To determine which sides of trapezoid A'B'C'D' would be parallel after trapezoid ABCD is rotated 90 degrees clockwise about the origin, let's analyze the effect of the rotation and the properties of trapezoids.
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Identifying Parallel Sides in Original Trapezoid ABCD: Typically, in a trapezoid, the top side (for instance AB) and the bottom side (CD) are parallel. After rotation, the same sides in trapezoid A'B'C'D' would become A'B' and D'A', thus they should still be parallel.
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Rotating by 90 Degrees: When you rotate points (x, y) by 90 degrees clockwise, their new coordinates become (y, -x). This applies to all vertices A, B, C, and D to find A', B', C', and D'.
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Identifying the New Coordinates: Without specific coordinates, we can still know that:
- If AB is parallel to CD in ABCD, after rotation, A'B' will be parallel to D'A' in A'B'C'D'.
- The new sides from the rotation may position B'C' and C'D' in such a way that they are not parallel, as they might not correspond to the original parallel sides.
Based on this reasoning, we can conclude the following about the options provided:
- I. A'B' and B'C': Not necessarily parallel as they're on different bases.
- II. D'A' and C'B': Not parallel (C'B' from the rotation does not match the orientation).
- III. A'B' and D'C': Not parallel (like above, C' doesn't correspond directly).
- IV. A'B' and D'A': Indeed parallel since these were originally the parallel sides AB and CD.
Thus, the correct choice that reflects the parallel sides post-rotation is:
D. III only