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# n Learning and Teaching Mathematics - Slaying a geometrical 'Monster' : finding the area of a crossed Quadrilateral

• ISSN : 1990-6811

### Abstract

Varignon's theorem states that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram. The parallelogram thus formed, known as the Varignon parallelogram, has the property that its area is half that of the original quadrilateral. In Figure 1 the Varignon parallelogram EFGH is formed by joining the mid-points of the sides of quadrilateral Proving that the midpoints of an arbitrary quadrilateral doing deed form a parallelogram can readily be accomplished using the midpoint theorem. In Figure 2, one of the diagonals of quadrilateral has been drawn in. Since H and E are the midpoints of the sides of triangle it follows that HE is parallel to Using a similar observation in triangle we have parallel to from which it follows that HE is parallel to Using the other diagonal of quadrilateral it can similarly be shown that is parallel to