When you click directly on a path segment, you select just that segment.
In the following demonstration movie, we will change the stroke colour of a circle so that each quadrant of the circle has a different stroke colour. To do this, we perform the following steps.
Click on the Direct Selection tool to make it the active tool.
Click on the border of the circle, between two anchor points. This selects just that segment of the circle.
Choose Edit - Cut.
Choose Edit - Paste in Front (or Paste in Back).
Choose the stroke colour.
Repeat the above steps for the remaining arcs of the circle.
