Question

which of the following transformations carry this regular polygon onto itself?
reflection across ( l )
rotation of ( 40 ^ { circ } ) clockwise
rotation of ( 120 ^ { circ } ) clockwise
rotation of ( 120 ^ { circ } ) counterclockwise

Explanation:

Step1: Analyze the properties of a regular triangle

A regular triangle (equilateral triangle) has rotational symmetry of order 3. The angle of rotational symmetry is \(\frac{360^{\circ}}{3}=120^{\circ}\). So, a rotation of \(120^{\circ}\) clockwise or counter - clockwise will map the triangle onto itself.

Step2: Analyze reflection

A regular triangle has 3 lines of symmetry. If line \(l\) is a line of symmetry (which it is, as it is drawn in a way that for an equilateral triangle it can be a median/altitude/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