Question

a student claims that a rotation of 180° about the center of a regular polygon carries the polygon onto itself. which statement best describes the students claim? a the students claim is true for all regular polygons. b the students claim is not true for any regular polygon. c the students claim is true for regular polygons with an odd number of sides. d the students claim is true for regular polygons with an even number of sides.

Explanation:

Step1: Recall rotation - symmetry property of regular polygons

A regular polygon with \(n\) sides has rotational symmetry of order \(n\). The angle of rotation \(\theta\) that maps a regular polygon onto itself is given by \(\theta=\frac{360^{\circ}}{k}\), where \(k = 1,2,\cdots,n\).

Step2: Check if 180 - degree rotation is possible

We want to know if \(\theta = 180^{\circ}\) can be a valid angle of rotation for a regular polygon. If \(\frac{360^{\circ}}{k}=180^{\circ}\), then \(k = 2\). A regular polygon will map onto itself under a 180 - degree rotation if and only if \(n\) (the number of sides) is even. When \(n\) is odd, a 180 - degree rotation will not map the polygon onto itself.

Answer:

D. The student's claim is true for regular polygons with an even number of sides.