If the centroid of a triangle with vertices $(4, p,-3),(-1,-1,2)$ and $(3,5,-8)$ is given by mid-point of $(1,4,-2)$ and $(q, 2,-4)$, then $p^2+q^2=$

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Aicharya · Accepted Answer

Centroid of triangle with vertices $(4, p,-3),(-1,-1,2),(3,5,-8)$ is $ \left(\frac{4-1+3}{3}, \frac{p-1+5}{3}, \frac{-3+2-8}{3}ight) $ which is $\left(2, \frac{4+p}{3},-3ight)$. Mid-point of $(1,4,-2)(q, 2,-4)$ is $\left(\frac{1+q}{2}, 3,-3ight)$ As both points are, $ \begin{array}{rlrl} & & 2 & =\frac{1+q}{2} 	ext { and } \frac{4+p}{3}=3 \ \Rightarrow & \quad p & =5, q=3 \ 	herefore \quad & p^2+q^2 & =25+9=34 \end{array} $

If the centroid of a triangle with vertices $(4, p,-3),(-1,-1,2)$ and $(3,5,-8)$ is given by mid-point of…

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