Question : The ratio between the number of sides of two regular polygons is 1 : 2 and the ratio between their interior angles is 2 : 3. The number of sides of these polygons is respectively:
Option 1: 6, 12
Option 2: 5, 10
Option 3: 4, 8
Option 4: 7, 14
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Correct Answer: 4, 8
Solution : Given, The ratio between the number of sides of two regular polygons = 1 : 2 And, the ratio between their interior angles = 2 : 3 Let the number of sides be $n$ and $2n$ respectively in two regular polygons. For a regular polygon, interior angle = $\frac{180^\circ\times (n-2)}{n}$ So, $\frac{(n-2)\times 180^\circ}{n}\div\frac{(2n-2)\times 180^\circ}{2n} = \frac{2}{3}$ Or, $\frac{2n-4}{4n-4}=\frac{1}{3}$ Or, $n=4$ and, $2n = 8$ Hence, the correct answer is 4, 8.
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