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Areas of Combination of Plane Figures

Let G1, G...

Question

Let $G_{1}$ , $G_{2}$ and $G_{3}$ be the centroids of the triangular faces OBC, OCA and OAB of the tetrahedron OABC. If $V_{1}$ denote the volume of the tetrahedron OABC and $V_{2}$ that of the parallelopiped with $O G_{1}$ , $O G_{2}$ and $O G_{3}$ as three concurrent edges, then

4 V_{1} = 9 V_{2}

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3 V_{1} = 4 V_{2}

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3 V_{1} = 2 V_{2}

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3 V_{2} = 2 V_{1}

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Solution

The correct option is B $4 V_{1} = 9 V_{2}$
$A (\to a)$
$B (\to b)$
$c (\to c)$
$a_{1} = \frac{\to b + \to c}{2}$
$a_{2} = \frac{\to c + \to d}{2}$
$a_{3} = \frac{\to a + \to b}{2}$
$V_{1} = \frac{1}{6} [\to a \to b \to c], V_{2} = [O \to G, O {\to G}_{2}]$
$V_{2} = \frac{1}{27} [\to b + \to c . \to c + \to a \to a + \to b]$
$= \frac{1 \times 2}{27} [\to a \to b \to c]$
$6 V_{1} = [\to a \to b \to c]$
$V_{2} = \frac{1}{27} \times 2 \times 6 V_{1}$
$9 V_{2} = 4 V_{1}$

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