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amitpaswan09
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Asked: 2023-04-01T04:30:25+00:00 In:

In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively see Figure. Show that the line segments AF and EC trisect the diagonal BD.

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In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively see Figure. Show that the line segments AF and EC trisect the diagonal BD.

Define vertically opposite angles.
Prove to corresponding sides of similar triangles
Class 9, EXERCISE 8.2, Page No: 151, Questions No:5.
Session 2023-2024 are based on CBSE Board.

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      In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively see Figure. Show that the line segments AF and EC trisect the diagonal BD.

      In quadrilateral ABCD,
      AB = CD [∵ Given]
      (1/2)AB = (1/2)CD
      ⇒ AE = CF [∵ E and F are the mid-points of AB and CD respectively]
      In Quadrilateral AECF,
      AE = CF [∵ Proved above]
      AE ∥ CF [∵ Opposite sides of a parallelogram]
      Hence, AECF is a parallelogram.
      In ΔDCQ,
      F is mid-point of DC [∵ Given]
      and FP ∥ CQ [∵ AECF is a Parallelogram]
      Hence, P is mid-point of DQ [ Converse of mid-point theorem]
      Hence, DP = PQ …(1)
      Similarly,
      In ΔABP,
      E is mid-point of AB [∵ Given]
      and EQ ∥ AP [∵ AECF is a parallelogram]
      Hence, Q is mid-point of PB [∵ Converse of mid-Point Theorem]
      Hence, PQ = QB …(2)
      From (1) and (2), we have
      DP = PQ = QB
      Hence, line segment AF and EC trisect BD.

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