Further Mechanics Elasticity Question (Old M3)

A particle of mass 3kg is attached to one end of a light elastic string, of natural lenght 1m and modulus of elasticity 14.7N. The other end of the string is attached to a fixed point. The particle is held in equilibrium by a horizontal force of magnitude 9.8N with the string inclined tot he vertical at an angle theta.

Questions asks to find theta and the extension in the string, thats fine. Then it says:

"If the horizontal force is removed, find the magnitude of the least force that will keep the string inclined at the same angle."

Solution bank tells me this force is perpendicular to the string, but why?

Triangle of forces:

The third side of your triangle is the new force, and when is it a minimum, i.e. when is its length a minimum? When it's perpendicular to the direction of T.