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(B) Let $A, B, C$ be the vertices of the triangle with coordinates $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$ and $(x_{3}, y_{3}, z_{3})$ respecti

Vedclass · Accepted Answer

$\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}, \frac{z_{1}+z_{2}+z_{3}}{3}\right)$

Vedclass · Answer

(B) Let $A, B, C$ be the vertices of the triangle with coordinates $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$ and $(x_{3}, y_{3}, z_{3})$ respectively.Let $D$ be the midpoint of $BC$. The coordinates of $D$ are $\left(\frac{x_{2}+x_{3}}{2}, \frac{y_{2}+y_{3}}{2}, \frac{z_{2}+z_{3}}{2}\right)$.The centroid $G$ divides the median $AD$ in the ratio $2:1$.Using the section formula,the coordinates of $G$ are $\left(\frac{2\left(\frac{x_{2}+x_{3}}{2}\right) + x_{1}}{2+1}, \frac{2\left(\frac{y_{2}+y_{3}}{2}\right) + y_{1}}{2+1}, \frac{2\left(\frac{z_{2}+z_{3}}{2}\right) + z_{1}}{2+1}\right)$.Simplifying this,we get $\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}, \frac{z_{1}+z_{2}+z_{3}}{3}\right)$.

Find the coordinates of the centroid of the triangle whose vertices are $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$ and $(x_{3}, y_{3}, z_{3})$.

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