your solution is confusing.Here is a suggested method.

    connect M midpoint of DE and C (centroid) with extended lenght.F lies on this line. FC = 2CM. Slope diagram of C and M = 2/1/2.Slope diagram of FC is twice that or 4/1. Working from point C one point left and 4 points up gives F (2,8)



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  4. #4
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    If ABC is a triangle with $\displaystyle $$A\left( {{x_1},{y_1}} \right),B\left( {{x_2},{y_2}} \right),C\left( {{x_3},{y_3}} \right)$$$ then its centroid is
    $\displaystyle $$G\left( {\tfrac{{{x_1} + {x_2} + {x_3}}}{3},\tfrac{{{y_1} + {y_2} + {y_3}}}{3}} \right)$$$
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  5. #5
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    Hello, darksoulzero!

    $\displaystyle \text{Triangle }D{E}F\text{ has vertices: }D(1,3)\text{ and }E(6,1)\text{, and centroid at }C(3,4).$
    $\displaystyle \text{Determine the coordinates of point }F.$
    Code:
    
             1
        F o-----+
           \    :
            \   :4
             \  :
              \ :
               \:0.5
           (3,4)o---+
        D       C\  :
        o         \ :2
      (1,3)   *    \:
                  M o
                (3.5,2)   *     E
                                o
                              (6,1)

    We have vertices $\displaystyle D(1,3)$ and $\displaystyle E(6,1)$, and centroid $\displaystyle C(3,4).$

    The midpoint of $\displaystyle DE$ is: $\displaystyle M(3\tfrac{1}{2},\,2).$

    The median to side $\displaystyle DE$ starts at $\displaystyle \,M$, passes through $\displaystyle \,C,$
    . . and extends to $\displaystyle \,F$, where: .$\displaystyle FC \,=\,2\!\cdot\!CM.$


    Going from $\displaystyle \,M$ to $\displaystyle \,C$, we move up 2 and left $\displaystyle \frac{1}{2}$
    Hence, going from $\displaystyle \,C$ to $\displaystyle \,F$, we move up 4 and left 1.

    Therefore, we have: .$\displaystyle F(2,8).$

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