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Discussion: Dynamic Geometry Exploration: Properties of the Midsegment of a Trapezoid tool
Topic: Midsegment of a rectangle?
Related Item: http://mathforum.org/mathtools/tool/15621/


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Subject:   RE: More on What is a Trapezoid
Author: Mathman
Date: Dec 10 2004
On Dec 10 2004, lanius wrote:
> I suppose it is legal to respond to your own post. :-)

I'd like
> to hear more about the original discussion that I was trying to
> generate here.

Consider these two definitions:
Def 1 A
> quadrilateral with at least one pair of parallel sides.
Def 2 A
> quadrilateral with EXACTLY one pair of parallel sides.  


For example, of course by definition1, the
> midsegment theorem holds for all parallelograms, but that conclusion
> is never drawn in high schools, because in US high schools,
> parallelograms aren't trapezoids.


Dear Cynthia.  I can't speak for what might be held up in any particular
textbook, but only for the logical development of the subject.  So please
consider:

A sequence might have a value of "at least" two, or another might have a value
of "at most" two. However, if something has a value of "exactly" two, then by
definition it also has a value of at least two, and at most two.  This is a
consequence of the exactness.

A trapezoid is a quadrilateral with a pair of parallel sides ...period.  There
is absolutely no need or increased significance in taking it further.  It is
exctly two, period.  That is the "necessary and sufficient" information to
define a trapezoid.

Now, if you are considering the parallelogram, that has more than one pair of
parallel sides, and so there is more to take into account.  As with numbers, if
a variable has a value of 5, then that's it; Finito.  However, if a variable may
take more than one value and has all values including 5 and greater, then it has
"at least" a value of 5.  So, the expressions are used differently in different
situations, not the same situation.  The parallelogram has "at least" one pair
of parallel sides, so it also is a trapezoid, as "5" is also a solution in the
above example.

Whereas the parallelogram is a trapezoid, it is also more than that.  The
parallelogram is a trapezoid **because it has ALL of the properties of the
tapezoid** ...and then some. The rectangle is a parallelogram because it has ALL
of the properties of the parallelogram ...and then some. The Rhombus is a
parallelogram, and also its diagonals intersect at 90 degrees.  That is, it is a
parallelogram ...and then some. ....and so on.  So, the parallelogram is indeed
a trapezoid since it has "at least" [in that case] one pair of parallel
sides.

What I'm driving at is that although it is true, when solving x - 2 = 3, you
don't usually say "x is at most 5", since "5" is the only option.  You need to
qualify only when there is more than one option.

I hope all of this makes sense.

David.

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