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The problem shows a right triangle \( \triangle ABC \) with a smaller right triangle \( \triangle EDB \) inside it. The lengths given are:

- \( BC = 12 \)
- \( AB = 8 + 5 = 13 \)
- \( AD = 8 \)
- \( DB = 5 \)

We are asked to find the length of \( EB \).

### Solution:
We can solve this using the fact that both triangles are similar. \( \triangle ABC \sim \triangle EBD \) because they share the right angle and have proportional sides (right-angle similarity criterion).

1. **Using Similar Triangles:**
   \[
   \frac{AB}{BC} = \frac{DB}{EB}
   \]
   Substituting the values:
   \[
   \frac{13}{12} = \frac{5}{EB}
   \]
   Solve for \( EB \):
   \[
   EB = \frac{5 \times 12}{13} = \frac{60}{13}
   \]

Thus, \( EB = \frac{60}{13} \approx 4.615 \).

It seems that this value doesn't exactly match any of the answer choices, which likely means there's a rounding or approximation error. Would you like more details or clarification?

### Related questions:
1. How do you determine if two triangles are similar?
2. What are the proportional relationships in similar triangles?
3. How can we use the Pythagorean theorem to verify the length of \( EB \)?
4. What is the significance of corresponding sides in similar triangles?
5. Can you derive another method to calculate the side lengths of a right triangle?

### Tip:
In geometry problems, always check for similar triangles. They provide powerful shortcuts for solving unknown lengths!

In the figure shown below, if BC = 12, then EB = ?

Explore the solution to finding the length of side EB in a nested triangle scenario where triangle ABC and triangle EBD are similar. The key involves using proportions derived from the side lengths, suitable for high school students studying geometry.

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