1. **State the problem:** Quadrilateral PQRS is dilated by a scale factor of $\frac{3}{4}$ to form quadrilateral P'Q'R'S'. We are given the length of side S'R' as 18 and need to find the original length of side RS. 2. **Recall the dilation formula:** When a figure is dilated by a scale factor $k$, the lengths of corresponding sides are related by: $$\text{Length of image side} = k \times \text{Length of original side}$$ 3. **Apply the formula:** Here, the scale factor $k = \frac{3}{4}$ and the image side S'R' length is 18. Let the original side RS be $x$. $$18 = \frac{3}{4} \times x$$ 4. **Solve for $x$:** $$x = \frac{18}{\frac{3}{4}} = 18 \times \frac{4}{3}$$ 5. **Simplify:** $$x = \cancel{18} \times \frac{4}{\cancel{3}} = 6 \times 4 = 24$$ 6. **Conclusion:** The length of side RS in the original quadrilateral is $24$ units.