Question

which of the following transformations carry this regular polygon onto itself?
rotation of 45° counterclockwise
rotation of 40° counterclockwise
rotation of 40° clockwise
rotation of 72° counterclockwise

Explanation:

To determine which rotations carry a regular polygon (here, a regular octagon, since it has 8 sides) onto itself, we use the formula for the angle of rotational symmetry of a regular \( n \)-sided polygon:

The angle of rotational symmetry is \( \frac{360^\circ}{n} \), where \( n \) is the number of sides. For a regular octagon, \( n = 8 \), so the angle of rotational symmetry is \( \frac{360^\circ}{8} = 45^\circ \).

Analyzing each option:

1. Rotation of \( 45^\circ \) counterclockwise:

Since \( 45^\circ \) is the angle of rotational symmetry for a regular octagon, rotating the octagon by \( 45^\circ \) (counterclockwise or clockwise) will map it onto itself. Thus, this transformation is valid.

1. Rotation of \( 40^\circ \) counterclockwise:

\( 40^\circ \) is not a multiple of \( 45^\circ \) (the angle of rotational symmetry). Rotating by \( 40^\circ \) will not align the octagon’s vertices with their original positions, so this transformation does not carry the octagon onto itself.

1. Rotation of \( 40^\circ \) clockwise:

Similar to the previous case, \( 40^\circ \) is not a multiple of \( 45^\circ \). Rotating by \( 40^\circ \) clockwise will also fail to map the octagon onto itself.

1. Rotation of \( 72^\circ \) counterclockwise:

\( 72^\circ \) is not a multiple of \( 45^\circ \) (e.g., \( 45^\circ \times 1 = 45^\circ \), \( 45^\circ \times 2 = 90^\circ \), etc.). Thus, rotating by \( 72^\circ \) will not align the octagon’s vertices, so this transformation is invalid.

Only the rotation of \( 45^\circ \) counterclockwise carries the regular octagon onto itself.

Answer:

rotation of \( 45^\circ \) counterclockwise