A
areas may or may not be the same because the scale is not given.
ex
B
The area of trapezoid A is greater than trapezoid A'. A rotation decreases the size of the original polygon. Therefore, the area is not preserved.
*x
C
The area of trapezoid A' is greater than trapezoid A. A rotation increases the size of the original polygon. Therefore, the area is not preserved.
Ax
D
The areas are equal. A rotation is a rigid transformation, which does not change the size of the original trapezoid. Therefore, the area is preserved.