To determine the length of \( BC \) in the given triangle \( ABC \), we need to apply some geometric principles and use the information provided.

### Given Information Recap:
- \( \angle A = 46^\circ \)
- \( \angle BED = 2x \)
- The length of \( ED = \frac{7x}{2} + 2 \)
- Points \( E \) and \( D \) bisect their respective sides of the triangle \( ABC \).

### Analysis:
1. **Use the Angle Bisector Theorem**:
   Since \( E \) and \( D \) bisect the sides \( AC \) and \( AB \), respectively, the Angle Bisector Theorem may be applied. The theorem states that the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides.

   Let's denote:
   - \( AB = c \)
   - \( AC = b \)
   - \( BC = a \)

   If \( D \) and \( E \) bisect \( AC \) and \( AB \), we have:
   \[
   \frac{BD}{DC} = \frac{AB}{AC} = \frac{c}{b}
   \]

2. **Use the Law of Sines**:
   Using the Law of Sines, we know that in \( \triangle ABC \):
   \[
   \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
   \]

   Substituting the known values:
   \[
   \frac{BC}{\sin 46^\circ} = \frac{b}{\sin B} = \frac{c}{\sin C}
   \]

3. **Further Relations**:
   Since \( \angle BED = 2x \), this angle affects the interior angles \( \angle ABC \) and \( \angle ACB \). However, without knowing more specifics about \( x \), \( b \), and \( c \), we need another piece of information to definitively calculate the length of \( BC \).

### Missing Information:
To solve for \( BC \) directly, we either need the value of \( x \) or additional side lengths or angles. Specifically:
- The length of one side or
- The value of another angle in \( \triangle ABC \) (such as \( \angle B \) or \( \angle C \)).

### Conclusion:
With the information provided, the exact length of \( BC \) cannot be calculated directly without additional values. If you can provide \( x \) or another side length, I can proceed to solve for \( BC \).

---

Would you like to focus on finding the value of \( x \) or provide any additional data? Here are some questions to explore:

1. How does the value of \( x \) influence the calculation of \( BC \)?
2. What would be the next steps if the length of \( AB \) were known?
3. How can trigonometric identities be used to solve for side lengths in \( \triangle ABC \)?
4. Can we estimate \( BC \) using approximate values for \( x \)?
5. What geometric properties of the bisected triangle could help solve for \( BC \)?

**Tip:** Always consider applying the Law of Sines or Cosines when solving for unknown sides or angles in triangles, especially when specific angle measures are given.

Learn how to calculate the length of BC in triangle ABC using geometric principles such as the Angle Bisector Theorem and the Law of Sines. This problem involves solving for BC with given angles and side lengths, and explores trigonometric identities. Suitable for high school students studying geometry and trigonometry.

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