Trapeziums - Non Congruent Tiling (18) and Oriented Containers

This post marks a meeting of two tracks we have been pursuing - Oriented containers and Non-congruent tiling.

The previous posts in those series are here and here. We also build on this earlier post.

General remark: I had thought that the analog of 'isosceles triangles among triangles' is 'kites among quadrilaterals'. Had a series on them here But then trapeziums (being oriented) too could be considered among quads.

Questions:

1. How does one find the smallest (in terms of area OR perimeter) trapezium that contains a given set of points? It is of course enough to solve the problem of finding the smallest trapezium containing the convex hull of the points (A trapezium has an orientedness due to its pair of parallel edges - a property that I had not noticed earlier!).

2. The same question as above with the trapezium restricted to being isosceles (the two non-parallel edges are of equal length)?

3. For which convex region are the orientations of the smallest area trapezium container and the smallest perimeter trapezium container most different?

4. Can the plane be tiled with mutually non-congruent trapeziums (equally, isosceles trapeziums) having (1)same area and same perimeter OR (2) having same area and same diameter OR (3) same perimeter and diameter?
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And just as with containing trapeziums, we can ask a bunch of questions about largest trapeziums contained within a given convex region.