The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational.
To cut a unit square into n (a finite number) triangles with all sides of rational length. For which values of n can it be done if at all?
To cut a unit square into n right triangles with all sides of rational length. For which values of n can it be done, if at all?
Remark: If one can find a finite set of 'Pythagorean rectangles' (rectangles whose sides and diagonal are all integers) that together tile some square (of integer side), that would answer this question.
3.To cut a unit square into n isosceles triangles with all sides of rational length. For which values of n can it be done, if at all?
Now, one can add the requirement of mutual non-congruence of all pieces to all these questions. Further, one can demand rationality of area of pieces or replace the unit square with other shapes (including asking for a triangulation of the entire plane into mutually non-congruent triangles all with finite length rational length sides)...
Note: From what has been shown by Yaakov Baruch in the discussion below, cutting the unit square into mutually non-congruent rational sided-right triangles can be done for all n>=4. Indeed, he has shown n=4 explicitly; for higher n, one can go from m non-congruent pieces to m+1 pieces by recursively cutting any of the m right triangular pieces n by joining its right angle to the hypotenuse to cut it into two smaller and mutually similar but non-congruent pieces. That basically settles questions 1 and 2 - the non-congruent pieces case. However, if we need all pieces to be non-congruent and non-similar, the n=4 answer has no obvious generalization to higher n.