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 Discussion: All Topics 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 13 2004
On Dec 13 2004, Annie wrote:
> On Dec 10 2004, lanius wrote:
> I suppose it is legal to respond to

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'll make another comment to make my point clear, then back away, having done
that with respect for the learned opinions of others.  I'm old-school as well
as having a few more recent tricks up my sleeve, and perhaps still difficult to
change, so forgive me for that.

If I'm given certain information about a figure I like to run through it in a
categorical manner in this sense:

I ask, how many sides does it have, and am answered, "Four."
I say "It is a quadrilateral."
And am done with that.

Is it a trapezoid?
I ask, "Does it have two sides parallel?", and am answered" Yes."
I say, "Yes, it is a trapezoid?"
And am done with that.

Is it a parallelogram?
...IF I ask, "Does it have the other pair of sides parallel?", and am answered,
"No."  Then I can not say it is a parallelogram.
And it remains a trapezoid, and a quadrilateral.

...IF I ask, "Does it have the other pair of sides parallel?", and am answered,
"Yes."
I say, "Yes, it is a parallelogram."
And am done with that.  However, that did not deny either that it is a trapezoid
or that it is a quadrilateral from the earlier questions.

Again, so far as I know, a figure is one defined if it has ALL of the properties
of the defined figure, "ALL" meaning those necessary and sufficient as fit its
formal description.  Why not consider the parallelogram as a general trapezoid
having one non-parallel side move towards parallelism with the the other?
Then, as with the definition of limit, there is no distinction in the limit.
This is a finite limit of slope, not such a thing as a circle becoming a line
"at infinity".

Best wishes, and Seasons Greetings.  I'm off to see the children and
grandchildren.  Those are the important things in life, aside from a good
pizza.

David.