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To find the length of side \( a \), we can utilize the fact that both triangles \( \triangle ABC \) and \( \triangle CDE \) share the angle at vertex \( C \). Additionally, these triangles have corresponding angles marked as equal, which suggests they are similar triangles.

Given the similarity of the triangles \( \triangle ABC \) and \( \triangle CDE \), we can set up a proportion between the corresponding sides:

\[
\frac{AB}{AC} = \frac{CE}{CD}
\]

Substitute the known values:

- \( AB = 7 \, \text{m} \)
- \( AC = a \)
- \( CE = 12 \, \text{m} \)
- \( CD = 18 \, \text{m} \)

Thus, the proportion becomes:

\[
\frac{7}{a} = \frac{12}{18}
\]

Now, let's solve this proportion for \( a \):

\[
\frac{7}{a} = \frac{2}{3}
\]

By cross-multiplying:

\[
7 \times 3 = 2 \times a
\]

\[
21 = 2a
\]

\[
a = \frac{21}{2} = 10.5 \, \text{m}
\]

Therefore, the length of side \( a \) is \( 10.5 \, \text{m} \).

Would you like further details or have any questions?

Here are 5 related questions to expand this concept:

1. How do you verify if two triangles are similar?
2. What other methods could be used to solve for \( a \)?
3. How do corresponding angles play a role in triangle similarity?
4. How would the calculation change if different sides were known?
5. Can this method be applied to non-right triangles?

**Tip:** In problems involving similar triangles, always compare corresponding sides and set up a proportion to solve for unknowns.

Learn how to find the length of side a using similar triangles and proportions. In this example, triangles ABC and CDE are similar, and the length of side a is calculated using a proportion of corresponding sides. This problem is ideal for students in Grades 8-10, covering key concepts in geometry and triangle similarity.

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