To find the radius of the original circle when a scale factor of \( \frac{1}{5} \) is used for dilation, you can set up the equation:
\[ \text{dilated radius} = \text{original radius} \times \text{scale factor} \]
Given a dilated radius of 18 cm and a scale factor of \( \frac{1}{5} \), you can rearrange the equation to find the original radius:
\[ \text{original radius} = \frac{\text{dilated radius}}{\text{scale factor}} \]
Substituting the values:
\[ \text{original radius} = \frac{18 , \text{cm}}{\frac{1}{5}} = 18 , \text{cm} \times 5 = 90 , \text{cm} \]
So, the radius of the original circle is 90 cm.