To find the length of $DE$ in the given figure, we can use the concept of similar triangles.

Step 1: Identify the triangles

In the given figure:

• Triangle $\triangle ABC$ is the larger triangle.
• Triangle $\triangle CDE$ is the smaller triangle.

Step 2: Use Similarity of Triangles

Since $DE \parallel AB$, triangles $\triangle CDE$ and $\triangle ABC$ are similar by the AA (Angle-Angle) criterion.

Step 3: Set Up the Proportion

Because the triangles are similar, the ratios of corresponding sides are equal: $\frac{DE}{AB} = \frac{CD}{AC} = \frac{CE}{BC}$

Given:

• $AB = 20.0 \, \text{m}$
• $CD = 12.3 \, \text{m}$
• $AC = 8.2 \, \text{m}$
• $CE = 10.2 \, \text{m}$
• $BC$ is not directly needed since we are only comparing corresponding sides that relate to $DE$ and $AB$.

We calculate the proportion: $\frac{DE}{20.0} = \frac{12.3}{8.2}$

Step 4: Solve for $DE$

$DE = \frac{12.3}{8.2} \times 20.0$

Let me calculate this for you.The calculated length of $DE$ is approximately 30.0 meters.

It appears there might be an error in interpreting the question, as none of the provided answer choices match this calculation. However, given the principles of similar triangles, this result seems mathematically correct based on the provided dimensions.

Would you like to discuss this further, or have any questions? Here are some related questions to explore:

1. How can we check the accuracy of our calculations in geometry problems?
2. What other methods can be used to verify the similarity of triangles?
3. How do proportions relate to similar figures in geometry?
4. What would happen if $DE$ was not parallel to $AB$?
5. Can you apply similar triangle principles to other geometric shapes?

Tip: When working with similar triangles, ensure all corresponding sides and angles are correctly identified to avoid calculation errors.