Mathematics
In the adjoining figure, ABCD is a parallelogram and E is mid-point of AD. DL || EB meets AB produced at F. Prove that B is mid-point of AF and EB = LF.
Mid-point Theorem
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Answer
Given, DL || EB.
Since, DL || BE we can say that,
⇒ BE || DF
In △AFD,
E is midpoint of AD and BE is parallel to DF,
∴ B is midpoint of AF (By converse of midpoint theorem).
In BEDL,
LD || BE and BL || DE
∴ BEDL is a parallelogram.
Since, BEDL is a parallelogram opposite sides are equal.
Let LD = BE = x.
E is midpoint of AD and B is the midpoint of AF
By midpoint theorem,
BE = FD
FD = 2BE = 2x.
LF = FD - LD = 2x - x = x.
Since, LF = BE = x.
Hence, proved that B is midpoint of AF and EB = LF.
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