In the given the length of arcs AB and BC are in the ratio `3:2.`
If `angle AOB = 96 ^(@)` find
`angle BOC`

Answer

To solve the problem, we need to find the angle BOC given the lengths of arcs AB and BC are in the ratio 3:2 and angle AOB is 96 degrees.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The lengths of arcs AB and BC are in the ratio 3:2.
- Angle AOB = 96 degrees.

2. **Use the Property of Circles:**
- The angles subtended by arcs at the center of the circle are in the same ratio as the lengths of the arcs. Therefore, if the ratio of the lengths of arcs AB and BC is 3:2, the ratio of the angles AOB and BOC will also be 3:2.

3. **Set Up the Ratio:**
- Let angle BOC be represented as \( x \).
- According to the ratio, we can write:
\[ \frac{\text{Angle AOB}}{\text{Angle BOC}} = \frac{3}{2} \]
- Substituting the known value:
\[ \frac{96}{x} = \frac{3}{2} \]

4. **Cross Multiply to Solve for x:**
- Cross multiplying gives:
\[ 3x = 2 \times 96 \]
- Simplifying the right side:
\[ 3x = 192 \]

5. **Divide to Find x:**
- Now, divide both sides by 3:
\[ x = \frac{192}{3} = 64 \]

6. **Conclusion:**
- Therefore, angle BOC = 64 degrees.

### Final Answer:
Angle BOC = 64 degrees.