Perimeter of a triangle with an angle of $120^\circ$ and sides in arithmetic progression

Question

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.

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!

Answer 1

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 $a+b+c$.

Answer 2

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 $$a+b+c=\frac{45}{7}l.$$