Question 7
Triangle $ABC$ has side lengths $AB = 120,$ $BC = 220,$ and $AC = 180.$ Lines $\ell_A,$ $\ell_B,$ and $\ell_C$ are drawn parallel to $\overline{BC},$ $\overline{AC},$ and $\overline{AB},$ respectively, such that the intersections of $\ell_A,$ $\ell_B,$ and $\ell_C$ with the interior of $\triangle ABC$ are segments of lengths $55,$ $45,$ and $15,$ respectively. Find the perimeter of the triangle whose sides lie on lines $\ell_A,$ $\ell_B,$ and $\ell_C.$