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To solve this problem, we need to use the fact that the two triangles, \( \triangle FIM \) and \( \triangle LAK \), are similar. When two triangles are similar, their corresponding sides are proportional.

Given:

1. \( \triangle FIM \) has sides \( FI = 4 \), \( IM = 5 \), and \( FM = 6 \).
2. \( \triangle LAK \) has sides \( LK = 9 \) and we need to find \( LA \).

Since the triangles are similar, we can set up a proportion between the corresponding sides:

\[
\frac{FI}{LA} = \frac{IM}{AK} = \frac{FM}{LK}
\]

We know \( FM = 6 \) and \( LK = 9 \), so we can find the ratio:

\[
\frac{FM}{LK} = \frac{6}{9} = \frac{2}{3}
\]

This tells us that the sides of \( \triangle FIM \) are to the sides of \( \triangle LAK \) in a ratio of \( \frac{2}{3} \).

Now, we need to find \( LA \). Since \( \frac{FI}{LA} = \frac{2}{3} \), we substitute \( FI = 4 \):

\[
\frac{4}{LA} = \frac{2}{3}
\]

Cross-multiplying to solve for \( LA \):

\[
4 \cdot 3 = 2 \cdot LA \implies 12 = 2 \cdot LA \implies LA = \frac{12}{2} = 6
\]

Thus, the length of \( LA \) is **6**.

### Answer: 
The correct choice is **6**. 

Would you like more details, or do you have any other questions? 

Here are 5 related questions for further practice:
1. How do you determine if two triangles are similar?
2. If another pair of triangles are similar, how do you find the length of a missing side?
3. What are the properties of similar triangles?
4. How would the answer change if the sides of \( \triangle FIM \) were doubled?
5. How does knowing the scale factor between two similar figures help in real-world applications?

**Tip:** Always ensure the corresponding sides and angles match up correctly when working with similar triangles.

What is the length of LA in the similar triangles FIM and LAK, given that FI = 4, IM = 5, FM = 6, and LK = 9?

Learn how to find the length of LA in the similar triangles FIM and LAK using proportions and the Similar Triangles Theorem. This problem is ideal for students in Grades 8-10 and covers key geometry concepts.

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