Question : Two regular polygons are such that the ratio between their number of sides is 1 : 2 and the ratio of measures of their interior angles is 3 : 4. Then the number of sides of each polygon is:
Option 1: 10 and 20
Option 2: 4 and 8
Option 3: 3 and 6
Option 4: 5 and 10
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Correct Answer: 5 and 10
Solution : Given that the sides of the two regular polygons are in the ratio 1 : 2. The interior angles of these polygons are in the ratio 3 : 4 Let the number of sides of the polygons be $n$ and $2n$ respectively. The sum of interior angles for a $n$-sided regular polygon = $\frac{(n-2)\times 180^\circ}{n}$ So, $\frac{(n-2)\times 180^\circ}{n}\div\frac{(2n-2)\times 180^\circ}{2n} = \frac{3}{4}$ Or, $\frac{n-2}{n-1}=\frac{3}{4}$ Or, $n=5$ and, $2n=10$ Hence, the correct answer is 5 and 10.
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