The number of sides of two regular polygons are as \(5:4\) and the difference between their angles is \(9^\circ\). Find the number of sides of the polygons.
Solution:
Let the number of sides of the polygons be:
\[ 5x \text{ and } 4x \]
Interior angle of a regular polygon:
\[ \frac{(n-2)\times180^\circ}{n} \]
First polygon angle:
\[ \frac{(5x-2)\times180^\circ}{5x} \]
Second polygon angle:
\[ \frac{(4x-2)\times180^\circ}{4x} \]
Given difference:
\[ \frac{(5x-2)\times180}{5x} – \frac{(4x-2)\times180}{4x} = 9 \]
\[ 180\left(\frac{5x-2}{5x}-\frac{4x-2}{4x}\right)=9 \]
\[ 180\left(\frac{4(5x-2)-5(4x-2)}{20x}\right)=9 \]
\[ 180\left(\frac{20x-8-20x+10}{20x}\right)=9 \]
\[ 180\left(\frac{2}{20x}\right)=9 \]
\[ \frac{18}{x}=9 \]
\[ x=2 \]
Therefore, the number of sides are:
\[ 5x=10 \]
and
\[ 4x=8 \]
Hence, the polygons are:
\[ \text{Decagon and Octagon} \]