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The problem presents a right triangle \( ABC \) with a perpendicular bisector \( \ell \) of segment \( AC \), intersecting at point \( D \). We are asked to find the value of angle \( x \) formed by the perpendicular bisector and segment \( BD \).

Here are the key observations:
- \( AC = 9 \), as the segments \( AD = 4 \) and \( DC = 5 \).
- \( \ell \) is the perpendicular bisector of \( AC \), meaning it divides \( AC \) into two equal parts and forms a right angle with \( AC \) at point \( D \).
- Given that \( BD \) and \( \ell \) are involved, we need to use trigonometric properties or relationships to find the angle \( x \).

Let's first apply the Pythagorean theorem to find the length of \( BD \).

\[
BD = \sqrt{AB^2 - AD^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3
\]

Now that we know the sides of triangle \( BDE \), where \( BD = 3 \) and \( DE = 5 \), we can use trigonometric ratios to find angle \( x \). Specifically, the tangent of angle \( x \) in right triangle \( BDE \) is:

\[
\tan(x) = \frac{DE}{BD} = \frac{5}{3}
\]

Thus,

\[
x = \tan^{-1}\left(\frac{5}{3}\right)
\]

I'll calculate the angle value next.The value of angle \( x \) is approximately \( 59.04^\circ \).

Would you like more details about the process or any specific part of the solution? 

Here are some related questions for further exploration:
1. How would the problem change if \( \ell \) were not the perpendicular bisector?
2. What trigonometric identity could be used to confirm the tangent result?
3. How can we verify the result using sine or cosine instead of tangent?
4. How do perpendicular bisectors relate to the circumcenter of a triangle?
5. How would this problem change if the values for the triangle sides were different?

**Tip:** Always check the relationships between angles and sides in right triangles using trigonometric ratios to simplify angle calculations.

En la figura, halle el valor de x si \( \ell \) es mediatriz de \( AC \).

Solve the geometric problem where line \( \ell \) is the perpendicular bisector of segment \( AC \) in a triangle. Using the Pythagorean theorem and trigonometric ratios, find the value of angle x. Suitable for high school students in grades 9-12.

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