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In this problem, we are asked to explain why segment \( EA \) is the same length as segment \( BC \).

### Key Points:
- \( A \) and \( B \) are the centers of two circles.
- The circles are intersecting.
- We are given two segments, \( EA \) and \( BC \), and need to prove their equality in length.

### Geometrical Insight:
1. **Circle Properties**: 
   - Since \( A \) is the center of the left circle, the segment \( EA \) is a radius of that circle.
   - Since \( B \) is the center of the right circle, the segment \( BC \) is a radius of the right circle.

2. **Symmetry of Construction**:
   - The two circles are congruent (since the radii are equal). Therefore, any radius of one circle must be congruent to the radius of the other circle.
   - Since both \( EA \) and \( BC \) are radii of their respective circles and the circles are congruent, \( EA = BC \).

### Conclusion:
- Since both \( EA \) and \( BC \) are radii of congruent circles, they must be of equal length.

Would you like further clarification on any of these points?

### Related Questions:
1. Why is symmetry important in circle-based geometric constructions?
2. How can you generalize the properties of congruent circles in a construction like this?
3. What is the relationship between the radius and other chords in a circle?
4. How can you prove that two circles are congruent in a geometric construction?
5. What are other ways to prove equality of segments geometrically, aside from symmetry?

**Tip**: Always look for properties of congruence and symmetry when dealing with problems that involve circles intersecting or sharing common features.

In the construction, A is the center of one circle, and B is the center of the other. Explain why segment EA is the same length as segment BC.

In this problem, A and B are the centers of two intersecting circles. Learn how symmetry and circle properties are used to prove that segment EA is equal in length to segment BC, utilizing the concept of congruent circles and radii. Suitable for Grades 9-10.

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