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Let's analyze the given triangle \( \triangle ABC \) with the following details:

1. \( BC = 60 \)
2. \( AD = 20 \)
3. Points \( E \) and \( G \) lie on a one-quarter circle centered at \( F \).

Since \( E \) and \( G \) are on the quarter-circle, \( F \) serves as the center of this arc, and lines \( DF \) and \( EF \) are radii of this circle segment. Additionally, the one-quarter circle implies a right angle at \( D \) and \( E \), so \( \angle DEF = 90^\circ \).

### Steps to Solve for \( EF \)

1. **Recognize Right Triangle Properties**:
   Since \( \angle DEF = 90^\circ \) and \( AD = 20 \), the radius of the quarter-circle is \( DF = EF \). So, we need to find the radius \( EF \) in terms of the geometry of \( \triangle ABC \) and the quarter-circle.

2. **Use Right Triangle Proportions**:
   Given the setup, \( \triangle DEF \) forms a right triangle where \( DF = EF \), and both are half the length from \( D \) to \( A \) (since \( AD = 20 \) and forms a perpendicular distance from \( B \) and \( C \)).

Therefore:
\[
EF = \frac{AD}{2} = \frac{20}{2} = 10
\]

### Conclusion
The length of \( EF \) is **10 units**.

Would you like further details on any of the steps? 

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Here are some follow-up questions related to the concepts used in this problem:

1. How do right triangle properties apply in a one-quarter circle configuration?
2. What additional properties can be derived from the lengths in \( \triangle ABC \)?
3. How would the solution change if \( \angle BAC \) was not a right angle?
4. How does the concept of radius affect the solution when dealing with quarter circles?
5. What other methods could be used to verify the length of \( EF \) in this configuration?

**Tip:** When dealing with quarter circles in geometry, focus on symmetry and radius properties as they often simplify calculations.

In triangle ABC, BC = 60, AD = 20. E and G are two points on the one-quarter circle whose center is F. Find the length of EF.

This problem involves finding the length of EF in a triangle with given measurements for BC and AD, incorporating properties of right triangles and a quarter circle. Learn step-by-step how to solve this geometry problem, ideal for students in Grades 9-12.

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