[Maths-Education] concentric and congruent circles
Kate Mackrell
katemackrell at sympatico.ca
Thu Jan 28 20:51:37 GMT 2010
I'm in the middle of quite an interesting experience regarding the
definition of these mathematical terms, which I thought might be worth
sharing.
I'm developing some files for an Asian educational software company,
and in the course of this development have been discovering that many
terms have slightly different definitions in different places. One
example is "adjacent angles". To me, these are angles on a straight
line, that add up to 180. To this company, these are angles with one
common arm and a common vertex and no specific sum. Another example
is of course the trapezoid/trapezium - does it have only one or at
least one pair of parallel sides? I've been quite happy to go along
with the the company's terminology (I rather prefer their definition
of adjacent angles), but we've now hit a sticking point on concentric
and congruent circles.
To me (and Wolfram, and lots of other authoritative online
dictionaries of mathematics), two circles are concentric if they have
the same centre. The radii of the circles are immaterial - hence two
congruent circles may or may not be concentric. To the people I'm
working for, concentric circles necessarily have different radii.
I've been quite puzzled by this, but am finally beginning to
understand that it might be about identity: if two circles precisely
overlap (i.e. are congruent and concentric), can we say that two
distinct circles exist? Experience with dynamic geometry software
gives me the automatic answer of "of course" - I can drag two separate
objects about the screen, make them coincide, and then drag them apart
again. I can also construct the two circles to be congruent - but to
have quite different properties. The centre of one circle could be
free to move anywhere, while the centre of the other circle might be
constrained to a line. The radius of one circle might change by
dragging on the circle - and this could determine the radius of the
other circle, but not vice-versa. I've never thought to question the
Euclidean definition of a circle until now - but, in a dynamic
geometry environment, it seems to me that it's simply not sufficient,
as it does not give me any way of distinguishing between two congruent
circles that happen to coincide.
On the other hand, the company wants to still call two coinciding
circles "congruent", which implies that there are still two circles in
a relationship with each other - are they just being inconsistent?
I also find the question interesting of when to consider it legitimate
for a mathematical term to have a flexibly determined meaning (such
as trapezoid/trapezium or adjacent angles) and when to not consider it
legitimate. "Concurrent" has not got any of the mathematical
inevitability of "-" x "-" = "+". I shouldn't be concerned about its
definition being changed. But I find that I am very unwilling to
change.
Thoughts, anyone?
Kate Mackrell
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