1. **State the problem:** We have two similar trapezoidal prisms. The smaller prism has side lengths 5 mm, 6 mm, and a top edge of 2 mm with height $x$ mm. The larger prism has side lengths 7 mm, 18 mm, and one slant height of 12 mm. We need to find the value of $x$ as a fraction in simplest form. 2. **Use the property of similar shapes:** Corresponding sides of similar shapes are proportional. This means the ratio of corresponding sides in the smaller prism equals the ratio of the corresponding sides in the larger prism. 3. **Set up the ratio:** We can compare the heights $x$ and 12 mm, and the corresponding side lengths 2 mm and 7 mm (top edges). $$\frac{x}{12} = \frac{2}{7}$$ 4. **Solve for $x$:** Multiply both sides by 12 to isolate $x$. $$x = 12 \times \frac{2}{7}$$ 5. **Calculate $x$:** $$x = \frac{24}{7}$$ 6. **Simplify the fraction:** $\frac{24}{7}$ is already in simplest form because 24 and 7 have no common factors other than 1. **Final answer:** $$x = \frac{24}{7}$$