Two tangents are drawn from a point P to a circle at Q and R. If O is the centre of the circle and `angle QOP = 40^@`, then what is the value (in degrees) of `angleQPR`?

Options

A. 60
B. 80
C. 90
D. 100

Answer

To solve the problem, we need to find the value of angle QPR given that two tangents are drawn from point P to a circle at points Q and R, and the angle QOP is 40 degrees.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let O be the center of the circle.
- Points Q and R are the points where the tangents touch the circle.
- The angles formed by the radius and the tangent at the point of contact are right angles. Therefore, angle OQP and angle ORP are both 90 degrees.

2. **Identifying Angles**:
- We know that angle QOP = 40 degrees.
- Since the tangents from a point outside the circle are equal, angle QOP = angle ROP = 40 degrees.

3. **Finding Angle AOP**:
- The angle AOP (where A is the point P) can be calculated as follows:
\[ \text{Angle AOB} = \text{Angle QOP} + \text{Angle ROP} = 40^\circ + 40^\circ = 80^\circ \]

4. **Using the Exterior Angle Theorem**:
- The angle QPR is an exterior angle for triangle OQP, which can be expressed as:
\[ \text{Angle QPR} = \text{Angle OQP} + \text{Angle OQP} \]
- Since angle OQP = 90 degrees, we can substitute:
\[ \text{Angle QPR} = 90^\circ - \text{Angle QOP} \]
- Therefore, we have:
\[ \text{Angle QPR} = 90^\circ - 40^\circ = 50^\circ \]

5. **Final Calculation**:
- Thus, the value of angle QPR is:
\[ \text{Angle QPR} = 100^\circ \]

### Conclusion:
The value of angle QPR is **100 degrees**.