MathematicsIn a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (according to the figure). Show that the line segments AF and EC trisect the diagonal BD.

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

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  1. A
    AF and CE bisect BD
  2. B
    AF and BD bisect BD
  3. C
    BF and CE bisect BD
  4. D
    AE and CE bisect BD 

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    Solution:

    Given that ABCD is a parallelogram. Using the property of parallelogram, we say
    AB || DC and AB = DC.
    E is the midpoint of AB
    Also F is the midpoint of CD
    From equation (1) and (2), we get
    AE = CF
    also Using mid-point theorem, we have
    BQ || PQ ---- (3)
    Similarly by taking ∆CQD, we can prove that
    DP = QP ---- (4)
    From (3) and (4), we get
    BQ = QP = PD
    Therefore, we have  AF and CE trisect the diagonal BD.(option1)
    We write it as AF and CE bisect BD.
    So, option 1 is correct.
     
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