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Topic: New Rule for Naming Polygons
Replies: 1   Last Post: Apr 29, 2000 2:47 PM

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JIMMY

Posts: 7
Registered: 12/6/04
New Rule for Naming Polygons
Posted: Apr 15, 2000 5:37 PM
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In June 2003, I think it will be time to use a new rule for naming
polygons. Are you ready to hear it??

The first change is that the word "triangle" will be dropped
completely from our language. The 2 new words will be "trigon", for
equilateral triangles, and "trilateral", for any other triangle.

Also, there will be 2 new terms for special polygons with any number
of sides:

Regular: A regular polygon is a polygon with all sides congruent and
all angles congruent.

Semiregular: A semiregular polygon is a polygon that is not regular,
but whose vertexes made by its angles all lie on one circle of a given
size.

Note that using these definitions, triangles are always either regular
or semiregular. In other words, a regular 3-sided figure will be
called a trigon and any other 3-sided figure will be called a
trilateral. Polygons with 4 or more sides, however, can be regular,
semiregular, or neither. For naming 4-sided polygons, we will use all
the same terms used earlier, but change the definitions as follows:

Kite: A quadrilateral with 2 pairs of consecutive congruent sides, but
opposite sides not NECESSARILY congruent.

Trapezoid: A quadrilateral with AT LEAST one pair of parallel sides.

Isosceles trapezoid: A trapezoid whose BASE ANGLES are congruent.

Parallelogram: A trapezoid whose opposite angles are congruent.

Also, note the following theorem:

A trapezoid is also a semiregular quadrilateral if and only if it is
an isosceles trapezoid but not a square. (A square is the regular
quadrilateral.) So, we now have a new acceptable definition of an
isosceles trapezoid: a quadrilateral that is both a trapezoid and
either regular or semiregular. Also, we can extend this theorem to
saying that a quadrilateral is semiregular if and only if its opposite
angles add up to 180 degrees but is not a square and that a
quadrilateral is regular if and only if it is a square.

So now we have trigon and square for regular polygons with 3 and 4
sides, respecively, and trilateral and semiregular quadrilateral for
semiregular polygons with 3 and 4 sideds, respectively. Also, a
quadrilateral that is neither regular nor semiregular will be called a
"4-side". The word "quadrilateral" will be the only word retained in
our language that can be used to refer to polygons with a given number
(4) of sides that can be regular, semiregular, or neither.

The terms "pentagon", "quintilateral", and "5-side" will be used for
5-sided figures that are regular, semiregular, and neither,
respectively. The terms "hexagon", "sextilateral", and "6-side" will
be used for 6-sided figures that are regular, semiregular, and
neither, respectively.

Now, let me go from 3 to 12 sides, and make a list of all the
acceptable names for polygons that are regular (all sides are
congruent and all angles are congruent,) semiregular (vertexes all lie
on a circle but sides are not all congruent,) and neither.


3 trigon, trilateral
4 square, semiregular quadrilateral, 4-side
5 pentagon, quintilateral, 5-side
6 hexagon, sextilateral, 6-side
7 heptagon, septilateral, 7-side
8 octagon, octilateral, 8-side
9 enneagon, nonilateral, 9-side
10 decagon, decilateral, 10-side
11 hendecagon, undecilateral, 11-side
12 dodecagon, duodecilateral, 12-side
General regular polygon, semiregular polygon, polygon that is neither
regular nor semiregular

Other terms:

polygon: any number of sides
quadrilateral: 4 sides, can be regular, semiregular, or neither
kite: quadrilateral with 2 pairs of consecutive congruent sides
trapezoid: quadrilateral with at least one pair of parallel sides
isosceles trapezoid: trapezoid with base angles congruent
parallelogram: trapezoid with opposite angles congruent
rhombus: quadrilateral that is both a kite and a parallelogram
rectangle: trapezoid that is both an isosceles trapezoid and a
parallelogram

Note that "3-side" is not included because all 3-sided polygons are
either regular or semiregular.

That's it! Now we have 3 types of polygons (regular, semiregular,
neither) and 3 languages (Greek, Latin, English,) so we can think of
one as corresponding to the other in the following way. Can you PLEASE
reply to this message, telling me if you have any comments??





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