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.