Is this possible? Basically the other vertex would be like its pivot.
Yes. In edit mode, select the pivot vertex and press the space bar to bring up the command selector dialog. Choose "Snap Cursor to Active" to set the 3D cursor to the position of this vertex.
Then press the . key to set the cursor as the centre of rotation. You can then select the vertex you want to move and press R twice to enter free rotation mode (or just once to rotate in the current view). Either way, the distance of the selected vertex from the centre of rotation will remain constant.