Given three parallel lines separated by distances a, b as shown, what is the minimum length of x+y as a function of ?
The quantity to minimise is , which is the same expression we needed to minimise in the previous post, except that this time we have instead of . We simply replace with in the answer given there, and find that the minimum length is
where is the positive root satisfying the cubic equation
Indeed, if we rotate the top line by about the “hinge” we obtain the same picture as last time and see that a and b represent the same quantity in both cases:
It is interesting how these seemingly different problems are identical, with a simple rotation connecting them.