Let V be the volume of a tetrahedron and V′ be the volume of another tetrahedron formed by the centroids of the previous tetrahedron. If V=kV′, then the value of k is
A
27.0
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B
27
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C
27.00
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Solution
Let OABC be the original tetrahedron. V=16[→a→b→c]
Let G,G1,G2,G3 be the centroids of the original tetrahedron. →G=→a+→b+→c3 −→G1=→a+→b3;−→G2=→b+→c3;−→G3=→a+→c3 −−−→G1G=→c3;−−−→G2G=→a3;−−−→G3G=→b3
V′= Volume of tetrahedron GG1G2G3 =16[−−−→GG1−−−→GG2−−−→GG3]=16⎡⎣→a3→b3→c3⎤⎦=16⋅127[→a→b→c]V′=V27 ⇒V=27V′ ⇒k=27