Question

which of the following transformations carry this regular polygon onto itself? image of a regular triangle with a line t, and options: rotation of 120° counterclockwise, reflection across t, rotation of 90° clockwise, rotation of 120° clockwise

Explanation:

Step1: Analyze the regular polygon (equilateral triangle)

A regular triangle (equilateral triangle) has rotational symmetry. The central angle for rotational symmetry is calculated as $\frac{360^\circ}{n}$, where $n = 3$ (number of sides). So $\frac{360^\circ}{3}=120^\circ$. So rotations of $120^\circ$ (clockwise or counter - clockwise) will map the triangle onto itself.

Step2: Analyze reflection

For an equilateral triangle, a reflection across a line that is parallel to a side (like line $t$ in the diagram, which is parallel to the base of the triangle and passes through the mid - points of the other two sides) will map the triangle onto itself because the triangle is symmetric with respect to such a line.

Step3: Analyze rotation of $90^\circ$

Since the rotational symmetry angle is $120^\circ$, a rotation of $90^\circ$ (which is not a multiple of $120^\circ$) will not map the equilateral triangle onto itself.

Answer:

• rotation of $120^\circ$ counterclockwise
• reflection across $t$
• rotation of $120^\circ$ clockwise