Question

which of the following transformations carry this regular polygon onto itself?rotation of 72° counterclockwiserotation of 90° counterclockwiserotation of 120° clockwiserotation of 120° counterclockwise

Explanation:

Step1: Recall rotational symmetry of regular polygons

For a regular \( n \)-sided polygon, the angle of rotational symmetry is given by \( \frac{360^\circ}{n} \). Here, the polygon is a regular triangle (equilateral triangle), so \( n = 3 \).

Step2: Calculate the rotational symmetry angle

The rotational symmetry angle for a regular triangle is \( \frac{360^\circ}{3}= 120^\circ \). This means that rotating the triangle by \( 120^\circ \) (or multiples of \( 120^\circ \)) clockwise or counterclockwise will map the triangle onto itself.

Step3: Analyze each rotation option

• Rotation of \( 72^\circ \): \( 72^\circ \) is not a multiple of \( 120^\circ \), so this rotation does not carry the triangle onto itself.
• Rotation of \( 90^\circ \): \( 90^\circ \) is not a multiple of \( 120^\circ \), so this rotation does not carry the triangle onto itself.
• Rotation of \( 120^\circ \) clockwise: Since \( 120^\circ \) is the rotational symmetry angle, this rotation carries the triangle onto itself.
• Rotation of \( 120^\circ \) counterclockwise: Since \( 120^\circ \) is the rotational symmetry angle, this rotation carries the triangle onto itself.

Answer:

• rotation of \( 120^\circ \) clockwise
• rotation of \( 120^\circ \) counterclockwise