# 'The Imperial triangles' trigonometry question

There is a very well-known family of right triangles in Mathematics called "The Imperial triangles". Let me explain you how they are defined:

We say $$T_n$$ is the $$n$$-th Imperial triangle if these two conditions are met:

• Its hypothenuse has length $$\frac{1}{n}$$
• Water boils at the second-to-longest side

This family has a very strange property: the shortest side is the same length on every triangle. How long exactly?

EDIT: I expected more from you. Come on, I've seen this community solve tougher ones! Let's give you a hint. Also, please stick to flat geometry (you know, like the shape of the Earth. OK. Actually, at the time they already knew the Earth wasn't flat)
Water boils at 100º, 90 of them coming from the right angle

• What does it mean to be "Water boils at the second-to-longest side"? And if the $n$ continues to grow, does it mean the shortest side will be less than $\frac{1}{n}$ and goes to $0$? Aug 22, 2019 at 15:39
• @athin there is no such thing in mathematics as an "imperial triangle" ;) Aug 22, 2019 at 15:41
• I think you're using hypotenuse incorrectly. Hypotenuse does not mean the longest side. It means the side that is opposite of a right angle which I don't think is happening here. Aug 22, 2019 at 15:41
• @WeatherVane Yep I'm aware of it, but still it's kinda impossible for normal math to solve this (unless I missed something here... XD). And in normal right triangle, hypotenuse is always the longest one, no? Aug 22, 2019 at 15:44
• @athin Please be open-minded! Don't just stick to the boring traditional rules of Math Aug 23, 2019 at 7:07

This is

spherical geometry

The interior angles can sum to more than 180°.
It is a right triangle, so one angle is 90°.
The side where water boils is opposite an angle of 100° (geddit?)
But I think that's a red herring.

One problem is that the smallest side has to be the same length in each triangle.
But it cannot be longer than half the smallest hypotenuse, which tends towards 0 in the series.

So the length of the shortest side is $$\frac {1} {2 n}$$

Edit following a comment from OP.

The boiling point of water isn't a red herring after all.
Each spherical triangle has two known angles, 90° and 100°, the third varies with $$n$$.

But my spherical angle geometry skills are no greater than the smallest hypotenuse.
So that's as far as I can go: perhaps I am on the right track, if not the required answer.

• The length is the same for all triangles, so it cannot depend on $n$ Aug 22, 2019 at 16:31
• It's not a complete answer. Perhaps that should have been capital $N$ (the limit of $n$). Aug 22, 2019 at 16:43
• Rules 6 and 7 of right angles in spherical trigonometry imply that $$\tan x=\cos(100°)\tan \frac 1n$$ where $x$ is the desired length, assuming that the sphere is of constant radius 1 (i.e. assuming that the radius does not depend on $n$, and that arc length and angle coincide). Seems to be a dead end, but +1 for the idea. Aug 22, 2019 at 17:40
• @ArnaudMortier... but can you progress the idea? Aug 22, 2019 at 17:44
• How is it that you there put the great Imperial triangles on the surface of s sphere? Back in the Imperial times, everyone knew the Earth was flat! Aug 23, 2019 at 7:10

Well, I am surprised about how bad at math all of you are! Come on, this is the Puzzling site! You can do better than this!

Water boils at the second-to-longest side (horizonal in the picture): 90 degrees come from the right angle. We still need 10 more degrees, which of course come from the other angle formed by that side.

Now that we know the smallest angle in 10º and that the hypothenuse has length $$\frac{1}{n}$$, we can calculate the length of the opposite side (the shortest side) as

$$\frac{\sin{(10)}}{n} = \frac{\sin{(X)}}{n} = siX = 6$$ The Empire I was refering to is of course the Roman Empire. Comments regarding tea are just disgusting!

• Hey David, some friendly advice (meant well): Probably would have been good to add more hints to the question before resorting to a self-answer (we're used to puzzles with this kind of pseudo-maths solution, just sometimes it needs to be clearer that that's what's expected). And watch out with the jokey sarcasm/insults - they're likely to get you downvotes!
– Stiv
Aug 27, 2019 at 10:18
• @Stiv I don't see how saying "you are bad at math" for not being able to solve a problem that has almost nothing to do with math is insulting. I am just pretending the problem is a math problem, then pretending people who cannot solve it are bad at math. Anyway my reputation on this site is the last and least of my concerns. Also, even though I posted the answer, you can still try it yourself anyway! Aug 27, 2019 at 10:59
• I don't know how you can say this has "almost nothing to do with maths" when the answer relies on subverting a mathematical relationship that only people with some knowledge of geometry will get. Instead of acknowledging that my connection with 100° was on course (and only in spherical geometry can a right triangle have another angle of 100°) all you can say is that my reference to tea is disgusting. Aug 27, 2019 at 17:44
• @WeatherVane I expected this community to be at least somewhat acceptant of humor and sarcasm, but it's just as depressing as any other. I think I'm done with this! Of course when I say "that was disgusting" I am not literally meaning that Aug 28, 2019 at 7:14
• I'm just going to leave this here. xkcd.com/169
– Rubio
Sep 2, 2019 at 2:52