If the origin is the centroid of the triangle | vedclass

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(A) Let the vertices of the triangle be $A(x, y, z)$,$B(-2, 3, 4)$,and $C(3, -1, 5)$.The centroid $G$ of a triangle with vertices $(x_1, y_1, z_1)$,$(

Vedclass · Accepted Answer

$(-1, -2, -9)$

Vedclass · Answer

(A) Let the vertices of the triangle be $A(x, y, z)$,$B(-2, 3, 4)$,and $C(3, -1, 5)$.The centroid $G$ of a triangle with vertices $(x_1, y_1, z_1)$,$(x_2, y_2, z_2)$,and $(x_3, y_3, z_3)$ is given by $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$.Given that the centroid is the origin $(0, 0, 0)$,we have:$\frac{x-2+3}{3} = 0 \Rightarrow x+1 = 0 \Rightarrow x = -1$$\frac{y+3-1}{3} = 0 \Rightarrow y+2 = 0 \Rightarrow y = -2$$\frac{z+4+5}{3} = 0 \Rightarrow z+9 = 0 \Rightarrow z = -9$Therefore,the third vertex is $(-1, -2, -9)$.

If the origin is the centroid of the triangle for which $(-2, 3, 4)$ and $(3, -1, 5)$ are two vertices,then the third vertex is

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