We add a little to Tiling the plane with pairwise non-congruent rational triangles

Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "yes", can such an equal area tiling be achieved with the perimeters of the triangles upper bounded?

(For any given possible area A, https://nyjm.albany.edu/j/1998/4-1p.pdf proves that there are indeed infinitely many rational triangles all with area A)

I couldn't figure out if the theorem(s) proved in https://web.archive.org/web/20210414080312/https://math.dartmouth.edu/~carlp/PDF/paper14.pdf would settle the above question.

Further question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same perimeter?

We add a little to Tiling the plane with pairwise non-congruent rational triangles

Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "yes", can such an equal area tiling be achieved with the perimeters of the triangles upper bounded?

Remark 1: For any given possible area A, https://nyjm.albany.edu/j/1998/4-1p.pdf proves that there are indeed infinitely many rational triangles all with area A. But from among triangles for a given A to select a set that tiles the plane might not be possible. If the answer to the question is "no", one could ask if mutually non-congruent rational triangles whose areas are all one of a finite set of values would tile the plane.

2. I couldn't figure out if the theorem(s) proved in https://web.archive.org/web/20210414080312/https://math.dartmouth.edu/~carlp/PDF/paper14.pdf are relevant here.

Further question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same perimeter?

We add a little to Tiling the plane with pairwise non-congruent rational triangles

Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "yes", can such an equal area tiling be achieved with the perimeters of the triangles upper bounded?

I couldn't figure out if the theorem(s) proved in https://web.archive.org/web/20210414080312/https://math.dartmouth.edu/~carlp/PDF/paper14.pdf would settle the above question.

Further question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same perimeter?

We add a little to Tiling the plane with pairwise non-congruent rational triangles

Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "yes", can such an equal area tiling be achieved with the perimeters of the triangles upper bounded?

(For any given possible area A, https://nyjm.albany.edu/j/1998/4-1p.pdf proves that there are indeed infinitely many rational triangles all with area A)

I couldn't figure out if the theorem(s) proved in https://web.archive.org/web/20210414080312/https://math.dartmouth.edu/~carlp/PDF/paper14.pdf would settle the above question.

Further question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same perimeter?