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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: Midsegment of a rectangle?
Author: Mathman
Date: Dec 9 2004
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

I have to ask, with all of the changes and recent trends in education ....Does
no-one study formal geometry anymore?

Draw a line parallel to one of the non-parallel sides, and you have a
parallelogram and a triangle.  It is shown formally that the line joining
midpoints in the parallelogram will be the same length as the shorter parallel
of the trapezoid.  In the triangle it is also readily shown formally that the
line joining the midpoints is half the base. Add them for the result.

>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)

They are all polygons, the triangle the simplest.  After that you have
quadrilaterals, which include the simplest having no special quality, then the
trapezoid with *at most* two parallel sides.  After that you have the
parallelograms which include the rectangle, square, and rhombus.  They are all
inter-connected, of course, but are distinguished by each's special quality.
The discovered properties of a lesser member are automatically those of the
higher, the latter gaining some other item that makes it distinct.  Their
definition uses the principle of "necessary and sufficient", and can be singular
or several.  The Rhombus can be thought of as a parallelogram with *one* pair of
adjacent sides equal [the "parallelogram" takes care of the rest], or a
parallelogram whose diagonals intersect at 90 degrees.  A square is a rectangle,
is a parallelogram, is a trapezoid, is a quadrilateral.  So the expectation of
similar results is not surprising after all?

To answer your last query, a parallelogram is a trapezoid, having "at least" one
pair of parallels AND being quadrilateral [so excluding the hexagon etc..]  The
Trapezoid has "at most" AND "exactly" one pair of parallels, being also a
quadrilateral [either/or].  Likewise, the parallelogram has exactly two pairs of
parallels, and it also has at most two pair. Either will do, since they
precisely define the trapezoid, a quadrilateral with .....pick one. The reason
is that there are no four sided figures with fewer, and those with more have at
least that many.  In fact, you might say more simply, "A trapezoid is a
quadrilateral with one pair of parallel sides." and qualify it no further.

These are interesting structures since they deal with similarity and congruence;
parallel pencils, and coincident pencils, and lead to all sorts of other
interesting topics in geometry.

David.

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