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To solve for the length \( x \) in the given diagram using the properties of similar triangles and cross-multiplying, follow these steps:

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

1. **Identify the triangles and their sides:**
   - The smaller triangle has legs of lengths \( x \) and 3.
   - The larger triangle has legs of lengths 5 and 7.5.

2. **Set up the proportion using similar triangles:**
   Since the triangles are similar (they share a common angle and both have a right angle), 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:
   \[
   \frac{x}{5} = \frac{3}{7.5}
   \]

3. **Solve the proportion for \( x \):**
   Cross-multiply to solve for \( x \):
   \[
   x \times 7.5 = 3 \times 5
   \]
   \[
   7.5x = 15
   \]
   \[
   x = \frac{15}{7.5}
   \]
   \[
   x = 2
   \]

Thus, the length \( x \) is \( 2 \).

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 double-check the corresponding sides in similar triangles to ensure the proportions are set up correctly.

Learn how to solve for length x in similar triangles using cross-multiplication and proportionality. Follow a step-by-step explanation with detailed calculations and examples. Suitable for high school students studying geometry and proportions.

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