Non-congruent tilings- 20: Rational triangles

The previous episode of this lengthy series is here .

Ref: this mathoverflow discussion .

Broad Question: to tile the plane into rational triangles (all side lengths rational) all mutually non-congruent.

Additional constraints: all triangles should have same area or same perimeter or...

A construction (tiling with mutually non-congruent triangles all of equal area with rationality of side lengths not insisted) presented in detail by Stan Wagon here , it appears, can also be made to yield a tiling of the plane with rational triangles with perimeter unbounded - it appears to give only mutually non-congruent rational triangles, no equality among their areas. A further result obtained by Frettloh (https://arxiv.org/pdf/1603.09132.pdf, more specifically figure 4 therein) indicates that tiling with rational triangles with bounded perimeter also can be got - again with no guarantee on equality of areas.

So, what we could ask for here is for a tiling of the plane into mutually non-congruent rational triangles all with (1)same area or (2) same perimeter or whatever.

An earlier discussion on mathoverflow is here .
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Variant: Rationality of triangles could also be defined as: all angles are irrational fractions of pi. What could one do with them?

Further question: It appears that any polygon with all angles rational can be cut into some finite number of rational angled triangles. Will the question of dividing an n-gon with rational angles into the least number of rational angled triangles have interesting optimization features?