Question

which of the following transformations carry this regular polygon onto itself? rotation of 36° clockwise rotation of 60° clockwise rotation of 120° counterclockwise reflection across l

Explanation:

Step1: Analyze the polygon

The figure is a regular triangle (equilateral triangle), so it has 3 sides. The central angle for a regular \( n \)-sided polygon is \( \frac{360^\circ}{n} \). For \( n = 3 \), the central angle is \( \frac{360^\circ}{3}=120^\circ \). So rotations by multiples of \( 120^\circ \) (clockwise or counterclockwise) will map the triangle onto itself. Also, reflections across lines of symmetry (like the line \( l \) shown, which is a line of symmetry for the equilateral triangle) will map it onto itself.

Step2: Evaluate each option

• Rotation of \( 36^\circ \) clockwise: \( 36^\circ \) is not a multiple of \( 120^\circ \), so this rotation will not map the triangle onto itself.
• Rotation of \( 60^\circ \) clockwise: \( 60^\circ \) is not a multiple of \( 120^\circ \), so this rotation will not map the triangle onto itself.
• Rotation of \( 120^\circ \) counterclockwise: Since \( 120^\circ \) is a multiple of \( 120^\circ \) (the central angle), this rotation will map the triangle onto itself.
• Reflection across \( l \): The line \( l \) is a line of symmetry (it passes through a vertex and the midpoint of the opposite side) for the equilateral triangle, so reflecting across \( l \) will map the triangle onto itself.

Answer:

rotation of \( 120^\circ \) counterclockwise, reflection across \( l \)