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AP EAMCET201920 Apr 2019Morning ShiftMathematicsVector AlgebraActual

A new tetrahedron is formed by joining the centroids of the faces of a given tetrahedron (O A B C ). Then the ratio of the volume of the new tetrahedron to that of the given tetrahedron is

Options

  1. A( 3 25 )
  2. B( 1 27 )
  3. C( 5 62 )
  4. D( 1 162 )

Correct answer

B. ( 1 27 )

Step-by-step solution

Let the position vectors of the vertices of tetrahedron (O A B C ) are ( O A = a , O B = b ) and ( O C = c ) So, volume of tetrahedron (O A B C= 1 6 [ a b ~ c ] ) Now, position vectors of vertices of new tetrahedron are ( a + b 3 , b + c 3 , c + a 3 ) and ( a + b + c 3 ), so, coterminous edge vectors of the new tetrahedron are ( a 3 , b 3 ) and ( c 3 ). ( ) Volume of new tetrahedron is ( 1 6 [ a 3 b 3 c 3 ] ) (= 1 6 27 [ a ~ b c ] ) So, the required ratio (= 1 27 ) Hence, option (2) is correct.

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