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The problem shows a graph with various points labeled \( L \), \( M \), \( N \), \( Q \), \( R \), and \( S \), along with lines connecting these points. The question asks which of the given statements about the ratios of line segments is true. The given slope is \( \frac{3}{2} \), and the points are positioned on the graph to form similar triangles.

Let's analyze the points and their coordinates:

- \( L = (-8, -6) \)
- \( M = (4, -6) \)
- \( N = (2, 0) \)
- \( Q = (4, 3) \)
- \( R = (6, 6) \)
- \( S = (8, 9) \)

Now, let's calculate the lengths of the line segments:

1. **Length of \( LM \)**:
   - \( L = (-8, -6) \) and \( M = (4, -6) \)
   - This is a horizontal segment, so its length is the difference in the x-coordinates:
     \[
     LM = 4 - (-8) = 4 + 8 = 12
     \]

2. **Length of \( MN \)**:
   - \( M = (4, -6) \) and \( N = (2, 0) \)
   - We use the distance formula:
     \[
     MN = \sqrt{(4 - 2)^2 + (-6 - 0)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32
     \]

3. **Length of \( RS \)**:
   - \( R = (6, 6) \) and \( S = (8, 9) \)
   - This is a vertical segment, so its length is the difference in the y-coordinates:
     \[
     RS = 9 - 6 = 3
     \]

4. **Length of \( QR \)**:
   - \( Q = (4, 3) \) and \( R = (6, 6) \)
   - This is a vertical segment, so its length is the difference in the y-coordinates:
     \[
     QR = 6 - 3 = 3
     \]

5. **Length of \( LN \)**:
   - \( L = (-8, -6) \) and \( N = (2, 0) \)
   - We use the distance formula:
     \[
     LN = \sqrt{(2 - (-8))^2 + (0 - (-6))^2} = \sqrt{(2 + 8)^2 + (6)^2} = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.66
     \]

6. **Length of \( QS \)**:
   - \( Q = (4, 3) \) and \( S = (8, 9) \)
   - We use the distance formula:
     \[
     QS = \sqrt{(8 - 4)^2 + (9 - 3)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21
     \]

Now let's analyze the statements in the options.

### Option A: \( \frac{LM}{MN} = \frac{RS}{QR} \)
- \( \frac{LM}{MN} = \frac{12}{6.32} \approx 1.9 \)
- \( \frac{RS}{QR} = \frac{3}{3} = 1 \)
- This is false.

### Option B: \( \frac{MN}{LM} = \frac{RS}{QR} \)
- \( \frac{MN}{LM} = \frac{6.32}{12} \approx 0.53 \)
- \( \frac{RS}{QR} = 1 \)
- This is false.

### Option C: \( \frac{LM}{LN} = \frac{QS}{QR} \)
- \( \frac{LM}{LN} = \frac{12}{11.66} \approx 1.03 \)
- \( \frac{QS}{QR} = \frac{7.21}{3} \approx 2.4 \)
- This is false.

### Option D: \( \frac{MN}{LM} = \frac{QR}{QS} \)
- \( \frac{MN}{LM} = \frac{6.32}{12} \approx 0.53 \)
- \( \frac{QR}{QS} = \frac{3}{7.21} \approx 0.42 \)
- This is false.

None of the options seem to hold. Would you like further details or help with additional questions? 

Here are 5 related questions:
1. How is the distance formula derived?
2. How do similar triangles help in solving geometric problems?
3. What is the significance of the slope in a line segment?
4. How do you calculate the slope between two points?
5. Can you generalize the relationship between slopes and line segments in geometric figures?

**Tip:** Always double-check the calculations when working with geometric problems to ensure accuracy!

Which of the following is true? Given that the line has a slope of 3/2 between points L and N, and points Q and S.

Learn how to analyze line segments on a graph where the slope is given as 3/2 between certain points. This problem covers the calculation of distances, the use of the slope formula, and verification of true ratio statements between line segments.

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