The chord of a circle is AB=23.1 mm and the radius is 34.6 mm. Calculate the central angle formed at AOB where O is the center of the circle

To calculate the central angle ∠AOB corresponding to the chord AB in a circle, where O is the center, we can follow a structured approach.

### Step 1: Use the chord length and radius relationship

The relationship between the chord length \( AB \), the radius \( r \), and the central angle \( \theta \) (in radians) is given by:

\[
\cos\left(\frac{\theta}{2}\right) = \frac{l}{2r}
\]

Where:
- \( l = 23.1 \, \text{mm} \) (the length of the chord AB)
- \( r = 34.6 \, \text{mm} \) (the radius of the circle)
- \( \theta \) is the central angle ∠AOB

### Step 2: Substitute the given values

Substitute the values into the equation:

\[
\cos\left(\frac{\theta}{2}\right) = \frac{23.1}{2 \times 34.6}
\]

\[
\cos\left(\frac{\theta}{2}\right) = \frac{23.1}{69.2}
\]

\[
\cos\left(\frac{\theta}{2}\right) \approx 0.3337
\]

### Step 3: Calculate \(\frac{\theta}{2}\)

Now, take the inverse cosine (arccos) of 0.3337 to find \(\frac{\theta}{2}\):

\[
\frac{\theta}{2} \approx \cos^{-1}(0.3337) \approx 70.53^\circ
\]

### Step 4: Calculate the full central angle \(\theta\)

Finally, multiply by 2 to find the full angle:

\[
\theta = 2 \times 70.53^\circ = 141.06^\circ
\]

### Final Answer:
The central angle ∠AOB corresponding to the chord AB is approximately **141.06°**.

Would you like any further details or have any other questions?

### Related Questions:
1. How does the central angle change if the chord length decreases?
2. What is the relationship between the central angle and the arc length?
3. How do you calculate the area of the segment formed by the chord and the arc?
4. What is the formula for the length of the chord given the radius and the central angle?
5. How can you find the distance from the chord to the center of the circle?

### Tip:
When working with circles, remember that the cosine of half the central angle relates directly to the ratio of the chord length to twice the radius. This relationship is crucial for many geometric calculations involving circles.

Learn how to calculate the central angle ∠AOB in a circle using the given chord length AB = 23.1 mm and radius 34.6 mm. Step-by-step solution with cosine rule application. Suitable for high school students.

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