Answer
I determined the problem.
The “cut path” function in Inkscape requires exactly two objects. If the user has 3 objects selected then the program does nothing.
In my case I had two concentric circles and the line, so I was selecting 3 objects.
The solution is the following:

Drag a guide from the ruler until it snaps to the center of the
circles 
Draw a rectangle that is larger than the circles above them

Extend the rectangle until it snaps to the guide (the rectangle
should now cover half the circles). 
Select the rectangle and one of the two circles

Do Path / Division to make a closed shape or Path / Cut to make an open shape
Attribution
Source : Link , Question Author : Tyler Durden , Answer Author : Tyler Durden