Question

which of the following transformations carry this regular polygon onto itself?rotation of 90° clockwiserotation of 30° clockwiserotation of 90° counterclockwiser reflection across l

Explanation:

1. First, identify the regular polygon. From the diagram, it's a regular octagon (8 - sided polygon).

• For a regular \( n \) -sided polygon, the angle of rotational symmetry is given by \( \frac{360^{\circ}}{n} \). For \( n = 8 \), the angle of rotational symmetry is \( \frac{360^{\circ}}{8}=45^{\circ} \). So, a rotation by a multiple of \( 45^{\circ} \) will map the octagon onto itself.
• A \( 90^{\circ} \) rotation (clockwise or counter - clockwise) is a multiple of \( 45^{\circ} \) (\( 90\div45 = 2 \)), so \( 90^{\circ} \) rotations (both clockwise and counter - clockwise) are valid. A \( 30^{\circ} \) rotation is not a multiple of \( 45^{\circ} \), so it is invalid.

1. For reflection: A regular octagon has 8 lines of symmetry. The line \( l \) in the diagram is a line of symmetry (it connects the mid - points of opposite sides, which is a line of symmetry for a regular octagon), so reflection across \( l \) will map the octagon onto itself.

Answer:

• rotation of \( 90^{\circ} \) clockwise
• rotation of \( 90^{\circ} \) counterclockwise
• reflection across \( l \)