splitting paths

Yes, here is one way of doing what you want

Draw a perfect Circle using the Circle and Elipse Tool (F5), when you drag your mouse:
>Drag it at a 45 degree angle and hold down the Ctrl Key (Ctrl+Drag) this will place a even ratio circle onto the canvas.
>Once it is drawn there will be a set of TRANSFORM nodes two white squares (top and left edge of the circle) and a white circle (right edge of the circle)

details are here:
http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Shapes-Arcs.html

>Drag the Circle Transform Node (end point)
   >Dragging inside the circle will create an ARC what you are looking for
   >Draggin outside the circle will create a SEGMENT (like a slice of pizza)
   >Dragging Inside with Ctrl (Ctrl+Drag) will snap the arc in steps.

You could also convert your circle to a path >Path >Object to Path

Then use the Edit Path by Nodes Tool (F2) and select a path node and use the 'Break path at selected nodes', 4th icon from the left of the Tool Controls Bar (Bar above the canvas), repeat this for the path node at the opposite edge of your circle.

When you break closed paths, you may want to use the >Path >Break Apart (Shift+Ctrl+K) to make separating (positioning) the paths easier for you.

details are here:
http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Paths-Editing.html#Paths-Editing-Node