If `V` be the volume of a tetrahedron and `V '` be the volume of another tetrahedran formed by the centroids of faces of the previous tetrahedron and `V=K V^(prime),t h e nK` is equal to a. `9` b. `12` c. `27` d. `81`

Options

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

Answer

c

Explanation

Consider a tetrahedron with vertices O (0,0,00 A (a, 0,0), B(0,b,0) and C (0,0,c) ,
volume V ` = 1/6 [veca vecb vecc]`
Now centroids of the faces OAB, OAC, OBC and ABC are ` G_(1)(a//3,b//3,0) , G_(2) (a//3,0, c//3), G_(3) (0, b//3, c//3) and G_(4) (a//3, b//3, c//3) ` respectively,
`G_(4)G_(1) = vecc//3, vec(G_(4)G_(2)) =vecb//3, vec(G_(4)G_(3))=veca//3`
volume of tetrahedron by centroids
`V' 1/6[veca/3 vecb/3 vecc/3]= 1/27 V`
`Rightarrow K = 27`