If (2,3,4) is the centroid of the tetrahedron for which (2,3,−1),(3,0,−2),(−1,4,3)are three vertices , then the distance of the fourth vertex from the origin is

A
$\sqrt{33}$
B
$2 \sqrt{33}$
C
$3 \sqrt{33}$
D
$4 \sqrt{33}$

Solution:

The centroid of the tetrahedron formed by the vertices $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}), (x_{4}, y_{4}, z_{4})$ is $G (\frac{x_{1} + x_{2} + x_{3} + x_{4}}{4}, \frac{y_{1} + y_{2} + y_{3} + y_{4}}{4}, \frac{z_{1} + z_{2} + z_{3} + z_{4}}{4})$

Hence, Fourth vertex is

$D = 4 G - A - B - C = (8 - 2 - 3 + 1, 12 - 3 - 4, 16 + 1 + 2 - 3) = (4, 5, 16)$

The distance between the fourth vertex and the centroid is $D G = \sqrt{16 + 25 + 256} = \sqrt{297}$

Therefore, the required distance is $3 \sqrt{33}$

If $(2, 3, 4)$ is the centroid of the tetrahedron for which $(2, 3, - 1), (3, 0, - 2), (- 1, 4, 3)$ are three vertices , then the distance of the fourth vertex from the origin is

Solution:

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