Given that triangles \( \triangle ABC \sim \triangle LMN \), we know that their corresponding sides are in proportion.
The sides of triangle \( ABC \) are:
- \( AB = 5 \)
- \( AC = 4 \)
- \( BC = 3 \)
And for triangle \( LMN \):
- \( LM = 7.5 \)
We need to find the length of \( LN \) corresponding to side \( AC \).
From the similarity of the triangles, the proportion of corresponding side lengths can be set up as:
\[ \frac{AB}{LM} = \frac{AC}{LN} \]
Substituting the known values:
\[ \frac{5}{7.5} = \frac{4}{LN} \]
To solve for \( LN \), we can cross-multiply:
\[ 5 \cdot LN = 4 \cdot 7.5 \]
Calculating the right side:
\[ 4 \cdot 7.5 = 30 \]
So we have:
\[ 5 \cdot LN = 30 \]
Now dividing both sides by 5:
\[ LN = \frac{30}{5} = 6 \]
Thus, the length of \( LN \) is
\[ \boxed{6} \]
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