Similar and non-Congruent Dissections

The following questions have been posted at https://mathoverflow.net/questions/389303/cutting-polygons-into-mutually-similar-and-non-congruent-pieces

1. Is there a non-rectangular polygon P that can be cut into a finite number of scaled down copies of P such that each copy is of a unique size (unique scale factor)?

2. Is there a polygon P that can be cut into a finite number of pieces all of which are mutually similar and of different sizes but are not similar to P?

updates to follow...