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To solve for the length \( x \) in the given diagram, we will use the properties of similar triangles. The diagram shows two right triangles, one larger and one smaller. Since they share a common angle and both have a right angle, the triangles are similar by the AA (Angle-Angle) similarity criterion.

### Steps to Solve for \( x \):

1. **Identify the triangles and their sides:**
   - The smaller triangle has legs of lengths 5 and 4.5.
   - The larger triangle has a leg of length 10 (sum of the two smaller legs) and another leg of unknown length \( x \).

2. **Set up the proportion using similar triangles:**
   Since the triangles are similar, the ratios of their corresponding sides are equal.
   \[
   \frac{\text{leg}_1 \text{ of smaller triangle}}{\text{leg}_1 \text{ of larger triangle}} = \frac{\text{leg}_2 \text{ of smaller triangle}}{\text{leg}_2 \text{ of larger triangle}}
   \]
   Substituting the known values, we get:
   \[
   \frac{5}{10} = \frac{4.5}{x}
   \]

3. **Solve the proportion for \( x \):**
   Cross-multiply to solve for \( x \):
   \[
   5x = 4.5 \times 10
   \]
   \[
   5x = 45
   \]
   \[
   x = \frac{45}{5}
   \]
   \[
   x = 9
   \]

Thus, the length \( x \) is \( 9 \).

Would you like more details or have any questions?

Here are some related questions:

1. What is the AA (Angle-Angle) similarity criterion for triangles?
2. How do you identify corresponding sides in similar triangles?
3. Can you solve for the hypotenuse of the smaller triangle?
4. What other methods can be used to solve for \( x \) in similar triangles?
5. How would the solution change if the given triangle was not a right triangle?
6. What is the significance of cross-multiplying in solving proportions?
7. How do you prove that two triangles are similar?
8. How can you verify the accuracy of your solution?

**Tip:** Always check if triangles are similar before setting up proportions to ensure the ratios are valid.

Learn how to solve for the length x in similar triangles using cross-multiplication. This problem involves applying the AA (Angle-Angle) similarity criterion and proportionality in triangles. Suitable for students in Grades 8-10.

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