Question

a regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°. how many sides does the polygon have?
○ 10
○ 12
○ 18
○ cut off

Explanation:

Step1: Recall rotational symmetry formula

For a regular polygon with \( n \) sides, the smallest angle of rotational symmetry is \( \frac{360^\circ}{n} \), and other angles of rotational symmetry are multiples of this smallest angle.

Step2: Find the greatest common divisor (GCD) of given angles

The given angles are \( 20^\circ \), \( 40^\circ \), and \( 80^\circ \). The GCD of these angles is \( 20^\circ \), which should be the smallest angle of rotational symmetry.

Step3: Calculate the number of sides

Using the formula \( \frac{360^\circ}{n}=\text{smallest rotational angle} \), substitute the smallest angle (\( 20^\circ \)):
\( n = \frac{360^\circ}{20^\circ} = 18 \)? Wait, no, wait. Wait, let's check again. Wait, if the smallest angle is \( 20^\circ \), then \( n=\frac{360}{20}=18 \)? But wait, let's check the options. Wait, maybe I made a mistake. Wait, no, wait, the angles given are \( 20^\circ \), \( 40^\circ \), \( 80^\circ \). Let's check for \( n = 18 \): smallest angle is \( 360/18 = 20^\circ \), then multiples are \( 40^\circ \) (220), \( 60^\circ \), \( 80^\circ \) (420), etc. Wait, but the options have 10, 12, 18, and another. Wait, maybe I miscalculated. Wait, let's check \( n = 18 \): 360/18=20. So rotational angles are 20k where k=1,2,...,17. So 20,40,60,80,... So 80 is 420, which is valid. Wait, but let's check other options. For \( n = 10 \): 360/10=36. So rotational angles are 36,72,108,... Not 20,40,80. For \( n = 12 \): 360/12=30. Rotational angles 30,60,90,... Not 20,40,80. For \( n = 18 \): 360/18=20. So rotational angles 20,40,60,80,... So 80 is included (420). So the answer should be 18? Wait, but wait, the problem says "possible angles of rotational symmetry of 20°, 40°, and 80°". So we need to find n such that 20,40,80 are multiples of 360/n. Let's denote \( \theta = 360/n \), so 20 = k\(\theta\), 40 = m\(\theta\), 80 = p\(\theta\), where k,m,p are positive integers. So \(\theta\) must be a divisor of 20,40,80. The GCD of 20,40,80 is 20, so \(\theta = 20\), so n = 360/20 = 18. So the number of sides is 18. Wait, but let's check the options. The options are 10,12,18, and another (maybe 36? But the visible options are 10,12,18, and a fourth). So 18 is an option. So the correct answer is 18.

Answer:

18 (corresponding to the option with 18, e.g., if the options are A.10, B.12, C.18, D...., then the answer is C. 18)