Question

which of the following transformations carry this regular polygon onto itself? reflection across l rotation of 90° counterclockwise rotation of 90° clockwise rotation of 45° counterclockwise

Explanation:

Step1: Analyze reflection across \( l \)

A reflection across the line \( l \) (which is a line of symmetry for the rectangle) will map each point of the rectangle to another point on the rectangle. So, reflection across \( l \) carries the rectangle onto itself.

Step2: Analyze rotation of \( 90^{\circ} \) counter - clockwise

A rectangle (a special case of a regular polygon with 4 sides) has rotational symmetry of order 4. A rotation of \( 90^{\circ}\) counter - clockwise about the center of the rectangle will map each vertex to another vertex and each side to another side of the rectangle. So, rotation of \( 90^{\circ}\) counter - clockwise carries the rectangle onto itself.

Step3: Analyze rotation of \( 90^{\circ} \) clockwise

A rotation of \( 90^{\circ}\) clockwise about the center of the rectangle will map each vertex to another vertex and each side to another side of the rectangle. So, rotation of \( 90^{\circ}\) clockwise carries the rectangle onto itself.

Step4: Analyze rotation of \( 45^{\circ} \) counter - clockwise

A rectangle does not have rotational symmetry of order 8. A rotation of \( 45^{\circ}\) counter - clockwise about the center of the rectangle will not map the rectangle onto itself. Because the angles between the sides of a rectangle are \( 90^{\circ}\), not \( 45^{\circ}\).

Answer:

reflection across \( l \), rotation of \( 90^{\circ}\) counter - clockwise, rotation of \( 90^{\circ}\) clockwise