How do ATC radars know where to position a primary target? Primary target: An aircraft not reporting mode-C, the only thing the controller has is the return on the radar.

When a controller reports a primary target as traffic to other aircraft, the controller does not have the altitude of the target. Given this, I conclude that ATC radar does not have the altitude (angle-up) to the target, and only provides azimuth.

So then without the altitude, how does the radar-system know where to put the target laterally on the screen?

Example, a radar picks up a target that is 10 miles from the station. If the target is 0 AGL, the proper position would be 10 miles out. However, if the target is at 15,000ft, the proper position would be 9.5 miles out.

Since the difference is so small, does the radar just put the target at 10 miles, and the FAA separation guidelines factor in the discrepancy?
Primary radar, even modern digital systems, do not give any possibility to the controller to determine altitude.  The digital system has that capability, but civil ATC does not.

In primary radar displays usually just the azimuth and range are plotted. So an aircraft at 15.000 ft 9.5 NM away will be plotted on 10 NM distance, just like an aircraft at 0 ft 10NM away. Usually the secondary radar is collocated with the primary so secondary targets can be displayed using the same projection (disregarding the altitude). 

If you now apply horizontal separation based on this projection, you will never bust the separation minima. It is a conservative way of providing separation, but when there is no altitude information it is the best possible way. 



Note that this does not only apply to primary targets, but also to Mode-A only targets