The chord of the circle AB = 23.1 mm and the radius r = 34.6 mm. Calculate the central angle ∠AOB corresponding to the chord AB, where O = the center of the circle.

To calculate the central angle ∠AOB corresponding to the chord AB in a circle, we can use the following steps:

### Step 1: Use the relationship between the chord length, radius, and the central angle.

The formula that relates these is:

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

Where:
- \(\theta\) is the central angle ∠AOB.
- \(l\) is the length of the chord AB.
- \(r\) is the radius of the circle.

### Step 2: Substitute the given values into the formula.

Given:
- \(l = 23.1 \, \text{mm}\)
- \(r = 34.6 \, \text{mm}\)

Substituting these values into the formula:

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

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

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

### Step 3: Solve for \(\theta\).

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

\[
\frac{\theta}{2} = \cos^{-1}(0.3337)
\]

\[
\frac{\theta}{2} \approx 70.53^\circ
\]

Finally, multiply by 2 to find the full central angle \(\theta\):

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

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

Would you like more details or have any questions?

### Related Questions:
1. How would the central angle change if the chord length was increased?
2. What would be the length of the chord if the central angle was 90°?
3. How can you calculate the area of the sector formed by the central angle?
4. What is the distance from the center to the midpoint of the chord?
5. How do you find the length of a chord given only the radius and the central angle?

### Tip:
Knowing the relationship between the chord, radius, and central angle is fundamental in solving various geometric problems related to circles.

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

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