Question

when you're rotating a trapezoid that's in quadrant 4 90 degrees, does the trapezoid goes in quadrant 2 and the coordiantes switch?

also, does the trapezoid goes in quadrant 1 when rotating 180 degrees

This is my answer? Is this right?

Answer 1

No,
if your trapezoid is in quadrant 4, then a rotation of 90ºwould put you in quadrant 1, (counterclockwise is the positive rotation in math).
Consider a possible point on the original trapezoid, lets say (5,-2)
Its image point would be (2,5)
So under a rotation of 90º any point (a,b) would have its image point (-b,a)

Similarly after a rotation of 180º, your fourth quadrant trap would land in the second quadrant, so any point (a,b) would have its image point become (-a,-b)

If you know anything about matrices, a 90º rotation has a rotation matrix of
0 -1
1 0
and a 180º rotation matrix is
-1 0
0 -1