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प्रश्न
The number of sides of two regular polygons are as 5 : 4 and the difference between their angles is 9°. Find the number of sides of the polygons.
उत्तर
Let the number of sides in the first polygon be 5x and the number of sides in the second polygon be 4x.
We know:
Angle of an n-sided regular polygon = \[\left( \frac{n - 2}{n} \right)180^\circ\]
Thus, we have:
Angle of the first polygon = \[\left( \frac{5x - 2}{5x} \right)180^\circ\]
Angle of the second polygon = \[\left( \frac{4x - 2}{4x} \right)180^\circ\]
Now,
\[\left( \frac{5x - 2}{5x} \right)180 - \left( \frac{4x - 2}{4x} \right)180 = 9\]
\[ \Rightarrow 180\left( \frac{4(5x - 2) - 5(4x - 2)}{20x} \right) = 9\]
\[ \Rightarrow \frac{20x - 8 - 20x + 10}{20x} = \frac{9}{180}\]
\[ \Rightarrow \frac{2}{20x} = \frac{1}{20}\]
\[ \Rightarrow \frac{2}{x} = 1\]
\[ \Rightarrow x = 2\]
Thus, we have:
Number of sides in the first polygon = 5x = 10
Number of sides in the second polygon = 4x = 8
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