Let $$G1$$,$$G2$$,$$G3$$ be the centroids of the triangular faces OBC,OCA,OAB of a tetrahedron OABC . If $$V1$$ denote the volume of the tetrahedron OABC and $$V2$$ that of the parallelopiped with $$OG1$$,$$OG2$$,$$OG3$$ as three concurrent edges, then :

Question

Let $$G1$$,$$G2$$,$$G3$$ be the centroids of the triangular faces OBC,OCA,OAB of a tetrahedron OABC . If $$V1$$ denote the volume of the tetrahedron OABC and $$V2$$ that of  the parallelopiped with $$OG1$$,$$OG2$$,$$OG3$$ as three  concurrent edges, then :

Infinity Learn · Accepted Answer

Taking O as the origin,let the position vectors of A,B and C be a→,b→ and c→ respectively. Then the position vectors $$ofG1$$,$$G2$$ and $$G3$$ are b→$$+c$$→$$3$$,c→$$+a$$→$$3$$ and a→$$+b$$→$$3$$  respectively. ∴                      $$V1$$  $$=$$  $$16$$ a→ b→ c→ and  $$V2=OG1$$→$$OG2$$→$$OG3$$→ Now,  $$V2=OG1$$→$$OG2$$→$$OG3$$→⇒                    $$V2$$  $$=$$ b→ $$+c$$→$$3$$ c→ $$+a$$→ $$3$$ a→ $$+b$$→$$3$$⇒                 $$V2$$  $$=127$$ b→ $$+c$$→  c→ $$+$$ a→ a→ $$+b$$→⇒                 $$V2$$  $$=227$$  a→ b→ c→⇒                 $$V2$$  $$=227$$ × $$6V1$$   ⇒  $$9$$ $$V2$$ $$=$$ $$4V1$$ Hence,   (a) is the correct answer.

$Let G_{1}, G_{2}, G_{3} be the centroids of the triangular faces OBC, OCA, OAB$ $of a tetrahedron OABC . If V_{1} denote the volume of the tetrahedron OABC and V_{2} that of$ $the parallelopiped with O G_{1}, O G_{2}, O G_{3} as three$ $concurrent edges, then :$

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Let G1,G2,G3 be the centroids of the triangular faces OBC,OCA,OAB of a tetrahedron OABC . If V1 denote the volume of the tetrahedron OABC and V2 that of the parallelopiped with OG1,OG2,OG3 as three concurrent edges, then :

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$Let G_{1}, G_{2}, G_{3} be the centroids of the triangular faces OBC, OCA, OAB$ $of a tetrahedron OABC . If V_{1} denote the volume of the tetrahedron OABC and V_{2} that of$ $the parallelopiped with O G_{1}, O G_{2}, O G_{3} as three$ $concurrent edges, then :$