https://r2.erweima.ai/mathbot/image/17248757986680etwim9c.jpg

To solve for \( x \) in this geometric problem, we can use the properties of similar triangles.

### Step-by-Step Solution:

1. **Identify the Similar Triangles**:
   - The triangles share a common angle, and both have right angles. Therefore, they are similar by the AA (Angle-Angle) similarity criterion.

2. **Set Up the Proportion**:
   - Using the corresponding sides of the similar triangles, we can set up the proportion. Denote the smaller triangle's sides as:
     - Hypotenuse \( x \)
     - One leg as \( 3.5 \)
     - The other leg as \( 6 \)

     And for the larger triangle:
     - Hypotenuse as \( 7 \)
     - One leg as \( 6 \) (same as in the smaller triangle)
     - The other leg as \( x \)

   Since the triangles are similar, the ratios of the corresponding sides are equal:
   \[
   \frac{x}{3.5} = \frac{7}{6}
   \]

3. **Solve for \( x \)**:
   - Cross-multiply to find \( x \):
   \[
   x \times 6 = 7 \times 3.5
   \]
   \[
   6x = 24.5
   \]
   \[
   x = \frac{24.5}{6}
   \]
   \[
   x \approx 4.08
   \]

Thus, the value of \( x \) is approximately 4.08 units.

Would you like further explanation or have any questions about this problem?

### Related Questions:
1. What are the key steps to identify similar triangles?
2. How can the Pythagorean theorem be used to solve right triangles?
3. Can the concept of similar triangles be applied to non-right triangles?
4. What are other methods to find the lengths of sides in a triangle?
5. How can proportionality help in solving real-life geometric problems?

### Tip:
When working with similar triangles, always remember to match corresponding sides correctly to ensure the proportion is set up properly.

Learn how to solve for x in a geometry problem involving similar triangles. This detailed solution uses the AA similarity criterion and proportions to find the value of x. Suitable for students in Grades 9-12.

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