Answer
The clothesline has a mass that is affected by gravity, and there is tension, so for the rope to remain at equilibrium, it must sag.
Work Step by Step
On a small increment of rope, there is a small amount of mass. The clothesline would have tension pulling on the rope and the weight pulling down. In order for all of the forces to add up to zero, the clothesline must be displaced downward and sag.
