To find the radius of the original circle given that the scale factor is \( \frac{3}{4} \) and the radius of the dilated circle is 9 inches, you can use the formula for dilation:
\[ \text{Radius of dilated circle} = \text{Scale factor} \times \text{Radius of original circle} \]
Let \( r \) be the radius of the original circle. Thus, we have:
\[ 9 = \frac{3}{4} \times r \]
To solve for \( r \), multiply both sides by the reciprocal of \( \frac{3}{4} \):
\[ r = 9 \times \frac{4}{3} \]
Calculating this:
\[ r = 9 \times \frac{4}{3} = \frac{36}{3} = 12 \]
Therefore, the radius of the original circle is \( \boxed{12} \) inches.
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