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Discussion: All Topics in Geometry
Topic: Is a rhombus a kite?


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Subject:   RE: Is a rhombus a kite?
Author: Mathman
Date: May 6 2005
On May  6 2005, subbu wrote:

Sorry, but there is an underlying principle of "necessary and sufficient".  The
"sufficient" part requires that there be no extraneous information in the
definition.  It also makes any additional information redundant. Basically, what
it means is that there is enough information, and only enough information to
accurately draw the figure being described, and only that figure.  Additional
information is then called "properties" of that figure.  For example, the
diagonals of a parallelogram are equal, but that need no be part of the
definition.

> Kite: Plane shape having two sets of equal sides and one set of
> opposite angles are equal

The equal sides need to be stated as being adjacent, otherwise they could be
drawn opposite, and the "plane shape" needs to be defined first and definitely
as a quadrilateral.  The above definition also fits a parallelogram [if only
four-sided] which does not necessarily have the kite shape.

Rhombus: parallelogram with four equal
> sides and equal opposite angles

A parallelogram with two adjacent sides equal is sufficient.  The rest follows
from that.

square: four equal sides and four
> right angles, opp angles that are parallel, two diagonals that
> bisect at right angles, four lines of symmetry

Four equal sides and one right angle is sufficient.

parallelogram: a
> quadrilateral with opposite sides that are parallel and of equal
> length and opp
angles that are equal.

A quadrilateral with opposite sides parallel is sufficient.

My definition for kite: is
> same as defined by Tusk. It must be an equilateral quadrilateral.

That implies that all sides are necessarily equal as in any other equilateral
figure.  It need not be entirely equilateral.  One pair of adjacent sides can be
equal, as is the other pair, but those two pairs are generally different from
each other, even if not so in a particular instance.  Think of a "traditional"
kite used for flying.  The bottom is usually longer than the top, but top and
bottom are formed from two usually non-identical equilateral triangles.  The
definition must be a general definition which holds true also for the particular
case.

David.



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