Question

a certain regular polygon is rotated 240° about its center, which carries the figure onto itself. which regular polygon could it be?
answer
pentagon
hexagon
decagon
square
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Explanation:

Step1: Recall rotation symmetry condition

A regular polygon with \(n\) sides maps to itself under rotation by \(\frac{360^\circ}{n}k\) (k integer). So \(240^\circ\) must be a multiple of \(\frac{360^\circ}{n}\).

Step2: Calculate n for each option

• Pentagon (\(n=5\)): \(\frac{360}{5}=72^\circ\); 240/72≈3.33 (not integer)
• Hexagon (\(n=6\)): \(\frac{360}{6}=60^\circ\); 240/60=4 (integer)
• Square (\(n=4\)): \(\frac{360}{4}=90^\circ\); 240/90≈2.67 (not integer)
• Decagon (\(n=10\)): \(\frac{360}{10}=36^\circ\); 240/36≈6.67 (not integer)

Answer:

hexagon