A circle has points P, Q, R on a circle ill such a way that angle PQR is 60° and angle QRP is 80°. Calculate the angle subtended by an arc QR at the centre.

Options

A. `40^(@)`
B. `80^(@)`
C. `100^(@)`
D. `120^(@)`

Answer

To solve the problem, we need to find the angle subtended by the arc QR at the center of the circle. We are given the angles PQR and QRP in triangle PQR.

### Step-by-Step Solution:

1. **Identify the Given Angles**:
- Angle PQR = 60°
- Angle QRP = 80°

2. **Use the Triangle Sum Property**:
- The sum of the angles in a triangle is always 180°.
- Therefore, we can write the equation:
\[ \text{Angle P} + \text{Angle Q} + \text{Angle R} = 180° \]
- Substituting the known angles:
\[ \text{Angle P} + 60° + 80° = 180° \]

3. **Calculate Angle P**:
- Combine the known angles:
\[ \text{Angle P} + 140° = 180° \]
- Now, isolate Angle P:
\[ \text{Angle P} = 180° - 140° = 40° \]

4. **Relate the Angle at the Center to the Angle at the Circumference**:
- The angle subtended by an arc at the center of the circle is twice the angle subtended at any point on the circumference.
- Therefore, the angle subtended by arc QR at the center (let's denote it as Angle O) is:
\[ \text{Angle O} = 2 \times \text{Angle P} \]
- Substituting the value of Angle P:
\[ \text{Angle O} = 2 \times 40° = 80° \]

5. **Conclusion**:
- The angle subtended by arc QR at the center is 80°.

### Final Answer:
The angle subtended by arc QR at the center is **80°**.