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Which regular polygon would carry onto itself after a rotation of
300^(@) about its center?
(A) decagon
(B) nonagon
(C) octagon
(D) hexagon

Which regular polygon would carry onto itself after a rotation of $300^{\circ}$ about its center? $\newline$ (A) decagon $\newline$ (B) nonagon $\newline$ (C) octagon $\newline$ (D) hexagon

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Identify and classify polygons

Full solution

Q. Which regular polygon would carry onto itself after a rotation of

300^{\circ}

about its center?

\newline

(A) decagon

\newline

(B) nonagon

\newline

(C) octagon

\newline

(D) hexagon

Calculate Angle of Rotation: To determine which regular polygon would carry onto itself after a rotation of $300$ degrees about its center, we need to find a polygon whose angle of rotational symmetry is a factor of $300$ degrees. The angle of rotational symmetry for a regular polygon is $360$ degrees divided by the number of sides ( $n$ ).
Check Divisibility for Each Polygon: For a regular polygon to carry onto itself after a $300$ -degree rotation, $360/n$ should be a divisor of $300$ . This means that $300$ divided by $(360/n)$ should be a whole number.
Decagon: Let's check each option: $\newline$ For a decagon ( $10$ sides), the angle of rotational symmetry is $360/10 = 36$ degrees. $300$ divided by $36$ is not a whole number.
Nonagon: For a nonagon ( $\newline$ $9$ sides), the angle of rotational symmetry is $360/9 = 40$ degrees. $300$ divided by $40$ is not a whole number.
Octagon: For an octagon ( $8$ sides), the angle of rotational symmetry is $360/8 = 45$ degrees. $300$ divided by $45$ is not a whole number.
Hexagon: For a hexagon ( $6$ sides), the angle of rotational symmetry is $360/6 = 60$ degrees. $300$ divided by $60$ is exactly $5$ , which is a whole number.
Hexagon: For a hexagon ( $6$ sides), the angle of rotational symmetry is $360/6 = 60$ degrees. $300$ divided by $60$ is exactly $5$ , which is a whole number.Therefore, a hexagon would carry onto itself after a rotation of $300$ degrees about its center because its angle of rotational symmetry is a factor of $300$ .

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