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Topic: Midsegment of a rectangle?
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Subject:   RE: More on What is a Trapezoid
Author: Alan Cooper
Date: Dec 14 2004
On Dec 13 2004, Annie wrote:
> As far as I know, the UCSMP books use the more inclusive definition.  > . . .
The topic of the defintion has been discussed a
> bit on the Forum's geometry.pre-college newgroup, and is summarized
> in this Dr. Math read:
> http://mathforum.org/library/drmath/view/54901.html.
For some reason this link didn't work for me from the discussion page although
it did work from the email version of the posting.

For the benefit of any who are following this without getting the emails (and
also because the "higher authorities" seem to be in support of my own view) let
me quote from Dr Math:

"This has actually been discussed quite thoroughly by professionals on
our geometry newsgroups. . . .The bottom line is that these professionals are
most comfortable with a definition like the first one you offer." (referring to
the inclusive one)

and from the referenced geometry newsgroup:

Subject:      Re: Trapezoid definition
Author:       Floor van Lamoen <f.v.lamoen@wxs.nl>
Date:         Wed, 09 Aug 2000 09:55:17 +0200

Hi,

My actual problem with those definitions is not that they are difficult
to deal with. In my honest opinion these definitions are unmathematical
in the sense that mathematics generalizes things.

The trapezoid is a weaker form of a rectangle (which is a weaker form of
a square), and as such theorems on geometric properties of trapezoids
naturally include rectangles (and squares). I am afraid that if one
teaches pupils to be precise on these exclusive definitions, one teaches
them to focus on the wrong things, and perhaps forget the important
concept of generalization.

We wouldn't like to use exclusive definitions for number sets like
Natural numbers, Integers, Rational numbers, Real numbers and Complex
numbers, do we? It's so good that those include each other! The use of
exclusive definitions of - for example - trapezoids, is rather the same.

Kind regards,
Floor van Lamoen.

John Conway wrote:
>
> On Tue, 8 Aug 2000, Floor van Lamoen wrote:
>
> > No, No!!
> >
> > One must call it the "Trapezoid-Rectangle-or-Square Rule" of course,
if
> > one really wants to use exclusive definitions.
>
>     Thanks, Floor!  This just goes to illustrate my point that it's
> so hard to work with the exclusive definitions that even the best of
> us (as I modestly term myself) can't actually manage to do it!
>
>     John Conway

Actually, perhaps Floor didn't get it right either!
in terms of consistently exclusive definitions it should be the
"Trapezoid-Parallelogram-Rectangle-or-Square Rule"

And similarly, the theorem discussed earlier in this topic area should be "If a
midsegment of a quadrilateral has length equal to the average of the lengths of
the two edges it separates, then the quadrilateral is a Trapezoid, or a
Parallelogram, or a Rectangle, or a Square"

A useful moral to this story may be that textbooks aren't always right and may
even be consistently (or at least predominantly) wrong.

cheers,(for "our" side),
   Alan




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