0
$\begingroup$

Let $\Delta_1, \Delta_2$ and $\Delta_3$ be three triangles.

Triangle $\Delta_1$ has sides equal to $x,y$ and $l_1$. Triangle $\Delta_2$ has sides equal to $y,z$ and $l_2$, and triangle $\Delta_3$ has sides $x,z$ and $l_3$. My question is:

For known values of $l_1, l_2$ and $l_3$, what is the relation between $x,y$ and $z$?

I cannot use the law of cosines because I have no information about angles. So, I do not know how to deal with this problem.

Thanks in advance for any help!

$\endgroup$
  • $\begingroup$ No more information about triangles ? $\endgroup$ – Khosrotash Feb 10 '17 at 11:27
  • 2
    $\begingroup$ The only information you can get is what you get from the $9$ triangle inequalities (three for each triangle). Are there particular numerical values of $l_1,l_2,l_3$ that interest you? $\endgroup$ – quasi Feb 10 '17 at 11:27
  • $\begingroup$ @quasi, No particular values of $l_1,l_2$ and $l_3$. $\endgroup$ – Alex Silva Feb 10 '17 at 12:47
  • $\begingroup$ @Khosrotash, I have no other inofrmation about the triangles. $\endgroup$ – Alex Silva Feb 10 '17 at 12:47
  • $\begingroup$ @Alex Silva: Where does the problem come from? Or what does it relate to? $\endgroup$ – quasi Feb 10 '17 at 12:49
1
$\begingroup$

You can recover the triangle identity for $\ell_1, \ell_2, \ell_3$, since you can glue the three triangles with the corresponding sides together and obtain a tetrahedron whose fourth side is the triangle with sides $\ell_1, \ell_2, \ell_3$.

$\endgroup$
  • 1
    $\begingroup$ Associating a tetrahedron with the three triangles is a cool idea. But I have 2 questions: (1) What do you mean by "the triangle identity for $\ell_1, \ell_2, \ell_3$"?; (2) The question asks for the relation between $x,y,z$ when $\ell_1, \ell_2, \ell_3$" are known. Aren't you reversing the question? $\endgroup$ – quasi Feb 10 '17 at 12:26
  • $\begingroup$ I do not understand either what you mean by "recover the triangle identity". Moreover, you cannot ensure that $l_1, l_2$ and $l_3$ form a triangle. $\endgroup$ – Alex Silva Feb 10 '17 at 12:49
  • $\begingroup$ @quasi Oh yeah, I've reversed the $\ell$'s and $x,y,z$..but still, you can glue those triangles into a tetrahedron. I mean all the three relations $\ell_i + \ell_j \leq \ell_k$.. $\endgroup$ – pepa.dvorak Feb 10 '17 at 13:02
  • $\begingroup$ But there will be some more information we can obtain, since if you maintain the idea of a tetrahedron, the assignment tells we have a given base of the solid and now, for given values of $x,y$ (i.e. for given another face of the solid), there will be a lower bound $b(x,y)$ of $z$ to close the tetrahedron. $\endgroup$ – pepa.dvorak Feb 10 '17 at 13:23
  • $\begingroup$ I don't think the three triangles can be glued together to form a tetrahedron if one of the triangles is excessively small. $\endgroup$ – Mick Feb 10 '17 at 13:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.