PA and PB are the tangent on a circle with centre O such that `angle APB = 42^@`. What will be the value of `angle AOB ?`

Options

A. `84^@`
B. `116^@`
C. `138^@`
D. `124^@`

Answer

To solve the problem, we need to find the value of angle AOB given that PA and PB are tangents to the circle at points A and B respectively, and that angle APB is 42 degrees.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have a circle with center O.
- Points A and B are points of tangency where the tangents PA and PB touch the circle.
- By the property of tangents, the radius at the point of tangency is perpendicular to the tangent line. Therefore, OA ⊥ PA and OB ⊥ PB.

2. **Identifying Angles**:
- Since OA is perpendicular to PA, angle OAP = 90 degrees.
- Similarly, since OB is perpendicular to PB, angle OBP = 90 degrees.
- We know that angle APB = 42 degrees.

3. **Using the Angles in Triangle APB**:
- The angles in triangle APB can be expressed as:
- Angle OAP + Angle APB + Angle OBP = 180 degrees.
- Substituting the known values:
- 90 degrees + 42 degrees + 90 degrees = 180 degrees.

4. **Finding Angle AOB**:
- The angle AOB can be found using the fact that the angles around point P add up to 360 degrees:
- Angle AOB + Angle APB = 180 degrees (since they are supplementary).
- Therefore, we can calculate angle AOB:
- Angle AOB = 180 degrees - Angle APB
- Angle AOB = 180 degrees - 42 degrees
- Angle AOB = 138 degrees.

5. **Final Answer**:
- Thus, the value of angle AOB is 138 degrees.