To find the radius of the original circle when a scale factor of \( \frac{3}{4} \) is applied to a circle with a radius of 9 inches, you can use the relationship between the radius of the original circle (denote it as \( r \)) and the radius of the dilated circle (which is given as 9 inches).
The relationship can be expressed as follows:
\[ \text{Dilated radius} = \text{Original radius} \times \text{Scale factor} \]
Substituting the known values:
\[ 9 = r \times \frac{3}{4} \]
To solve for \( r \), you can rearrange the equation:
\[ r = \frac{9}{\frac{3}{4}} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ r = 9 \times \frac{4}{3} \]
Now calculate \( r \):
\[ r = 9 \times \frac{4}{3} = \frac{36}{3} = 12 \]
Thus, the radius of the original circle is \( \boxed{12} \) inches.