
Re: Is a rectangle a square?
Posted:
Sep 28, 2004 4:09 PM


On 28 Sep 04 05:21:38 0400 (EDT), Kit wrote: >Pamela Paramour wrote: >>Is a square a rectangle? ... >> If you refer to Webster, ... > >Here is a nice story realy happend in german tv, sorry for my bad >english. > >In the german quizshow "Wer wird MillionÃ¤r" (Who becomes a >millionaire) from January, 31 2003 the 8000Euro question was: >Every rectangle is: >(a) a rhombus >(b) a square >(c) a trapezoid >(d) a parallelogram. > >In this show _allways_ exactly one answer is (has to be) correct. >The candidate was so confused, she didn't know if c or d is thw right >answer, so she skipped the question and went home (with "just" 4000 >Euro). In the following days the broadcast station got tons of mails, >letters and phone calls. Nearly all "mathematicians" regarded c _and_ >d as correct. The broadcast station told, that they looked up in three >different encyclopaedias, all three saying that trapezoids have only >one pair of parallel sides. Taking this definition only d is correct. > >That's the problem. Who is right: More than 90 percent of the >mathematicians saying a parallelogram is also an trapezoid or three >encyclopeadias saying the opposite? > >The Solomonian solution. In the next week the candidate got a "new" >8000Euroquestion.
You can't really say "who's right." It's just a question of how one defines "trapezoid." In all American textbooks (except the University of Chicago geometry textbook), a trapezoid is defined as a quadrilateral with exactly one (or at most one) pair of parallel sides. A parallelogram is defined as a quadrilateral with 2 pairs of parallel sides.
Professor Conway, on the other hand, defines a trapezoid as a quadrilateral with "at least" on pair of parallel sides. Using that definition, the set of parallelograms is clearly a subset of the set of trapezoids. Thus, using that definition, every rectangle is a trapezoid, and also a parallelogram (as is the case with the usual definition).
A similar problem exists with the definition of the kite. Most writers say it is a quadrilateral in which AT MOST one diagonal is the perpendicular bisector of the other. Conway would say that it is a quadrilateral in which AT LEAST one diagonal is the perpendicular bisector of the other. So using Conway's definition, every rhombus is a kite.
There have been many, many message threads here on this issue. Logically, there is a great deal to be said for Prof. Conway's position.

