Mathematics
In the given figure, D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Prove that ADEF is a parallelogram.
Mid-point Theorem
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Answer
Given,
D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Thus,
AD = DB, AF = FC and BE = EC
By mid-point theorem,
The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.
Since, D and E are the mid-points of AB and BC respectively.
⇒ DE || AC
⇒ DE || AF …..(1)
Since, F and E are the mid-points of AC and BC respectively.
⇒ FE || AB
⇒ FE || AD …..(2)
In quadrilateral ADEF,
AD // FE and DE // AF
Since, opposite sides of quadrilateral ADEF are parallel.
∴ ADEF is a parallelogram.
Hence, proved that ADEF is a parallelogram.
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