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The problem provided asks to find the value of \( x \), given that \( \ell \) is the perpendicular bisector of \( AC \). Let's break this down step by step:

### Analysis:
1. We are given a right triangle \( ABC \) with points \( A \), \( B \), and \( C \) on the triangle, and point \( D \) lies on \( AC \), such that \( \ell \) is the perpendicular bisector of \( AC \).
   
2. Given the lengths:
   - \( AD = 4 \)
   - \( DC = 5 \)
   - The total length of \( AC = AD + DC = 9 \)

3. Since \( \ell \) is the perpendicular bisector of \( AC \), we know that:
   - \( D \) is the midpoint of \( AC \), so \( AD = DC \), which would mean \( AD = DC = \frac{AC}{2} = \frac{9}{2} = 4.5 \).

However, the given values for \( AD \) and \( DC \) are 4 and 5, respectively. This seems inconsistent with the property of the perpendicular bisector. One possible explanation is that this is a right triangle configuration, and we can use geometric relationships.

### Steps to find \( x \):

Since \( \ell \) is the perpendicular bisector, and point \( E \) is on the bisector, triangle \( BED \) should have a right angle at \( E \).

1. First, identify key relationships between the sides using trigonometric functions.
2. Apply the Pythagorean theorem if necessary to solve for the unknown angle \( x \).

Let me now calculate the angle \( x \).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you have any further questions or want a detailed breakdown of the process, feel free to ask.

Here are five related questions for deeper exploration:
1. What is the significance of a perpendicular bisector in a triangle?
2. How can we apply trigonometric functions to solve for angles in triangles?
3. How does the Pythagorean theorem help in solving for unknown triangle sides?
4. What properties of a right triangle are helpful in solving geometric problems?
5. Can the perpendicular bisector be used to find other unknowns in this configuration?

**Tip:** In a right triangle, the trigonometric function tangent (\( \tan \)) is often useful when you have the lengths of the opposite and adjacent sides.

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

Learn how to find the value of angle x in a triangle when given a perpendicular bisector. This problem involves the Perpendicular Bisector Theorem, Pythagorean Theorem, and trigonometric relationships. Ideal for students in grades 9-12.

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