PA and PB are two tangents to a circle with centre O, from a point P outside the circle. A and B are points on the circle. If `angleAPB =100^(@)` , then OAB is equal to:

Options

A. `25^(@)`
B. `20^(@)`
C. `35^(@)`
D. `50^(@)`

Answer

To solve the problem, we need to find the angle OAB given that PA and PB are tangents to a circle from point P, and the angle APB is 100 degrees.

### Step-by-Step Solution:

1. **Draw the Circle and Tangents**:
- Draw a circle with center O.
- Mark point P outside the circle.
- Draw tangents PA and PB from point P to the circle, touching the circle at points A and B respectively.

2. **Label the Angles**:
- We know that angle APB = 100 degrees.
- Since PA and PB are tangents to the circle, the angles between the radius and the tangent at the point of contact are 90 degrees. Thus, angle OAP = 90 degrees and angle OBP = 90 degrees.

3. **Use the Tangent-Secant Angle Theorem**:
- According to the theorem, the angle formed between the two tangents from an external point (angle APB) is equal to twice the angle formed at the center of the circle (angle OAB).
- Therefore, angle OAP + angle OAB = 90 degrees.

4. **Calculate Angle OAP**:
- Since angle APB = 100 degrees, we can find angle OAP:
\[ \text{Angle APO} = \frac{1}{2} \times \text{Angle APB} = \frac{1}{2} \times 100 = 50 \text{ degrees} \]

5. **Apply the Triangle Sum Theorem**:
- In triangle AOP, we have:
\[ \text{Angle OAP} + \text{Angle AOP} + \text{Angle PAM} = 180 \text{ degrees} \]
- Substituting the known values:
\[ 90 + 50 + \text{Angle PAM} = 180 \]
- This simplifies to:
\[ \text{Angle PAM} = 180 - 90 - 50 = 40 \text{ degrees} \]

6. **Find Angle OAB**:
- Now, we know that:
\[ \text{Angle OAP} = \text{Angle PAM} + \text{Angle OAB} \]
- Substituting the known values:
\[ 90 = 40 + \text{Angle OAB} \]
- Rearranging gives:
\[ \text{Angle OAB} = 90 - 40 = 50 \text{ degrees} \]

### Conclusion:
Thus, the angle OAB is equal to 50 degrees.