Question

If $V$ be the volume of a tetrahedron and $V^{{}^{\prime }}$ be the volume of another tetrahedran formed by the centroids of faces of the previous tetrahedron and $V=K{V}^{\prime }$, then $K$ is equal to

• A. $9$
• B. $12$
• C. $27$
• D. $81$

Answer 1

Answer: (C)

$27$

Consider a tetrahedron with vertices $O(0,0,0),A(a,0,0),B(0,b,0)$ and $C(0,0,c)$.
Volume $V=\frac{1}{6}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$

Now centroids of the faces $OAB,OAC,OBC$ and $ABC$ are $G_1(\frac{a}{3},\frac{b}{3},0),G_2(\frac{a}{3},0,\frac{c}{3}),G_3(0,\frac{b}{3},\frac{c}{3})$ and $G_4(\frac{a}{3},\frac{b}{3},\frac{c}{3})$, respectively.

$\overrightarrow{G_4{G}_1}=\frac{\overrightarrow{c}}{3},\overrightarrow{G_4{G}_2}=\frac{\overrightarrow{b}}{3},\overrightarrow{G_4{G}_3}=\frac{\overrightarrow{a}}{3}$

Volume of tetrahedron by centroids
$V^{{}^{\prime }}=\frac{1}{6}\left[\frac{\overrightarrow{a}}{3}\frac{\overrightarrow{b}}{3}\frac{\overrightarrow{c}}{3}\right]=\frac{1}{27}V$
$\Rightarrow V=27V^{{}^{\prime }}$
$\Rightarrow K=27$