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MHT CET20239 May 2023Evening ShiftMathematicsProperties of TrianglesActual

If D, E and F are the mid-points of the sides B C , CA and AB of triangle ABC respectively, then AD + 2 3 BE + 1 3 CF =

Options

  1. A1 2 AB
  2. B1 2 AC
  3. C1 2 BC
  4. D2 3 AC

Correct answer

B. 1 2 AC

Step-by-step solution

Let the position vector of A, B, C, D, E, F be a , b , c , d , e , f respectively. aligned d & = b + c 2 , e = c + a 2 , f = a + b 2 & Now, AD + 2 3 BE + 1 3 CF & = d - a + 2 3 ( e - b )+ 1 3 ( f - c ) & = b + c 2 - a + 2 3 ( c + a 2 - b )+ 1 3 ( a + b 2 - c ) & = b + c -2 a 2 + c + a -2 b 3 + a + b -2 c 6 & = 3 c -3 a 6 & = 3 6 ( c - a ) & = 1 2 AC aligned

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