# Identification of a quadrilateral as a trapezoid, rectangle, or square

Yesterday I was tutoring a student, and the following question arose (number 76):

My student believed the answer to be J: square. I reasoned with her that the information given only allows us to conclude that the top and bottom sides are parallel, and that the bottom and right sides are congruent. That's not enough to be "more" than a trapezoid, so it's a trapezoid.

Now fast-forward to today. She is publicly humiliated in front of the class, and my reputation is called into question once the student claims to have been guided by a tutor. The teacher insists that the answer is J: square ("obviously"... no further proof was given).

1. Who is right? Is there a chance that we're both right?

2. How should I handle this? I told my student that I would email the teacher, but I'm not sure that's a good idea.

• You are right. You can construct many counterexamples by changing the angle between the two sides of length 9. – onetoinfinity May 19 '15 at 22:26
• You are aware that whoever created the question most likely intended the answer to be "square" and just wasn't meticulous enough to notice that with too few angles specified it need not be one, I suppose? – Daniel Fischer May 19 '15 at 22:33
• Tell your student if this is for an exam, the answer can be a square (depends on the mindset of the teacher). If this is for his/her own learning purposes, the answer is a trapezoid. – achille hui May 19 '15 at 23:06
• Tell the truth. You based your answer on the fact that the angles of the right side are unspecified, and the fact that the angles on the left are explicitly noted as being right angles. This leaves you with a question the student should ask the teacher: if the angles on the left side are explicitly stated as right angles, why is the right side not explicitly stated if the intended answer is a square? In other words: the question is confusing and poorly presented (assuming the teacher is "correct"). Use this as an opportunity to teach the student to think and explain/defend their answer. – jpmc26 May 20 '15 at 0:21
• Another issue with this poorly-constructed question is that if the figure is considered a square, it is then also a rectangle and a trapezoid, so only one answer could be considered wrong. – Kimball May 20 '15 at 3:55

Of course, you are right. Send an email to the teacher with a concrete example, given that (s)he seems to be geometrically challenged. For instance, you could attach the following pictures with the email, which are both drawn to scale. You should also let him/her know that you need $5$ parameters to fix a quadrilateral uniquely. With just $4$ pieces of information as given in the question, there exists infinitely many possible quadrilaterals, even though all of them have to be trapezium, since the sum of adjacent angles being $180^{\circ}$ forces the pair of opposite sides to be parallel.

The first one is an exaggerated example where the trapezium satisfies all conditions but is nowhere close to a square, even visually.

The second one is an example where the trapezium visually looks like a square but is not a square.

Not only should you email the teacher, but you should also direct him/her to this math.stackexchange thread.

Good luck!

EDIT

Also, one might also try and explain to the teacher using the picture below that for the question only the first criterion is met, i.e., only one pair of opposite sides have been made parallel.

• Note: "Trapezium" (UK) = "Trapezoid" (US). Another confusion. – GEdgar May 21 '15 at 12:04
• I like your first picture. Your second will only really help if there is confusion about the definitions of the shapes involved. I don't think that's the case. – Teepeemm May 21 '15 at 15:04
• @Teepeemm Judging from the way the instructor seemed to have answered the question, it does seems the case that (s)he is confused about the definitions of the shapes involved. – Adhvaitha May 21 '15 at 15:49
• I disagree. I think the second picture only helps if someone thinks the first picture is a square. The fundamental problem the teacher and the test makers assumed that another side or angle were congruent just because they appeared to be. – Teepeemm May 21 '15 at 17:36
• I think the first picture would be better if it had a 9.1 and an 8.9 or whatever it ends up being with an 89 degree angle or so, and the other angles marked. – Random832 May 22 '15 at 2:36

Clearly the figure is a trapezoid because you can construct an infinite number of quadralaterals consistent with the given constraints so long as the vertical height $h$ obeys $0 < h \leq 9$ inches. Only one of those infinite number of figures is a square.

I would email the above statement to the teacher... but that's up to you.

As for the "politics" or "pedagogy" of drawing a square, but giving conditions that admit non-square quadralaterals... well, I'd take this as a learning opportunity. The solution teaches the students that of course any single drawing must be an example member of a solution set, but need not be every example of that set. In this case: a square is just a special case of a trapezoid.

The solution goes further and reveals that the vertexes (apparently) lie on a semi-circle... ("obvious" to a student). A good followup or "part b" question would be to prove this is the case.

• Right, and I showed this to my student yesterday. I'm wondering if the class is just at that place where, if you draw [what kids call a square], then it's a square. – The Chaz 2.0 May 19 '15 at 22:27
• @TheChaz2.0 If that's the case, then it's pretty silly to give any angles or any side lengths, as it's just as "obviously" a square with or without that information. Or so I would think... – pjs36 May 19 '15 at 22:29
• What software did you use to make the diagram? – GFauxPas May 20 '15 at 1:00
• Top notch picture... is proof in of itself that the bottom and right can be the same length, the left angles can be 90(deg) and the figure not a square. Made peffect sense the second I saw it. – WernerCD May 20 '15 at 12:53
• That picture itself is an amazing answer. – Barry May 21 '15 at 2:27

FWIW, this question appears to come from a diagnostic test which can be perused at http://web.archive.org/web/20161228001122/http://www.mathmatuch.com/presentations/diagnostic_test.pdf -- where the official answer is given as J (the square). So it's not just the teacher who is wrong.

(Remark: I found the site by googling on "identify the figure shown" and "trapezoid" then looking for "76" and "J" in the results.)

Update Feb. 17, 2020: The original link (above) is now dead. By googling again on "identify the figure shown" and "trapezoid," I did find another version of the diagnostic test at https://blevinshornets.org/teachers/wp-content/uploads/sites/6/2018/12/diagnostic_test-7th.pdf with the same error. Interestingly, a similar test can be found at http://stpatrickschoolstoneham.org/wp-content/uploads/2018/06/Rising-6th-grade-math-packet.pdf but problem 76 there is somewhat different and has no error.

• +1 Interesting, when I look up the publisher wiki's entry, it has been ranked the $10^{th}$ worst company to work for in the state. Not sure whether there has anything to do with the quality of its test. – achille hui May 19 '15 at 23:46
• The sad thing is that lifes are destroyed by such tests set up by mathematical morons. – Christian Blatter May 20 '15 at 15:32
• This is one of many problems with standardized tests, and I say this as someone who is very good at taking standardized tests. What's really frustrating is when you're taking a standardized test and the correct answer isn't any of the ones presented. I remember this happening on a teacher certification test I took quite a while back. It was a simple physics problem but the correct answer wasn't one of the four listed. I re-read the question multiple times to make sure there wasn't some detail I missed, but, no, just a bad question. (I'll note that the test was heavily biology-centric.) – Ben Hocking May 20 '15 at 21:56
• Q73 asks for the length of a frog, which is positioned against a ruler. But not positioned in any way that would facilitate determining the length. This would be a fun test to answer facetiously! EDIT: Oh, as @DavidK answers. – OJFord May 21 '15 at 9:28
• Ironically here is one of their glassdoor reviews from an employee: glassdoor.com/Reviews/… "Drastically undervalue people's experience and input. Middle Management has been promoted from various positions within education or sales and doesn't have solid foundation in product development and launch to manage teams well. Marketing leadership lacks insight, leadership, and notion of team development. Too much focus by middle managers to checking off boxes, and doing things because they think they should be done. " – don bright May 25 '15 at 13:08

The point is that mathematically, you can't tell from the picture. It might be this:

It is easy enough to describe a construction of this with compass and straightedge, so it is definitely a legitimate geometric figure by any reasonable definition.

The same "diagnostic test" from which this came (thanks to Barry Cipra for finding it) has numerous other zingers like this where assumptions are made based on the fact that you can't tell from the picture whether two segments are equal or whether two angles are equal, so we assume they are equal.

Who measures the lengths of frogs that way?

• I glanced though the test and had exactly the same thought. – WoJ May 20 '15 at 15:03
• Yeah, it's a really poor test. – Reinstate Monica -- notmaynard May 20 '15 at 16:23
• How the OP talks about this is a little worrying. The OP spotted a gotcha that even the test makers didn't spot but appears to be in some kind of conflict of wills with a child who's at the level of being asked to determine the length of the frog that is so serious that they're worried about their reputation. At that age if my teacher had brought up say, some example involving imaginary numbers technically allowed by a mistake in the phrasing of the question and actually got into a serious conflict with me I'd have only been put off math, not educated. – Murphy May 20 '15 at 16:54
• If the tutor's answer delved into mathematics beyond what the question asked about, this would be a different question. The test clearly expects knowledge of what makes something a mere trapezoid rather than a rectangle, and clearly indicates just enough information to establish a trapezoid (unless you choose to regard the right-angle symbols as superfluous). So we're left trying to guess what answer the test makers wanted, based on which rules of math they arbitrarily chose to think about that day. That is what I think is likely to turn students off math. – David K May 20 '15 at 17:15

I can't help but say something... As noted in other answers, this is clearly a trick question, playing on deliberately misleading visuals, and potentially on delicate (non-universal!) semantic conventions. (I am disturbed by the idea that, for example, a "square" is not a "rectangle", because, supposedly, "rectangle" only refers to (actual) rectangles that are not squares, etc.)

The element(s) of "arbitrary/capricious authority" that enter in both the context and in responses is completely unsurprising, but also chronically upsetting to me. Such episodes advertise the apparent utility of mathematics for creating and enforcing arbitrary, unfathomable rules, as well as highlighting the specific irrationality of "external, uncommunicative, ineffable" authority. Really ugly.

Let's admit to the kids that the picture was drawn to look like the dang thing was a square. Seriously! It's not a klutz's drawing of a real thing, it's a test question. It's not that we have to wonder about the verisimilitude of an inadequate reporter, but, rather, to wonder about the ulterior motives of people at ETS in New Jersey, etc.

In particular, instead of the too-popular traditional rather sub-verbal responses to such questions (is it an X, or not?), there should always be sufficient room to explain/address the genuine issue, as opposed to merely-semantic, merely artifactual. That is, we should teach kids to write prose that says "well, the picture makes the figure look like a square... the given data wouldn't itself physically require that it be a square, but what bumpus would draw a thing to look like a square if it wasn't?..."

(Seriously, very many peoples' physical intuition is excellent, but then we consistently prank them so that they think that there's scant connection to mathematics, which is completely false. We should teach kids to trust their physical intuition at least as a first approximation! Math is not perversity!)

But, yes, for multiple-choice tests, ABSOLUTELY tell your kids to deconstruct the stupid things, and imagine what the test-maker was thinking. For that matter, we should admit to the kids that those test-makers have a streak of mean prankishness that they (the kids) should be aware of. Too bad.

• Though I understand where you are coming from, aren't there also mathematical situations where data genuinely look like something they aren't (for example, aliasing from undersampling)? – user88319 May 21 '15 at 1:37
• @Strants, oh, yes, indeed, sometimes things misrepresent themselves, and it's good to be alert to that. For that matter, sometimes one deceives oneself via inadvertent semantic boo-boos. But, unlike a contrived testing situation, in those cases we behave as though there were a genuine underlying truth, rather than a truth accessible only to a capricious authority. – paul garrett May 21 '15 at 11:58
• My issue is that three of the four options are correct answers. This isn't a case of a trick question or a kid (or tutor) over-thinking it. This is a bad question. – Kevin May 21 '15 at 12:02
• @Kevin, yes, also it is a bad question, but/and to analyze it as though there were underlying truth is a waste of energy. – paul garrett May 21 '15 at 12:17
• If the "correct" answer was trapezoid then this would be an example of a "trick question". But since the test's answer key indicates that the answer should be square, it's not — only a very bad question. – hobbs May 21 '15 at 14:50

I agree with @davidgstork re: the first question.

As for your second question, it's important to get the word out that it is a trapezoid, but you'll need to draw a few diagrams that actually show the trapezoids that conform to the conditions. (As they say on standardized tests all the time, just because it looks like a whatever, doesn't mean it automatically is a whatever unless specifically told.) I'd e-mail diplomatically, of course, but with clear diagrams.

In our 5th grade math group, our teacher came across this multiple choice problem that we thought about for a short period of time. We discussed the answer, which we concluded would be a trapezoid, square, or a rectangle. We also discussed what the tutor should do about this problem. We had varying answers, from the tutor having a conference with the teacher with/without the student to the tutor just letting it go. We also thought this was a very cool situation and we all could relate to having us being right and the teachers being wrong.We thought it could be a good idea to include the students when the tutor discusses the problem with the teacher because the students could state their thinking about the problem.

Of course the mathematically correct answer is as the OP and others have stated. However, if you showed this drawing to an architect, engineer or carpenter they would probably assume that all sides are 9 inches - you don't usually indicate all measures, but assume that the missing lengths are equal to the opposite sides.

• Exactly! Such questions are perverse. We should not teach students to be paranoid about "gotchas", which are not at all the primary issues and ideas in mathematics. Many pranks and traps are not even of much secondary interest, except to illustrate that the test maker(s) do not know any genuine mathematical questions to ask. Tsk! :) – paul garrett May 20 '15 at 18:25
• If you want to be good at maths, you do need to be paranoid about "gotchas". They come up all the time in the real world by themselves! – Brice M. Dempsey May 21 '15 at 8:00
• I don't think that's true at all. Admittedly I'm a mere 'engineer in training' - but if you showed that diagram to me I'd think "Right - these are 9in, these are 90deg, and I'm free to do whatever's most convenient to me with the others. Done: i.stack.imgur.com/rWjZM.png". This happens a lot. And it's usually good. When it isn't, it's the fault of the specification; not the work. – OJFord May 21 '15 at 9:36
• Naturally, when you are doing something practical, you will make a lot of assumptions to get a conclusion. But, make "one too many" assumption and your project will break apart. This is why mathematics exists in a first place, to have a tool that allows us to investigate real life problems and find errors in our "natural thinking". This is an exercise in "seeing" beyond your senses, finding multitude of possibilities instead of one. This is a quality everyone should cultivate. – Ennar May 21 '15 at 18:55
• @Ennar, I absolutely do not see any disconnect between "mathematical thinking" and "natural thinking". – paul garrett May 21 '15 at 22:01

1. Who is right? Is there a chance that we're both right?

2. How should I handle this? I told my student that I would email the teacher, but I'm not sure that's a good idea.

1. As explained by other answers, you are right.

2. Rather than feel humiliated, the student could explain why it is a square, showing counterexamples, and feel like a genius. I would also call the teacher, and I will explain why:

First, as for the student: Part of being a tutor is making sure the student understands the "why." This is a great opportunity for them to learn to stand-up against peer pressure (and faculty!).

Second, if you approach the teacher carefully, they might give the student the opportunity to explain to the class the source of the misconception. That would boost their confidence and be a learning experience for all. If you think there is a chance for confrontation:

You are aware that whoever created the question most likely intended the answer to be "square" and just wasn't meticulous enough to notice that with too few angles specified it need not be one, I suppose? – Daniel Fischer♦ 22 hours ago

This angle allows to have the student say "Well, the author of the test intended... but..." which doesn't come-off as confrontational.

• well good for you so magnanimous and generous i don't see any moral obligation to be non-confrontational though. it's like henry mccord in this one episode of madam(e) secretary where his wife was being subjected to some senate hearing or something but then it was like the way it was done reeked of dishonesty. therefore, henry mccord, when subpoenaed refused to be honest and intended to commit perjury. this was despite that he was a religious, philosophy or ethics professor. – BCLC Dec 1 '20 at 1:06
• but 1stly, the student was humiliated? i think BOTD is off the table. 2ndly, i mean, additionally, others even point out like the writer goes through the trouble of putting those 2 angles there. i mean obviously this isn't kindergarten teaching or anything (see next comment) so there's really some deduction to be done. it's not like you just look and that's that. 3rdly the whole 'OBVIOUS'. i mean come on. anyone who uses 'obvious' really deserves scrutiny in this case, where the case is like...it's like when people ask you a trick question but then they're the ones who are wrong. – BCLC Dec 1 '20 at 1:08
• related? math.stackexchange.com/q/2249741 and matheducators.stackexchange.com/questions/13700/ and matheducators.stackexchange.com/questions/13831/ and korean.stackexchange.com/questions/3394 and chinese.stackexchange.com/questions/29214 – BCLC Dec 1 '20 at 1:08

It's a trick question. The answer, as you concluded, is a trapezoid.

The figure is deliberately drawn to look like a square to fool the unwary.

The best procedure to teach your student how to determine and prove such things for herself. Once she can prove to herself the figure is a trapezoid, your "reputation" is irrelevant. Math does not depend on reputations, it depends on proof (thank god).

• Apparently it also fooled the people who created the test... – TravisJ May 20 '15 at 19:54
• @TravisJ ? The people who designed the problem know the correct answer. It is only this local teacher who got it wrong. – Tyler Durden May 20 '15 at 21:25
• Apparently not only the local teacher – Daniel Fischer May 20 '15 at 21:36
• "To fool the unwary"? Genuine mathematics is not created by pranksters to try to fool each other. Genuine problems are often difficult enough without nincompoops creating fake issues. This is why I am ever more disaffected with "school mathematics", despite having considerable enduring affection for real mathematics. I am not at all surprised that many kids grow up with a bad feeling about math, since it is too-often used as a way to prove to kids that the caprices of authority ... win every time... and that there's no sense to it, and that they're trying to prank you. Ugly. – paul garrett May 20 '15 at 22:39
• @TylerDurden - That's a pretty significant typo, to match what's drawn rather than what's specified. In fact, an answer key having a typo invalidates the entire concept of multiple choice testing. – Bobson May 22 '15 at 11:23

In Singapore math questions of this nature, they almost always preface the question statement with the phrase: "not drawn to scale".

That phrasing might seem redundant, but in cases like this, it becomes so very important. Even if the figure is printed as a perfect square, the disclaimer that the figure is not to scale means that no conclusions at all should be assumed from studying its general shape. Not even the acuteness/obtuseness of angles should be assumed from line segment orientation. Only angles, sides and relationships that are explicitly defined may be assumed in solving such questions.

If that phrase had been included, I would have absolutely no hesitation in stating that the only possible correct answer is "trapezoid" (or "trapezium" as we refer to it over here) and that "square" is totally wrong.

However, without that disclaimer, a case, however weak it may sound to a rigorous mathematician, may be made that the evidence of the senses (and actual measurement) indicate that it's a square, therefore the answer "square" is also acceptable! In fact, three of the answers now become admissible - trapezoid, rectangle and square, again invalidating the expected single choice format.

So either way you look at it this is a very poor question.

With regard to the other point about the student being humiliated for giving the "most correct" answer, tell her there's no shame in it, and she shouldn't feel bad about it. I know those are hollow words as I've been in the same position myself many years ago, having been embarrassed by my (completely correct) answer being dismissed by a Physics teacher who didn't know what he was talking about (he later sent out an erratum to correct his error, without apologising to me (or even acknowledging my correctness)). Did that hurt? Yes. Did I survive? Certainly.

We must remember that teachers are human, with very human foibles. They are certainly fallible.

The right vertical side of 9 inch is free to rotate about any of two vertices 1. upper-right vertex & 2. lower-right vertex without changing any of the conditions provided. This rotation shows that the quadrilateral is a trapezoid (having two right angles & two parallel sides not necessarily equal in length).

Thus, the resulting figure (given here) is generally a trapezoid not a square. It will be a square only if the angle between the sides of 9 inch is $90^o$ as an additional condition for this question. Assuming this condition (although not given here) some answer it as a square that is absolutely wrong.

1. Who is right? Is there a chance that we're both right?

1. How should I handle this? I told my student that I would email the teacher, but I'm not sure that's a good idea.

Unless your pupil is in a great danger (losing scholarship or similar), I think you should not contact the teacher.

Instead, talk with your pupil, explain honestly the situation, tell her/him that similar situations are frequent, and that she/he should be proud of seeing better/deeper than the teacher, but at the same time do not disrespect the teacher.

• Why would she respect the teacher who is too narrow-minded to admit mistake or cannot comprehend it? Everyone can make mistake, there is no shame in that, but the teacher that can't appreciate being corrected by pupil should not be respected. I am a maths teacher myself. – Ennar May 21 '15 at 18:40
• In my view, being a teacher is a sacred role, and being a pupil the same. Both teacher and pupil should have mutual respect, EVEN if one of them is just plain wrong, and EVEN if the opposite side shows disrespect. Don't forget that no man is perfect. On the other hand, role of the tutor is more practical. This means closer relation to pupil, and in a way more direct communication. @Ennar – VividD May 21 '15 at 21:32
• I agree that there should be mutual respect between a teacher and a pupil, but I cannot and will not expect from my pupils to respect me if I can't show them the same. This actually extends to anything in my classroom, all rules apply to them as they apply to me. If I'd violate any rule, I cannot hold it against them if they follow my lead. A teacher must be a role model and has to accept that any action is copied by his/her pupils. But, not to to stray too much, should the pupil in question be courteous to her teacher? Yes. Should she respect him? No. – Ennar May 21 '15 at 21:53
• In this situation, there is no reason whatsoever to respect the teacher. He publicly humiliated the pupil. Respect has to be earned. – Syd Kerckhove May 22 '15 at 8:55
• I agree that this teacher does not deserve respect. However, it is not disrespectful to someone to point out their error. So even if they did deserve respect, they should know about their error. – Joshua Meyers Mar 1 '17 at 21:03

It could not be only square because if it is square then it is a rectangle too and if it is rectangle then it is trapezium too. And I don't think so that it is multiple correct type. But it is sure that it cannot be a triangle as it is a four sided figure.
Yes you were right. By the given information it can't be proved that it is rectangle or square so the answer I think must be F.) trapezoid

One might also point out that a square is also a special case of trapezoid, rectangle, rhombus, quadrilateral and parallelogram, so any of those answers would technically be correct, assuming the object in question was, indeed, a square.

However, I think what everyone is missing is that while in a classroom the "right" answer is the one the teacher is looking for, even if that answer isn't the "correct" one... I've found in my experience that teachers can be obstinate, and aren't likely to be pleased by being corrected by a student in front of the class. Better (for ones academic success) to do so after class and let the teacher correct him/herself in a future class, or not...

• The problem wasn't to have the teacher corrected in the first place but that the student with the right answer was humiliated in front of the class. – JFS May 20 '15 at 18:27