1. **Problem Statement:** We have two triangles, RST and UVT, with UV parallel to RS. Given lengths are UV = 18.2, VT = 14.5, VS = 17.5, and RU = x. We need to find the value of x. 2. **Concept Used:** When a line is drawn parallel to one side of a triangle, it creates similar triangles. Here, triangle UVT is similar to triangle RST. 3. **Similarity Ratios:** Since UV is parallel to RS, the sides are proportional: $$\frac{RU}{RS} = \frac{UV}{RS} = \frac{VT}{ST}$$ 4. **Assigning Lengths:** We know UV = 18.2, VT = 14.5, VS = 17.5, and RU = x. Since UV is parallel to RS, and UV corresponds to RS, and VT corresponds to ST, we can write: $$\frac{x}{x + 17.5} = \frac{18.2}{18.2 + 14.5}$$ 5. **Calculate Denominators:** $$x + 17.5$$ is the full length RS. $$18.2 + 14.5 = 32.7$$ is the full length of UV + VT. 6. **Set up the proportion:** $$\frac{x}{x + 17.5} = \frac{18.2}{32.7}$$ 7. **Cross multiply:** $$32.7x = 18.2(x + 17.5)$$ 8. **Distribute:** $$32.7x = 18.2x + 318.5$$ 9. **Subtract 18.2x from both sides:** $$32.7x - 18.2x = 318.5$$ $$14.5x = 318.5$$ 10. **Solve for x:** $$x = \frac{318.5}{14.5} \approx 21.97$$ 11. **Round to nearest tenth:** $$x \approx 22.0$$ **Final answer:** $x = 22.0$