Newtonian mechanics

[kolahal_b]

KK2.22) A uniform rope of weight W hangs between two trees. The ends of the rope are the same height, & each of them make equal angle θ (acute) with the trees. Find the tension at the middle of the rope.

I did in the following way: considering equilibrium of the left/right portion of the rope, horizontal component of tension at the end, which is the horizontal component of reaction of the tree on the rope, must be equal in magnitude to the horizontal tension in the middle. Therefore, T(end) sinθ=T(mid); it appears to be correct & matches the criterion of the solution.

But normally we do equate forces considering a single point of action, such as C.G. of a body, in Newtonian mechanics. Am I justified in carrying the problem out in this way? Am I effectively equating horizontal forces acting at a single point?