The radius of a circle is 8 cm and O is its centre. If `angle AOB=60^(@)` and AB is a chord, then what is the length of the chord AB?

Answer

To find the length of the chord AB in a circle with radius 8 cm and a central angle AOB of 60 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Radius of the circle (OA and OB) = 8 cm
- Angle AOB = 60 degrees

2. **Understanding the Triangle:**
- Triangle AOB is formed by the two radii OA and OB and the chord AB.
- Since OA and OB are both radii of the circle, they are equal in length.

3. **Using the Properties of Triangles:**
- In triangle AOB, we know that the sum of the angles in a triangle is 180 degrees.
- Therefore, we can set up the equation for the angles:
\[ \angle AOB + \angle OAB + \angle OBA = 180^\circ \]
- Let the angles OAB and OBA be equal (let's call them x). Thus, we have:
\[ 60^\circ + x + x = 180^\circ \]
- Simplifying this gives:
\[ 60^\circ + 2x = 180^\circ \]
\[ 2x = 180^\circ - 60^\circ = 120^\circ \]
\[ x = 60^\circ \]

4. **Conclusion about the Triangle:**
- Since all angles in triangle AOB are 60 degrees, triangle AOB is an equilateral triangle.

5. **Finding the Length of the Chord AB:**
- In an equilateral triangle, all sides are equal. Therefore:
\[ AB = OA = OB \]
- Since OA and OB are both radii of the circle, we have:
\[ AB = 8 \text{ cm} \]

### Final Answer:
The length of the chord AB is **8 cm**.

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