Understanding the hierarchical classification of quadrilaterals through the ordered relation according to diagonal properties

ABSTRACT

This article's aim is to suggest a supplementary learning environment to understand the hierarchical classification of quadrilaterals for high school or higher degree learners. Three diagonal properties, ‘being congruent’, ‘being perpendicular’ and ‘dividing each other in particular ratio,’ and all possible combinations of these properties, were used to construct the quadrilaterals in a dynamic geometry environment. According to the diagonal properties, 15 quadrilaterals could be constructed and an order relation was constituted on 16 quadrilaterals including the quadrilateral that did not have any diagonal property. The definition of order relation is ‘any quadrilateral Q_i is included by another quadrilateral Q_j, if and only if Q_i has all diagonal properties of Q_j.’ According to this relation, an ordered relation diagram was created, and it was found that this relation was not well ordered. After the dynamic geometry construction of each quadrilateral, observations about the diagonal properties of special quadrilaterals were noted. Furthermore, the conditions under which a quadrilateral can be concave are examined. This alternative approach to the construction of quadrilaterals provided an opportunity to define quadrilaterals with more economical and less confusing way than using angle and side properties. For example, ‘a Kite is a quadrilateral whose diagonals are perpendicular and at least one of the diagonals bisects the other’ and ‘a Trapezoid is a quadrilateral whose diagonals divide each other in same ratio.’

KEYWORDS:

• Quadrilaterals
• diagonals
• hierarchical classification
• dynamic geometry
• ordered relation on the set of quadrilaterals

Acknowledgments

I would like to present my special thanks to reviewer(s) for their valuable critics and suggestions. I also would like to present my special thanks for her detailed language review to my colleague Dr Evrim Erbilgin.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Learners should be involved in the construction process of each case in a dynamic geometry environment. A completed version for each case, which is designed in GeoGebra by the author, can be accessed from http://www.geogebra.org.