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30. In a parallelogram ABCD, E and F the mid points of sides AB and CD respectively Show that theline segments AF and EC trisect the diagonal BD.CD |
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Answer» Given ABCD is a parallelogramHence AB || CD⇒ AE || FCAlso AB = CD (Opposite sides of parallelogram ABCD)⇒AE = FC (Since E and F are midpoints of AB and CD)In quadrilateral AECF, one pair of opposite sides are equal and parallel. ∴ AECF is a parallelogram.⇒ AF || EC (Since opposite sides of a parallelogram are parallel) In ΔDPC, F is the midpoint of DC and FQ || CPHence Q is the midpoint of DQ by converse of midpoint theorem.⇒ DQ = PQ → (1)Similarly, in ΔAQB, E is the midpoint of AB and EP || AQHence P is the midpoint of DQ by converse of midpoint theorem.⇒ BP = PQ→ (2)From equations (1) and (2), we getBP = PQ = DQHence, the line segments AF and EC trisect the diagonal BD of parallelogram ABCD. |
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