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Applications Scalar Triple Product

Let the volum...

Question

Let the volume of tetrahedron $A B C D$ is $81$ cubic units and $G_{1}$ , $G_{2}$ , $G_{3}$ are centroids of triangular faces $A B C, A B D$ and $A C D$ respectively, then volume of tetrahedron $A G_{1} G_{2} G_{3}$ is -

3

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6

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\frac{81}{4}

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54

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Solution

The correct option is D $54$
Let $A (\to 0)$ , $B (\to a)$ , $C (\to b)$ , $D (\to c)$

$G_{1} = \frac{\to a + \to b}{3}$ , $G_{2} = \frac{\to a + \to c}{3}$ , $G_{3} = \frac{\to b + \to c}{3}$

Volume of $A G_{1} G_{2} G_{3} = \frac{1}{6} ⎡ ⎣ \frac{\to a + \to b}{3} \frac{\to b + \to c}{3} \frac{\to c + \to a}{3} ⎤ ⎦$

$= \frac{2}{27} \times \frac{1}{6} [\to a \to b \to c] = \frac{2}{27} \times 81$

$= 6$

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