Question

for what angles of rotation does the figure map onto itself?
a. an equilateral triangle
rotate the figure in the tool to the right by dragging the slider.
find the angles of rotation about the center that map \\( \triangle a b c \\) onto itself.

• a rotation of \\( 120^{\circ} \\) creates an identical image with vertex \\( b \\) at the top.
• a rotation of \\( 240^{\circ} \\) creates an identical image with vertex \\( c \\) at the top.

Explanation:

Step1: Understand rotational symmetry

Rotational symmetry of a figure is the property where the figure maps onto itself after a rotation about a center by an angle. For a regular polygon with \(n\) sides, the angle of rotation \(\theta=\frac{360^{\circ}}{n}\).
For an equilateral triangle \(n = 3\), so \(\theta=\frac{360^{\circ}}{3}=120^{\circ}\).

Step2: Find all angles of rotation

The angles of rotation that map the figure onto itself are multiples of the basic angle of rotation \(\theta\) within the range \(0^{\circ}<\alpha\leq360^{\circ}\).
If \(\theta = 120^{\circ}\), then the angles are \(120^{\circ}\times1 = 120^{\circ}\) and \(120^{\circ}\times2=240^{\circ}\) (since \(120^{\circ}\times3 = 360^{\circ}\), but a \(360^{\circ}\) rotation is a full - turn and is considered a trivial case in non - full - turn rotational symmetry discussions for polygons).

Answer:

The angles of rotation about the center that map an equilateral triangle onto itself are \(120^{\circ}\) and \(240^{\circ}\).