Discussion:  All Topics in Geometry 
Topic:  Is a rhombus a kite? 
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Subject:  RE: Is a rhombus a kite? 
Author:  David Chandler 
Date:  May 6 2005 
widely used in physics problem solving is to look at extreme cases. What
happens as the sides of a kite become closer and closer to being equal? (I know
modern math types like to insist that sides are congruent, not equal, but I have
a rebellious streak in me.) Under such circumstances a kite morphs into a
rhombus. Therefore for me it is "useful" to think of a rhombus as a limiting
case of a kite where the sides become equal. In the same way it is useful to
see a parallelogram as a limiting case of a trapezoid where the legs become
parallel. Other such situations include circles and parabolas as limiting cases
of ellipses where the eccentricity goes to 0 or 1, respectively; squares as
limiting cases of both rectangles and rhombuses. Whether or not we insist on a
name change when we reach the limit is not terribly relevant. It is much more
important for students to see (and to learn to look for) the connections.
There is a simple demo that can be done with Geometer's Sketchpad where a
trapezoid, parallelogram, rectangle, rhombus, square, and kite are all
constructed and presented initially as squares. Overtly they are squares, but
they behave differently when the vertices are dragged. It makes sense to name
the shapes according to their behavior, so a square whose only invariant
property, under distortion, is that one pair of opposite sides remains parallel,
would be a trapezoid, etc.
 
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