# 4 equations, 4 variables. Why can't I solve it? **URL:** \ **Category:** Factual Questions\ **Tags:** science-math\ **Created:** [May 20, 2025, 1:19pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582 "2025-05-20T13:19:10Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [May 24, 2025, 8:52pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/41 "2025-05-24T20:52:30Z")
None of the solutions above reference a side length. Nor can there be a single solution no matter what you do. It is clearly scale invariant. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 24, 2025, 9:04pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/42 "2025-05-24T21:04:42Z")
There is a question— since people seem unsatisfied with some of the solutions presented— to find the “shortest”, or the “most elegant” proof, and that is something that absolutely _can_ be searched for on a computer these days. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [May 24, 2025, 9:07pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/43 "2025-05-24T21:07:22Z")
**Pleonast** and **Chronos** ’s solutions both start with “assign an arbitrary side length 1”. Obviously the value doesn’t actually matter, but you need _some_ value to plug into the law of sines, etc. It just falls out at the end. My question is, basically, is there an identity of some kind that _doesn’t_ reference a side length that provides the extra constraint in this problem? ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [May 24, 2025, 9:42pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/44 "2025-05-24T21:42:06Z")
> [@Dr.Strangelove](#): > > **Pleonast** and **Chronos** ’s solutions both start with “assign an arbitrary side length 1”. Obviously the value doesn’t actually matter, but you need _some_ value to plug into the law of sines, etc. That’s just a convenience. You could equally say something like, “Let a denote the length of this side” and express all the other lengths as multiples of a. You’re really dealing with the ratios of the side lengths, which is what trig is all about, after all. Having said that, I don’t see what the most elegant proof would be. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [May 24, 2025, 9:47pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/45 "2025-05-24T21:47:45Z")
I get all that. My point is that many or most problems of this nature don’t need to reference side length in any way. They can be solved with a small set of identities like that the angles in a triangle add up to 180 or that opposite angles on crossing lines are the same. At first glance, this problem appears to be of that nature. There are lots of opportunities to use the angle-only identities. But it turns out they aren’t enough. ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [May 24, 2025, 9:58pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/46 "2025-05-24T21:58:39Z")
> [@Dr.Strangelove](#): > > So what’s special about this problem? Are there more subtle angle identities that we’re ignoring? Can problems like this be placed into two different classes, one that’s solvable without referencing side lengths and one that isn’t? > > [@Stranger\_On\_A\_Train](#): > > > > Geometrically, you have no parallel lines or line lengths so you can’t use the law of sines or congruency, and nothing about how this problem is described gives right triangles. However, if you notice that you have two angles of a triangle (50º and 80º) so you know the third angle is 50º, so you have an isosceles triangle (two sides are the same length, and bisecting between them gives you two mirrored right triangles…the figure is misleading to say the least). I haven’t worked through the problem but I believe that you can use that to define the relative lengths of the sides and from there calculate W, X, Y, and Z using algebra and trigonometric relationships of the bisected right triangle. Stranger ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [May 24, 2025, 11:02pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/47 "2025-05-24T23:02:47Z")
> [@Dr.Strangelove](#): > > simply verifying the various triangle identities isn’t enough to prove that you have a rigid object. Some other constraint is needed on top of that. I’m probably out of my depth here and these are topics I haven’t thought rigorously about in decades … Do you mean “rigid” in the conventional sense or is there some geometric special use of the term I’m missing? Seems to me all problems of this general nature assume the figure is 2D planar and not elastic in any way. Otherwise the degrees of freedom just explode. I certainly understand your larger objection that it seems underspecified _assuming_ you’re not going to dig into actual trig functions to resolve it. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 24, 2025, 11:21pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/48 "2025-05-24T23:21:57Z")
> [@Dr.Strangelove](#): > > I get all that. My point is that many or most problems of this nature don’t need to reference side length in any way. They can be solved with a small set of identities like that the angles in a triangle add up to 180 or that opposite angles on crossing lines are the same. I mean, in principle, you can avoid referencing side lengths. Every ratio of side lengths @Pleonast and I used could be replaced with the appropriate trig expression. But that’s just the same thing with extra steps. It does feel like there ought to be a simpler way to do it; I agree with that. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [May 24, 2025, 11:24pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/49 "2025-05-24T23:24:36Z")
I meant rigid in the sense that all angles and side length ratios are determined. Imagine that you have lengths of infinitely extensible rod and then some clamps that hold the ends at specific angles (also, this is Euclidean space and we’re keeping everything planar), as well as some that connect rod ends but allow a variable angle. If you build a triangle out of these, with two constrained angles, then the whole system becomes rigid outside of scale (you can still make each rod 2x as long, say). But if you build a quadrilateral with opposite angles constrained to 90 degrees, then the whole thing is not rigid in the same way. The remaining angles can be changed, and the side lengths can be any ratio you want. It’s not just the same shape with different scales. I think I was wrong, though. The problem here is that line segment that connects the vertices with the X/Y angles and the W/Z angles. You can give them the wrong value and you can’t prove that it’s the wrong answer with non-trig tools (it seems). But I’m not sure if that makes it non-rigid, though. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [May 24, 2025, 11:29pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/50 "2025-05-24T23:29:35Z")
> [@Chronos](#): > > Every ratio of side lengths @Pleonast and I used could be replaced with the appropriate trig expression. I think the question can just be rephrased as “can it be solved without trig?” And if not, why not? What makes this problem special vs. otherwise similar problems that can be solved without referencing lengths (length ratios or otherwise)? ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [May 24, 2025, 11:35pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/51 "2025-05-24T23:35:07Z")
I’d suggest the answer is they cherry-pit-picked the given angles to leave the problem underspecified in angle-space. A more typical problem cherry-picks the givens to ensure there is a path to the solution wholly in angle-space, even if it’s obscure enough to keep clever folks digging for a bit before they find it. Whether that underspecification was deliberate is a different & harder question. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 24, 2025, 11:44pm UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/52 "2025-05-24T23:44:34Z")
Isn’t “angle-space” the same as “side-length ratios” like @Chronos says? To make it concrete, imagine constructing a circle circumscribing the given triangle and specify its diameter to be 1. Then the length of each side in those units is simply equal to the sine of the opposite angle. Furthermore, “angle space” is the same as “sines-of-angles space” via right triangles as pointed out by @Stranger_On_A_Train . So nothing is underspecified. It remains to figure out the shortest/best proof, though. I am actually curious about running this one through a computer. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 25, 2025, 12:05am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/53 "2025-05-25T00:05:29Z")
> [@DPRK](#): > > “angle-space” the same as “side-length ratios” Or, more fundamentally, that is expressed by similar triangles having the lengths of their sides in proportion. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [May 25, 2025, 12:45am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/54 "2025-05-25T00:45:51Z")
Oh yes. The trig functions are operators to convert between “angle-space” and “side ratio space”. And yes, the problem isn’t truly underspecified in the sense that there’s no way to compute the answer. It’s readily computable as has been shown upthread. Just introduce some “sideness” into the problem, take advantage of the trig operators, then the answer comes out and along the way the injected side info “cancels out” so there’s no loss of generality when you’ve gotten to the other end of the solution pipeline. The “debate”, if there is one, is whether this problem _should_ have an angles-only route to a solution. It doesn’t. But should it if it’s to fit in the class of traditional basic triangle geometry brain teasers? That to me was what @Dr.Strangelove was getting at with his post just above mine. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [May 25, 2025, 12:58am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/55 "2025-05-25T00:58:23Z")
> [@LSLGuy](#): > > The “debate”, if there is one, is whether this problem _should_ have an angles-only route to a solution. It doesn’t. But should it if it’s to fit in the class of traditional basic triangle geometry brain teasers? And if not… why? If you look again at the original diagram, _every_ angle except for XYZW can be easily worked out. If you plug in an incorrect solution (that fits the angular constraints), how do you work out that it’s wrong? Say, X=50, Y=50, W=30, Z=80. You can show it’s wrong with trig. But seemingly not without it (I did try using @Stranger_On_A_Train 's suggestion of using the isosceles property, but I didn’t get anywhere). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 25, 2025, 1:06am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/56 "2025-05-25T01:06:44Z")
Of course, it should be kept in mind that, whenever a human says that the Universe should be such-and-such a way, that’s actually a statement about us humans, and not actually a statement about the Universe. ---
**Author:** ![Snarky\_Kong](https://avatars.discourse-cdn.com/v4/letter/s/a183cd/32.png) [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)\ **Post date:** [May 25, 2025, 1:09am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/57 "2025-05-25T01:09:18Z")
Being able to solve it without trig seems to imply that the ratios of sides can be changed, keeping other knowns constant, without changing the answer. However that seems to necessarily change angles. So I vote no for answers without trig. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 25, 2025, 1:17am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/58 "2025-05-25T01:17:48Z")
Not really. For a much simpler problem, consider a triangle with two angles given, and the third angle to be determined. The ratios of sides are fixed there, too, but one never needs to use trig to solve it. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 25, 2025, 1:36am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/59 "2025-05-25T01:36:19Z")
If (hypothetically, but think of classical problems like trisecting the angle) the problem does _not_ have a certain kind of solution, that should be proof-theoretically formalizable and proven or disproven. I really think the solution space (number of auxiliary constructions needed) in this case is small enough that a computer can brute-force it. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [May 25, 2025, 1:57am UTC](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582/60 "2025-05-25T01:57:27Z")
I was using _should_ in the sense of the gamesmanship of folks who write problems for textbooks, quizzes, and internet brain teasers. This one seems inordinately hard which makes me wonder whether the test-writer goofed and intended XYZW to be soluble using angle-only reasoning. Or, as I suggested upthread whether somewhere along the chain of plagiarizing from the original, some key tidbit of auxiliary info got lost or garbled along the way. So this goofed up revised version is unintentionally insoluble using only angles, whereas the original was soluble, and intentionally so. You’re spot-on that the Universe cares not a fig for our puny human _shoulds_. I now think @Dr.Strangelove is suggesting a deeper sort of “Why?” question than I was, namely: “What’s pathological about this particular collection of facts that prevents an angles-only solution?” What can we learn or say about classes of problems that do, or do not, admit of an angles-only solution? And other than brute force, how would we recognize which class any given problem belongs to? [Previous page](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582.md?page=2) [Next page](https://boards.straightdope.com/t/4-equations-4-variables-why-cant-i-solve-it/1018582.md?page=4)