2 equal tangents PA and PB are drawn from an external point P on a circle with centre O. What is the length of each tangent, if P Is 12 cm from the centre and the angle between the tangents is `120^@` ?

Options

A. 24 cm.
B. 6 cm.
C. 8 cm.
D. Cannot be determined

Answer

To solve the problem, we need to find the length of the tangents PA and PB drawn from an external point P to a circle with center O. Given that the distance from P to O is 12 cm and the angle between the tangents PA and PB is 120 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a circle with center O.
- Point P is outside the circle, and two tangents PA and PB are drawn from point P to points A and B on the circle.
- The angle ∠APB is given as 120 degrees.

2. **Bisect the Angle**:
- The line segment OP bisects the angle ∠APB.
- Therefore, ∠APO = ∠BPO = 60 degrees (since 120 degrees divided by 2 equals 60 degrees).

3. **Identify Right Triangles**:
- The segments OA and OB are perpendicular to the tangents PA and PB at points A and B, respectively.
- This means we can analyze the right triangle OAP (or OBP).

4. **Apply Trigonometry**:
- In triangle OAP, we know:
- OP (hypotenuse) = 12 cm
- ∠APO = 60 degrees
- We need to find the length of the tangent PA.
- Using the cosine function:
\[ \cos(60^\circ) = \frac{PA}{OP} \]
- We know that \(\cos(60^\circ) = \frac{1}{2}\).

5. **Set Up the Equation**:
- Substitute the known values into the equation:
\[ \frac{1}{2} = \frac{PA}{12} \]

6. **Solve for PA**:
- Rearranging gives:
\[ PA = 12 \times \frac{1}{2} = 6 \text{ cm} \]

7. **Conclusion**:
- Since PA and PB are equal (both are tangents from the same external point), we conclude:
\[ PB = PA = 6 \text{ cm} \]

### Final Answer:
The length of each tangent PA and PB is **6 cm**.

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