Question

1. which figure will not carry onto itself after a 120 - degree rotation about its center?

a) regular octagon
b) equilateral triangle
c) regular hexagon
d) regular nonagon

Explanation:

Step1: Calculate the rotational symmetry angle formula

The formula for the angle of rotational symmetry of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). If \(120^{\circ}\) is a multiple of \(\frac{360^{\circ}}{n}\) (i.e., \(\frac{360^{\circ}}{n}\times k = 120^{\circ}\), where \(k\) is a positive integer), the polygon will map onto itself after a \(120^{\circ}\) rotation.

Step2: Analyze each option

• Option A: Regular octagon (\(n = 8\))

\(\frac{360^{\circ}}{8}=45^{\circ}\). And \(120\div45=\frac{8}{3}\), not an integer.

• Option B: Equilateral triangle (\(n = 3\))

\(\frac{360^{\circ}}{3} = 120^{\circ}\). Here \(k = 1\) (since \(120\div120=1\)).

• Option C: Regular hexagon (\(n=6\))

\(\frac{360^{\circ}}{6}=60^{\circ}\). And \(120\div60 = 2\) (where \(k = 2\)).

• Option D: Regular nonagon (\(n = 9\))

\(\frac{360^{\circ}}{9}=40^{\circ}\). And \(120\div40=3\) (where \(k = 3\)).

Answer:

A. regular octagon