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The question provided in the image involves solving for the length of EF, given the angles AOC = 0.75 radians and EOF = 0.9856 radians in a geometric diagram involving a circle. Let me guide you through the solution:

### Key Information:
- The diagram shows a circle with center \( O \), and points \( A \), \( B \), \( C \), \( D \), and \( E \) on the circle.
- \( AOC = 0.75 \) radians and \( EOF = 0.9856 \) radians.
- \( OA \), \( OB \), \( OC \), \( OD \), \( OE \), and \( OF \) are radii of the circle.
- The radius of the circle appears to be 10 units, since point \( A \) is located at (-10, 0) and point \( F \) is at (10, 0).

### Step-by-Step Solution:
1. **Understanding the Geometry**: 
   Since we are dealing with a circle and the points \( A \), \( C \), and \( F \) lie on the circumference, angles \( AOC \) and \( EOF \) subtend arcs of the circle. \( EF \) is the chord formed by the angle \( EOF \).

2. **Using the Law of Cosines for Chords**:
   The formula for the length of a chord \( L \) in terms of the angle \( \theta \) subtended at the center and the radius \( r \) is:
   \[
   L = 2r \sin\left(\frac{\theta}{2}\right)
   \]
   where \( r \) is the radius of the circle and \( \theta \) is the angle subtended by the chord at the center.

3. **Substituting Values**:
   - For chord \( EF \):
     - \( r = 10 \)
     - \( \theta = 0.9856 \) radians

   Now, using the formula:
   \[
   L = 2 \times 10 \times \sin\left(\frac{0.9856}{2}\right)
   \]

Let me calculate this.The length of the chord \( EF \) is approximately 9.46 units.

Would you like further details on this solution or have any other questions? Here are some related questions you might consider:

1. How do you calculate the area of the sector subtended by angle \( EOF \)?
2. What would be the length of chord \( AC \) using the given angle \( AOC \)?
3. How is the central angle related to the arc length in a circle?
4. Can we calculate the area of triangle \( EOF \) using the chord length and radius?
5. What is the general formula for the length of a chord given any angle in radians?

**Tip:** The sine function plays a crucial role in circular geometry, especially when calculating chord lengths and sector areas.

The Angle AOC = 0.75, EOF = 0.9856. Identify the length of EF.

Find the length of chord EF in a circle given the central angles AOC and EOF. This solution involves applying the chord length formula and trigonometry principles to solve geometry problems. Suitable for high school students (Grades 10-12).

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