Question

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.

 

Answer 1

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