To solve this problem, we will use the concept of similar triangles. Since $DE \parallel AB$, the two triangles $\triangle ABC$ and $\triangle DEC$ are similar. The corresponding sides of similar triangles are proportional.

Part (a) Find the length of $DE$:

Given:

• $AD = 8.80 \, \mathrm{m}$, $DC = 13.2 \, \mathrm{m}$, $AB = 25.0 \, \mathrm{m}$

The ratio of the smaller triangle's side to the larger triangle's corresponding side is:

$\text{Ratio} = \frac{AD}{AC} = \frac{AD}{AD + DC} = \frac{8.80}{8.80 + 13.2} = \frac{8.80}{22.0} = 0.4$

Using this ratio, the length of $DE$ is:

$DE = \text{Ratio} \cdot AB = 0.4 \cdot 25.0 = 10.0 \, \mathrm{m}$

Thus, $DE = 10.0 \, \mathrm{m}$.

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Part (b) Find the length of $BC$:

Given:

• $CE = 10.2 \, \mathrm{m}$, $DE = 10.0 \, \mathrm{m}$ (from part a)

Using the same ratio $\frac{AD}{AC} = 0.4$, the length of $BC$ (the larger triangle's side corresponding to $CE$) is:

$BC = \frac{CE}{\text{Ratio}} = \frac{10.2}{0.4} = 25.5 \, \mathrm{m}$

Thus, $BC = 25.5 \, \mathrm{m}$.

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Final Answers:

(a) $DE = 10.0 \, \mathrm{m}$
(b) $BC = 25.5 \, \mathrm{m}$

Let me know if you want further clarifications! Here are related questions to expand understanding:

1. Why are the two triangles similar when $DE \parallel AB$?
2. How does the proportionality principle work in similar triangles?
3. How can the solution change if the parallel condition is removed?
4. What real-world applications involve similar triangles?
5. Can you prove the similarity of the triangles using angles?

Tip: Always confirm the parallel condition when solving problems involving similar triangles.