Which of the following sets of transformations will map rectangle $ABCD$ onto itself?

• A. Only a rotation of $90°$ about the center
• B. A rotation of $180°$ about the center and reflections over both diagonals
• C. A rotation of $180°$ about the center and reflections over the lines through the midpoints of opposite sides
• D. Only reflections over the diagonals

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the symmetry properties of a rectangle. A rectangle has two lines of symmetry through the midpoints of opposite sides, but generally does not have symmetry over its diagonals unless it is a square.
2. Step 2: Consider rotations about the center of the rectangle. A rotation of 180\degree about the center maps the rectangle onto itself because opposite vertices swap places, preserving the shape.
3. Step 3: Analyze a 90\degree rotation about the center. For a rectangle (not a square), a 90\degree rotation will not map the rectangle onto itself because the side lengths are different and the shape orientation changes.
4. Step 4: Examine reflections over the diagonals. Since the diagonals of a rectangle are not lines of symmetry (unless it is a square), reflecting over the diagonals will not map the rectangle onto itself.
5. Step 5: Examine reflections over lines through the midpoints of opposite sides. These lines are axes of symmetry for a rectangle, so reflecting over these lines will map the rectangle onto itself.