Q4 of 24 Page 157

Let us prove that, the quadrilateral formed by joining midpoints of consecutive sides of a parallelogram is a parallelogram.


In ΔACD, as E and H are the midpoints of AC and CD respectively.


So by applying the theorem:-


The line segment joining the midpoints of two side of a triangle is parallel to the third side and equal to half of it, we get,


and EH || AD ……… (1)


Also, by using above theorem in ΔABD, as F and G are midpoints of AB and BD respectively,


and FG || AD ……… (2)


From equations (1) and (2) we see that, EH = FG and EH || AD.


As both the above conditions are sufficient for a parallelogram,


⇒ EFGH is a parallelogram


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