To solve the problem, we need to identify all transformations that map the rectangle $ABCD$ onto itself. The following transformations are provided:

1. Rotation by 90 degrees clockwise using center P:

   • A rectangle is only mapped onto itself by 180° rotations (or multiples of 180°) since its sides must align with the original orientation. Thus, a 90° rotation will not map the rectangle onto itself.
   • This transformation does not map the rectangle onto itself.

2. Rotation by 180 degrees clockwise using center P:

   • A 180° rotation will swap opposite corners (i.e., A with C, and B with D), and the rectangle will look the same as the original.
   • This transformation maps the rectangle onto itself.

3. Reflection across line m:

   • Line $m$ appears to be the perpendicular bisector of the rectangle, splitting it symmetrically along the midpoints of sides AB and CD. A reflection across this line will map point A to B, and D to C, effectively mapping the rectangle onto itself.
   • This transformation maps the rectangle onto itself.

4. Reflection across diagonal AC:

   • Reflecting across diagonal AC will swap points B and D, but since a rectangle is not symmetric along its diagonals, this transformation does not map the rectangle onto itself.
   • This transformation does not map the rectangle onto itself.

5. Translation by the directed line segment from A to B:

   • A translation along the directed line segment from A to B would shift the entire rectangle. Since the position of the rectangle changes, it does not map onto itself.
   • This transformation does not map the rectangle onto itself.

Correct transformations:

• B: Rotation by 180 degrees clockwise using center $P$
• C: Reflection across line $m$

Would you like further details on any of the transformations or concepts?

Related Questions:

1. How do we mathematically determine symmetry in shapes like rectangles?
2. What are the general rules for rotations mapping shapes onto themselves?
3. Why is reflection across line $m$ valid but not across diagonal $AC$?
4. How do translations typically affect geometric figures?
5. Can the same transformations be applied to other polygons, such as squares?

Tip:

For many geometric transformations, recognizing symmetry is key to determining which operations will map a figure onto itself. Symmetry lines and rotation centers are especially important.