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(A) Let the third vertex of the triangle be $A(x, y, z)$.The given vertices are $B(2, 4, 6)$ and $C(0, -2, -5)$.The centroid $G$ of a triangle with ve

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

Vedclass · Answer

(A) Let the third vertex of the triangle be $A(x, y, z)$.The given vertices are $B(2, 4, 6)$ and $C(0, -2, -5)$.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 $G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$.Given that the centroid is the origin $(0, 0, 0)$,we have:$(0, 0, 0) = \left(\frac{2+0+x}{3}, \frac{4-2+y}{3}, \frac{6-5+z}{3}\right)$.Equating the coordinates:$\frac{2+x}{3} = 0 \implies x = -2$.$\frac{2+y}{3} = 0 \implies y = -2$.$\frac{1+z}{3} = 0 \implies z = -1$.Thus,the third vertex is $(-2, -2, -1)$.

Column $I$	Column $II$
$(A)$ The centroid of the triangle formed by $(2, 3, -1)$,$(5, 6, 3)$,$(2, -3, 1)$ is	$(p)$ $(2, 2, 2)$
$(B)$ The circumcentre of the triangle formed by $(1, 2, 3)$,$(2, 3, 1)$,$(3, 1, 2)$ is	$(q)$ $(3, 1, 4)$
$(C)$ The orthocentre of the triangle formed by $(2, 1, 5)$,$(3, 2, 3)$,$(4, 0, 4)$ is	$(r)$ $(1, 1, 0)$
$(D)$ The incentre of the triangle formed by $(0, 0, 0)$,$(3, 0, 0)$,$(0, 4, 0)$ is	$(s)$ $(3, 2, 1)$

Find the third vertex of a triangle whose centroid is the origin and two vertices are $(2, 4, 6)$ and $(0, -2, -5)$.

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