Timeline for Defining an Isosceles Trapezoid
Current License: CC BY-SA 4.0
26 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2024 at 8:44 | audit | Suggested edits | |||
| Jul 30, 2024 at 8:44 | |||||
| Jul 3, 2024 at 13:22 | vote | accept | Suamere | ||
| Jul 2, 2024 at 22:45 | history | edited | Geoffrey Trang | CC BY-SA 4.0 | fixed obvious typos |
| Jul 2, 2024 at 7:58 | history | edited | ryang | CC BY-SA 4.0 | deleted 101 characters in body; edited title |
| Jul 1, 2024 at 19:04 | comment | added | Blue | @Suamere: Comments aren't for discussion, so this may be my last reply ... I'm just gonna say "It depends". Besides, even when consensus exists "in academia", an author shouldn't assume every reader will be familiar with it, so it helps to be explicit about what you mean in the here-and-now. You could do this w/disclaimers: "This trap formula works for parallel sides $a\neq b$." You can also explicitly limit the scope of usage: "For the purposes of this discussion (or even this portion of it, or even just this formula), a 'trap' is taken to mean blah-blah-blah". Etc. ... Good luck! | |
| Jul 1, 2024 at 18:44 | comment | added | Geoffrey Trang | @Suamere Of course, a square is a rectangle. The only problem is with the "equal legs" definition of an isosceles trapezoid. With that definition, any parallelogram (with either pair of parallel sides chosen to be the "bases" and the other pair the "legs") would be an isosceles trapezoid. But in fact, rectangles are the only parallelograms that should be called isosceles trapezoids. | |
| Jul 1, 2024 at 17:54 | comment | added | Vasili | I've seen trapezoid with exactly one pair of parallel sides defined as strict trapezoid and the isosceles trapezoid definition is given for strict trapezoids only. Your example would not be an isosceles trapezoid under these definitions. | |
| Jul 1, 2024 at 17:48 | comment | added | Suamere | @GeoffreyTrang But that's where my illustrative question comes into play. By your reasoning, a Square is not a Rectangle. It's a Square only. | |
| Jul 1, 2024 at 17:45 | comment | added | Suamere | @Blue Thanks again! I was hoping somebody would bring up the area formula. And your tone seems to agree with my stance that "my formula doesn't work if you define the shape that way" isn't a good reason not to define a shape some way. The shape is the shape, and formulas should take the shape's properties into account. Perhaps a non-symmetric (p-gram) isos trap should simply have a different formula. That doesn't preclude it from matching the definition of an isos trap. Yes? | |
| Jul 1, 2024 at 17:40 | comment | added | Blue | Opinion-wise, I tend to be an inclusionist, so that I can name things based on minimal info. If I have a quad with one pair of parallel opposite sides, but don't (or can't) know how their lengths compare, or what's up w/the other sides, I may feel comfortable in calling it a trapezoid ... especially if the unknown properties are subject to change (as in a "dynamic" family). So, is a parallelogram an isos trap? Sure! Well, maybe. After all, in this old answer, I derive a trap area formula using an argument that fails for p-grams. Go figure. | |
| Jul 1, 2024 at 17:38 | answer | added | ryang | timeline score: 3 | |
| Jul 1, 2024 at 17:16 | comment | added | Blue | @Suamere: "I need exacting rules." You won't find them. :) Definitions are context-dependent. I, for one, tend to allow vertices of a triangle to be collinear, so that my preferred rendition of the Triangle Inequality is $a+b\geq c$; others disagree ... and I'm often one of them. My usage can hinge on whether I'm considering a dynamic family of triangles that might reasonably include "flat" members, vs discussing a fixed geometric figure that simply doesn't make sense with flats and where the "strict" Triangle Inequality helps. | |
| Jul 1, 2024 at 17:13 | comment | added | Geoffrey Trang | If "the legs" are defined to mean "choose any pair of parallel sides, then the other two sides are called the legs" in the case of a parallelogram, then any parallelogram would meet the "equal legs" definition of an isosceles trapezoid. An isosceles trapezoid should instead be defined to mean either an exclusive trapezoid with equal legs or a rectangle. A non-rectangular parallelogram is not an isosceles trapezoid. | |
| Jul 1, 2024 at 17:09 | comment | added | Suamere | @Malady I suppose you're all right. This is a lot of discussion just to get to a definition, for which it appears to be far too fluid. In that case, I've added my followup question as you previously suggested. | |
| Jul 1, 2024 at 17:09 | history | edited | Suamere | CC BY-SA 4.0 | Defining an Isosceles Trapezoid |
| Jul 1, 2024 at 17:05 | comment | added | Malady | @Suamere define them in the way which is convenient for your interesting question. There is no council to appeal to. | |
| Jul 1, 2024 at 16:59 | comment | added | Suamere | @Vasili Agreed. And I'd say that the bases (top and bottom) are the two sides that meet the original requirement of "at least one set of parallel sides". The "legs" then are the left and right sides, whether they be parallel or not. So inclusivity works fine here. | |
| Jul 1, 2024 at 16:57 | comment | added | Suamere | @Blue Very fair point. As an Engineer, I need exacting rules, if not explicitly stated ambiguity. But you could be right that not enough people agree. Love the analogy. Hoping for any respectable answer at all though, with any source from something beyond an elementary teaching website or wikipedia. Anything collegiate or professional. | |
| Jul 1, 2024 at 16:57 | comment | added | Vasili | You have to be careful with an inclusive definition because what are legs and bases of a parallelogram? Of course, you can further define that legs and bases apply only to trapezoids with one pair of parallel sides. | |
| Jul 1, 2024 at 16:54 | comment | added | Blue | In these situations, I think back on the wisdom of Humpty Dumpty in Lewis Carroll's Through the Looking Glass: "When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean — neither more nor less." ... There is no naming authority in mathematics, so the literature is replete with competing definitions and conventions. (It's worth noting that we can't even decide whether zero is a natural number.) A thoughtful author will take care to precisely define a term that might have multiple meanings. | |
| Jul 1, 2024 at 16:53 | comment | added | Suamere | I love that, but I feel it's leading the witness. Or tainting the results. I'm extremely torn about doing that. I'd rather have people say Oh, in THAT case yeah, the definition is wrong. But only after nailing down (or allowing ambiguousness) is solidly determined here. | |
| Jul 1, 2024 at 16:51 | history | edited | J. W. Tanner | edited tags | |
| Jul 1, 2024 at 16:51 | comment | added | Malady | Maybe write your follow-up question in your original post! That might give some context for what you need these definitions for. | |
| Jul 1, 2024 at 16:47 | comment | added | Suamere | @Malady Thanks! I do plan to use the accepted answer/reference as a proof for a different follow up question. Assuming the accepted answer agrees with what you and I said. In a way, I hope or expect the accepted answer to have some exclusivity, which will stop me from doing a different follow up question. :/ | |
| Jul 1, 2024 at 16:45 | comment | added | Malady | In “real” mathematics, ambiguous terms are defined as needed. Don’t worry too much about teaching precisely “correct” definitions. However, in my schooling I learned the inclusive definitions. I learned that squares are rectangles. | |
| Jul 1, 2024 at 16:42 | history | asked | Suamere | CC BY-SA 4.0 |