Prove that the quadrilateral formed by joining the mid-points of the pairs of consecutive sides of a quadrilateral is a parallelogram.
Given: In quadrilateral ABCD, E, F, G and H are the mid-points of AB, BC, CD and DA respectively.
To Prove: EFGH is a parallelogram
Construction: Join AC
Proof: In ΔABC, E is the mid-point of AB and F is the mid-point of BC.
... EF =
AC and EF ïï AC...(i)
Similarly, HG =AC and HG ïï AC...(ii)
From eqns (i) and (ii), we get
HG ïï EF and HG = EF ...(iii)
... EFGH is a parallelogram (One pair of opposite sides of a quadrilateral are equal and parallel.)
To Prove: EFGH is a parallelogram
Construction: Join AC
Proof: In ΔABC, E is the mid-point of AB and F is the mid-point of BC.
... EF =
AC and EF ïï AC...(i)Similarly, HG =
AC and HG ïï AC...(ii)From eqns (i) and (ii), we get
HG ïï EF and HG = EF ...(iii)
... EFGH is a parallelogram (One pair of opposite sides of a quadrilateral are equal and parallel.)
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