D, E and F are respectively the midpoints of sides AB, BC and CA of ΔABC. Find the ratio of the areas of ΔDEF and ΔABC.

B A D F C E


Answer:

1:4

Step by Step Explanation:
  1. In ΔABC, D and F are the midpoints of sides AB and CA respectively.

    Therefore, DF||BC [ By midpoint theorem ]

      DF||BE.

    Similarly,   EF||BD.

    Therefore, BEFD is a parallelogram.

    B=EFD,EF=BD=12AB and DF=BE=12BC.

    Also, ECFD is a parallelogram.

      EDF=C.
  2. Now, in ΔDEF and ΔCAB, we have EFD=Band  EDF=C Thus, the ratio of the areas of \Delta DEF and \Delta ABC is 1:4.

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