What determined the common 9’ x 12’ room size? Twelve feet is a lumber dimension, but why nine feet and not eight?
WAG:
A 9’ x 12’ room follows the a[sup]2[/sup]+b[sup]2[/sup]=c[sup]2[/sup] formula:
9[sup]2[/sup]+12[sup]2[/sup]=c[sup]2[/sup]
81+144=c[sup]2[/sup]
225=c[sup]2[/sup]
15=c
I believe any room with square corners would be two right triangles. 
A 3/4/5 triangle (as Mr. Blue Sky mentioned) does make it easier to square things up (since there is no need to calculate irrational square roots), but I’m pretty sure that the house are almost always squared up at the perimeter when the floor plates are placed. Wall layout is usually just measured from an outside wall.
I suspect that 9’X12’ is popular because anything smaller seems too small. For what it’s worth, most of the older houses I’ve seen had 12’X14’ (or so) master bedrooms and 10’X10-1/2’ (or so) children’s bedrooms.
Sorry Mr. Sky, I misunderstood.
I’ve seen 3x4x5 framing squares used to layout corners.