If the orthocentre and the centroid of a tria | vedclass

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(A) We know that in any triangle,the circumcentre $(S)$,centroid $(G)$,and orthocentre $(O)$ are collinear,and the centroid divides the line segment j

Vedclass · Accepted Answer

$(6,2,5)$

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(A) We know that in any triangle,the circumcentre $(S)$,centroid $(G)$,and orthocentre $(O)$ are collinear,and the centroid divides the line segment joining the circumcentre and the orthocentre in the ratio $1:2$.Let the circumcentre be $S(x, y, z)$. Given $O(-3, 5, 2)$ and $G(3, 3, 4)$.Using the section formula,the centroid $G$ is given by:$G = \left(\frac{1 \cdot O + 2 \cdot S}{1+2}\right) = \left(\frac{-3+2x}{3}, \frac{5+2y}{3}, \frac{2+2z}{3}\right)$Equating this to the given centroid $(3, 3, 4)$:$\frac{-3+2x}{3} = 3 \Rightarrow -3+2x = 9 \Rightarrow 2x = 12 \Rightarrow x = 6$$\frac{5+2y}{3} = 3 \Rightarrow 5+2y = 9 \Rightarrow 2y = 4 \Rightarrow y = 2$$\frac{2+2z}{3} = 4 \Rightarrow 2+2z = 12 \Rightarrow 2z = 10 \Rightarrow z = 5$Thus,the circumcentre is $(6, 2, 5)$.

If the orthocentre and the centroid of a triangle are $(-3,5,2)$ and $(3,3,4)$ respectively,then its circumcentre is

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