1. **State the problem:** We have two similar triangles, $\triangle ABC \sim \triangle ADE$, and we need to find the length $u = AE$. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are proportional. This means: $$\frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE}$$ 3. **Identify corresponding sides:** - $AB$ corresponds to $AD$ - $BC$ corresponds to $DE$ - $AC$ corresponds to $AE$ 4. **Given lengths:** - $BC = 24$ m - $DE = 51$ m - $AB = 34$ m - $AE = u$ (unknown) 5. **Set up the proportion using corresponding sides $BC$ and $DE$ to find the scale factor:** $$\frac{BC}{DE} = \frac{24}{51}$$ 6. **Use the scale factor to find $AD$ corresponding to $AB$:** Since $\frac{AB}{AD} = \frac{24}{51}$, then $$AD = \frac{51}{24} \times AB = \frac{51}{24} \times 34$$ 7. **Calculate $AD$:** $$AD = \frac{51}{24} \times 34 = \frac{51 \times 34}{24} = \frac{1734}{24} = 72.25$$ 8. **Use the proportion for $AC$ and $AE$:** Since $\frac{AC}{AE} = \frac{BC}{DE} = \frac{24}{51}$, and $AE = u$, we need $AC$ to find $u$. 9. **Find $AC$ using triangle $ABC$:** $AC = AB + BC = 34 + 24 = 58$ m 10. **Set up the proportion for $AC$ and $AE$:** $$\frac{AC}{AE} = \frac{24}{51} \Rightarrow \frac{58}{u} = \frac{24}{51}$$ 11. **Solve for $u$:** $$58 \times 51 = 24 \times u$$ $$2958 = 24u$$ $$u = \frac{2958}{24} = 123.25$$ **Final answer:** $$u = 123.25 \text{ meters}$$