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The problem asks which proportion can be used to calculate the length of \( LM \) in centimeters, given that triangle \( HJK \) is similar to triangle \( LMN \).

### Key information:
- Triangle \( HJK \) is similar to triangle \( LMN \), meaning their corresponding sides are proportional.
- \( HJ = 4.5 \, \text{cm} \), \( JK = 6 \, \text{cm} \), and \( HK = 7.5 \, \text{cm} \).
- \( LM = ? \), \( MN = 9 \, \text{cm} \), and \( LN = 11.25 \, \text{cm} \).

To solve this, we need to identify the corresponding sides between the two triangles:
- \( HJ \) corresponds to \( LM \),
- \( JK \) corresponds to \( MN \),
- \( HK \) corresponds to \( LN \).

### Steps:
1. **Set up the proportion based on similar triangles**:
   Since \( \frac{HJ}{LM} = \frac{JK}{MN} = \frac{HK}{LN} \), we can use the ratios of corresponding sides to solve for \( LM \).

2. **Use \( HJ \) and \( LM \)**:
   The proportion involving \( HJ \) and \( LM \) is:
   \[
   \frac{HJ}{LM} = \frac{JK}{MN}
   \]
   Substituting the known values:
   \[
   \frac{4.5}{LM} = \frac{6}{9}
   \]
   Simplifying the right-hand side:
   \[
   \frac{4.5}{LM} = \frac{2}{3}
   \]

   Cross-multiply to solve for \( LM \):
   \[
   4.5 \times 3 = 2 \times LM
   \]
   \[
   13.5 = 2 \times LM
   \]
   \[
   LM = \frac{13.5}{2} = 6.75 \, \text{cm}
   \]

Thus, the proportion that can be used is:

\[
\boxed{\frac{6}{7.5} = \frac{LM}{9}} \quad \text{(Option B)}.
\]

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Would you like a detailed breakdown or have further questions? Here are some related questions for further practice:

1. How do you know that triangles are similar based on their sides?
2. What other proportions could be used in this problem?
3. How can you find unknown sides using similar triangles in general?
4. What is the difference between congruent and similar triangles?
5. How would the proportion change if we were solving for another side, like \( LN \)?

**Tip**: Always ensure that you're matching the corresponding sides correctly when working with similar triangles.

Which proportion can be used to calculate the length of LM in centimeters, given that triangle HJK is similar to triangle LMN?

Learn how to find the length of a side in similar triangles using proportions. This problem involves identifying corresponding sides and setting up the correct proportion to solve for the unknown side length. Suitable for students in Grades 8-10.

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