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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:   More on What is a Trapezoid
Author: lanius
Date: Dec 10 2004
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.  

I think most US high school text books define a trapezoid with definition 2.
Does anyone know of a US high school textbook that doesn' t define trapezoids in
that way (or equivalant to that)? Does anyone know the history of when or why
that definition came to be used in schools, at least in the US? Is definition 1
used in other countries in current K-12 textbooks?

I am also in agreement that definition 1 makes much more sense hierarchially, so
why is the use of the other definiton so widespread?  Drawing Venn diagrams to
illustrate the set of quadrilaterals using both definitions really illustrates
the differences.
 
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.

I hope this makes sense. I'd like to respond to the spirit of posts here. We are
a heterogenous community, coming here at various levels of mathematical
knowledge. I appreciate so much those of you who know much more mathematics than
I do, but still support me in my learning. I think you are most likely fabulous
teachers. I believe we all have much to learn from each other. Isn't that what
this kind of discourse is all about. Ok, I'll come off my soapbox now. Thanks,
-- cynthia


On Dec  8 2004, lanius wrote:
> Hello,
Check out the highlighted tool on
> http://mathforum.org/mathtools
> ( http://mathforum.org/mathtools/tool/15621/ )It's a really nice Java
> applet that explores the midsegment of a Traqezoid. I was playing
> with this and started wondering, does this also hold for a
> rectangle? Does it hold for all parallelograms? Does it hold for all
> convex quadrilaterals?

I remember having discussions with
> mathematicians that trapezoids should include the family of
> parallelograms. Do you agree? Then should it also include the family
> of triangles (given the applet at the bottom of the page)

What
> would be gained by defining trapezoids as quadrilaterals with *at
> least* one pair of opposite sides parallel? What is lost? And
> similarly, what is gained by defining trapezoids as quadrilaterals
> with *exactly* one pair of opposite sides parallel. What is lost?
> Isn't this how K-12 textbooks currently define trapezoid?

Thanks,
> cynthia

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