InterviewSolution
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InterviewSolution
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Let `G_(1),G_(2),G_(3)` be the centroids of the triangular faces `OBC, OCA, OAB` of a tetrahedron `OABC`. If `V_(1)` denote the volume of the tetrahedron `OABC` and `V_(2)` that of the parallelopiped with `OG_(1),OG_(2),OG_(3)` as three concurrent edges, thenA. `4V_(1)=9V_(2)`B. `9V_(1)=4V_(2)`C. `3V_(1)=2V_(2)`D. `3V_(2)=2V_(1)` |
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Answer» Correct Answer - A Taking `O` as the origin let the position vectors of `A,B` and `C` be `veca,vecb` and `vecc` respectively. Thenthe position vectors of `G_(1),G_(2)` and `G_(3)` are `(vecb+vecc)/3,(vecc+veca)/3` and `(veca+vecb)/3` respectively. `:.V_(1)=1/6[(veca,vecb, vecc)]` and `V_(2)=[(vec(OG_(1)),vec(OG_(2)),vec(OG_(3)))]` `V_(2)=[(vec(OG_(1)),vec(OG_(2)),vec(OG_(3)))]` `implies V_(2)=[((vecb+vecc)/3,(vecc+veca)/3,(veca+vecb)/3)]` `impliesV_(2)=1/27[(vecb+vecc,vecc+veca,veca+vecb)]` `impliesV_(2)=2/27[(veca,vecb,vecc)]impliesV_(2)=2/27xx6V_(1)=9V_(2)=4V_(1)` |
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