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.

# 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.

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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