Thank you Nilo for pointing me in a direction that was productive. I used the software AutoHotkey. A recorder wasn't necessary.
The following script does the job. I set the width of the windows to 350px. My screens have a total width of 1600px*2=3200px, so the left edges of the windows have X coordinate 3200-350=2850.
; start Mathematica and put it in the corner with the shortcut Windows key+C
Run C:\Users\Andrew\AppData\Roaming\Microsoft\Internet Explorer\Quick Launch\User Pinned\TaskBar\Wolfram Mathematica 8.lnk
IfWinNotActive, Untitled-1, , WinActivate, Untitled-1,
WinMove, ,, 2850, 75, 350, 300
WinMove, Wolfram Mathematica,, 0,0, 350
WinMove, Wolfram Mathematica,, 2850, 0
Run command I used the taskbar shortcut because I have Mathematica pinned to my taskbar and running the .exe created a new taskbar icon beside the pinned one instead of just activating the pinned icon. Note that if Mathematica is already running when this script is executed the result may not be as expected.