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Determine the lengths of the unknown sides in the following pairs of similar triangles.

To determine the lengths of the unknown sides in the given pairs of similar triangles, we can use the property that corresponding sides of similar triangles are proportional.

### Analyzing the Triangles:

1. **Larger Triangle:**
   - Sides: \( x \), 6, 9
   - Given side ratios: \( \frac{x}{2} = \frac{6}{y} = \frac{9}{3} \)
  
2. **Smaller Triangle:**
   - Sides: 2, \( y \), 3

### Step 1: Use the proportion based on the sides that are already known.
  
\[
\frac{9}{3} = 3
\]

This ratio applies to all corresponding sides.

### Step 2: Calculate \( x \) using the ratio.

\[
\frac{x}{2} = 3 \implies x = 2 \times 3 = 6
\]

### Step 3: Calculate \( y \) using the ratio.

\[
\frac{6}{y} = 3 \implies y = \frac{6}{3} = 2
\]

### Final Values:
- \( x = 6 \)
- \( y = 2 \)

The unknown sides are \( x = 6 \) and \( y = 2 \).

Would you like further details or have any questions about this solution?

### Related Questions:
1. How do you determine if two triangles are similar?
2. Can similar triangles have sides that are not proportional?
3. What is the significance of the scale factor in similar triangles?
4. How does the Pythagorean Theorem relate to similar triangles?
5. What happens if one of the triangles in the pair is a right triangle?

### Tip:
When working with similar triangles, always ensure that you correctly identify corresponding sides before setting up proportions.

Learn how to determine the lengths of unknown sides in similar triangles using proportional relationships. This problem involves analyzing corresponding sides in similar triangles and applying ratios to find the missing lengths. Suitable for high school students studying geometry.

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