If the centroid of a tetrahedron OABC is (1, | vedclass

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(B) The coordinates of the centroid $G$ of a tetrahedron with vertices $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$, and $(x_4, y_4, z_4)$

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$\sqrt{107}$ units

Vedclass · Answer

(B) The coordinates of the centroid $G$ of a tetrahedron with vertices $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$, and $(x_4, y_4, z_4)$ is given by $(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4})$.Given $O(0, 0, 0)$, $A(a, 2, 3)$, $B(1, b, 2)$, $C(2, 1, c)$ and centroid $G(1, 2, -1)$.Equating coordinates:$\frac{0+a+1+2}{4} = 1 \implies a+3 = 4 \implies a = 1$.$\frac{0+2+b+1}{4} = 2 \implies b+3 = 8 \implies b = 5$.$\frac{0+3+2+c}{4} = -1 \implies c+5 = -4 \implies c = -9$.Thus, $P = (1, 5, -9)$.The distance of $P$ from the origin $(0, 0, 0)$ is $\sqrt{1^2 + 5^2 + (-9)^2} = \sqrt{1 + 25 + 81} = \sqrt{107}$ units.

If the centroid of a tetrahedron $OABC$ is $(1, 2, -1)$, where $O$ is the origin, $A(a, 2, 3)$, $B(1, b, 2)$, and $C(2, 1, c)$ are the other vertices, then the distance of the point $P(a, b, c)$ from the origin is...

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