Question

which of the following transformations carry this regular polygon onto itself?rotation of 30° counterclockwiser reflection across lrotation of 120° clockwiserotation of 120° counterclockwise

Explanation:

Step1: Analyze rotation angle for regular triangle

A regular triangle (equilateral triangle) has rotational symmetry. The formula for the angle of rotational symmetry of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For \(n = 3\), the angle of rotational symmetry is \(\frac{360^{\circ}}{3}=120^{\circ}\). A rotation of \(120^{\circ}\) (clockwise or counter - clockwise) will map the triangle onto itself. A rotation of \(30^{\circ}\) will not, since \(30^{\circ}\) is not a multiple of \(120^{\circ}\) (the fundamental rotational symmetry angle for an equilateral triangle).

Step2: Analyze reflection

A reflection across a line of symmetry (in an equilateral triangle, there are 3 lines of symmetry). If line \(l\) is a line of symmetry (a median, altitude, or angle bisector), then reflection across \(l\) will map the triangle onto itself.

Answer:

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