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Angle between Two Planes

A tetrahedron...

Question

A tetrahedron $(T_{1})$ has a fixed volume, $V_{1}$ . Another tetrahedron $(T_{2})$ with volume $V_{2}$ is drawn by joining the centroids of the faces of $T_{1} .$ If $V_{1} = k V_{2},$ then the possible value(s) of $k$ is/are

9

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27

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\frac{9}{2}

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18

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Solution

The correct option is B $27$

Let the vertices of $T_{1}$ are $O (0, 0, 0), A (a, 0, 0), B (0, b, 0)$ and $C (0, 0, c)$
Now, centroids of faces of $T_{1}$
i.e., $O A B, O A C, O B C and A B C$ are $P (\frac{a}{3}, \frac{b}{3}, 0), Q (\frac{a}{3}, 0, \frac{c}{3}),$ $R (0, \frac{b}{3}, \frac{c}{3})$ and $S (\frac{a}{3}, \frac{b}{3}, \frac{c}{3})$ respectively.
$P S = \frac{\to c}{3}, Q S = \frac{\to b}{3}, R S = \frac{\to a}{3}$
$V_{1} = \frac{1}{6} [\to a \to b \to c] V_{2} = \frac{1}{6} ⎡ ⎣ \frac{\to a}{3} \frac{\to b}{3} \frac{\to c}{3} ⎤ ⎦ = \frac{1}{27 \times 6} [\to a \to b \to c] \Rightarrow k = 27$

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