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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