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To solve for the length of \( EF \) in the given figure, we can approach it geometrically using the information provided and trigonometric methods.

### Given Data:
- Angle \( \angle AOC = 0.75 \) radians
- Angle \( \angle EOF = 0.9856 \) radians
- The diagram represents a semi-circle centered at \( O \) with radius 10 (since points \( A \) and \( B \) appear to be symmetric about the origin, and the coordinates suggest the radius).

### Steps to solve for the length of \( EF \):
1. **Identify the radius of the circle:**
   Since the x-coordinates of points \( A \) and \( B \) are \( -10 \) and \( 10 \), respectively, this suggests the radius of the circle is \( r = 10 \) units.

2. **Use trigonometry for chord length:**
   The length of a chord in a circle can be determined by the formula:
   \[
   \text{Chord Length} = 2r \sin\left(\frac{\theta}{2}\right)
   \]
   where \( r \) is the radius and \( \theta \) is the central angle subtended by the chord.

3. **Find the length of \( EF \):**
   The central angle subtended by chord \( EF \) is given as \( \angle EOF = 0.9856 \) radians. Using the formula for the chord length:
   \[
   EF = 2 \times 10 \times \sin\left(\frac{0.9856}{2}\right)
   \]

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

Would you like further details or have any additional questions?

Here are some related questions you might find useful:
1. How can we find the coordinates of points \( E \) and \( F \)?
2. What is the relationship between the radius and the central angle for determining chord lengths?
3. How do we apply trigonometry to calculate chord lengths in general?
4. Can this method be applied to any circle, regardless of its radius?
5. What would the length of \( EF \) be if \( \angle EOF \) was different?

**Tip:** For quick chord length calculations, remember that as the central angle increases, the chord length also increases up to a maximum when the angle is \( \pi \) radians.

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

Find the length of the chord EF using the angles AOC and EOF in a circle with radius 10 units. This problem involves geometry and trigonometry concepts, using the formula for chord length based on the central angle. Suitable for Grades 9-11.

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