1
$\begingroup$

I need to find the perimeter of the following triangle in terms of $\ell$. The only extra info I have is the three sides form an arithmetic sequence.

enter image description here

I've been tutoring maths on these topics for a couple years now and I am still struggling with this problem. I'm sure there's something rather simple I can't see. Any help is appreciated!

$\endgroup$
  • $\begingroup$ I am pretty sure that there are not sufficient amount of information. We can also let the angle 120 to be a variable point, so I suppose the total perimeter wont be fixed. $\endgroup$ – sentheta Feb 2 at 0:52
  • $\begingroup$ It seems to me that this triangle is similar to triangle with sides 3, 5, 7. $\endgroup$ – richrow Feb 5 at 12:52
5
$\begingroup$

Hint

It is reasonable to assume that the sides of the triangle, have a form $a=3l-2d$, $b=3l-d$ and $c=3l$ for some $d>0$. You can use the law of cosines as $$c^2=a^2+b^2-2ab\cos {2\pi\over 3}=a^2+b^2+ab$$and by substitution, first find $d$ then $a+b+c$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ why is it reasonable to assume sides of the triangle have that form at all? $\endgroup$ – jimjim Feb 2 at 1:32
  • 4
    $\begingroup$ Because this can be the only case that the sides form an arithmetic sequence alongside the side $3l$ being the largest side of the triangle. $\endgroup$ – Mostafa Ayaz Feb 2 at 1:34
1
$\begingroup$

Let $a$, $b$ and $3l$ be sides-lengths of the triangle.

Thus, $2b=a+3l,$ or $a=2b-3l$ and $$(2b-3l)^2+b^2+b(2b-3l)=(3l)^2$$ or $$7b^2-15bl=0,$$ which gives $$(a,b)=\left(\frac{9l}{7},\frac{15l}{7}\right)$$ and $$a+b+c=\frac{45}{7}l.$$

| cite | improve this answer | |
$\endgroup$

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.