Proving centroid of a triangle from medians?

[typomaster75]

The triangle XYZ has co-ordinates X(0, 0), Y(4, 4), and Z(8, -4), I have figured out the median equations:

. . .Median of X

. . .Myz = A = (4 + 8, 4 + (-4) / 2

. . .Myz = A = (6, 0)

. . .mxa = 0 - 0 / 6 - 0

. . .mxa = 0 / 6

. . .y - 0 = 0 / 6(x - 6)

What do I do next, as the slope is 0/6? I know that the centroid is (4, 0); I am just proving it.

 

[soroban]

Hello, typomaster75!

You've found all the necessary information.
You just have to state the conclusion clearly.

The triangle ABC has coordinates $\displaystyle A(0,0),\;B(4,4),\;C8, -4)$
Find the centroid.

        |       B(4,4)
        |       *
        |P    *  *
       (2,2)o     *
        | *        *   Q
   ---A-*-----------o(6,0)---
   (0,0)|   *        *
        |       o     *
        |       R   *  *
        |    (4,-2)     *
        |               C(8,-4)

The midpoint of $\displaystyle AB$ is: $\displaystyle P(2,2)$
The midpoint of$\displaystyle BC$ is: $\displaystyle Q(6,0)$
The midpoint of $\displaystyle AC$ is: $\displaystyle R(4,-2)$

The median $\displaystyle AQ$ is the horizontal line: $\displaystyle y\,=\,0$

The median $\displaystyle BR$ is the vertical line: $\displaystyle x\,=\,4$

The centroid is the intersection of the medians (any two).
. . Therefore, the centroid is: $\displaystyle (4,0)$