1. **Problem statement:** We have two triangles, UVT and RST, where UV is parallel to RS. We need to find the length $x = RU$. 2. **Key concept:** 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:** Corresponding sides of similar triangles are proportional. So, $$\frac{RU}{RT} = \frac{UT}{ST} = \frac{VT}{TS}$$ 4. **Given lengths:** - $UT = 18.2$ - $VT = 14.5$ - $RS = 17.5$ - $RU = x$ 5. **Find $RT$ and $ST$:** Since $RS = RT + TS$ and $UV$ is parallel to $RS$, the segments correspond such that $RT = RU + UT = x + 18.2$ and $TS = VT = 14.5$. 6. **Set up proportion:** Using the ratio of sides, $$\frac{RU}{RT} = \frac{UT}{ST}$$ Substitute values: $$\frac{x}{x + 18.2} = \frac{18.2}{17.5}$$ 7. **Solve for $x$:** Cross-multiply: $$x \times 17.5 = 18.2 \times (x + 18.2)$$ $$17.5x = 18.2x + 18.2 \times 18.2$$ $$17.5x - 18.2x = 331.24$$ $$-0.7x = 331.24$$ $$x = \frac{331.24}{-0.7} = -473.2$$ 8. **Check for error:** Negative length is not possible, so re-examine the setup. The correct approach is to use the ratio of the smaller triangle to the larger triangle sides: $$\frac{RU}{RT} = \frac{UT}{ST}$$ But $RT = RU + UT = x + 18.2$ and $ST = VT + TS = 14.5 + 17.5 = 32$ 9. **Correct proportion:** $$\frac{x}{x + 18.2} = \frac{18.2}{32}$$ Cross-multiplied: $$32x = 18.2(x + 18.2)$$ $$32x = 18.2x + 331.24$$ $$32x - 18.2x = 331.24$$ $$13.8x = 331.24$$ $$x = \frac{331.24}{13.8} \approx 24.0$$ **Final answer:** $x \approx 24.0$ (rounded to the nearest tenth).