In the given figure, if PA and PB are tangents to the circle with centre O such that `angle APB = 54^(@)`, then `angle` OAB equals

Options

A. `36^(@)`
B. `18^(@)`
C. `27^(@)`
D. `36^(@)`

Answer

To solve for the angle OAB given that PA and PB are tangents to the circle with center O and that angle APB is 54 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- PA and PB are tangents to the circle.
- Angle APB = 54 degrees.

2. **Use the Property of Tangents**:
- The lengths of tangents drawn from an external point to a circle are equal. Therefore, PA = PB.

3. **Set Up the Angles**:
- Let angle OAP = angle OBP = α (since PA = PB, the angles opposite to equal sides are equal).

4. **Apply the Triangle Angle Sum Property**:
- In triangle APB, the sum of angles is 180 degrees.
- Therefore, we can write:
\[ \angle OAP + \angle OBP + \angle APB = 180^\circ \]
- Substituting the known values:
\[ α + α + 54^\circ = 180^\circ \]
- This simplifies to:
\[ 2α + 54^\circ = 180^\circ \]

5. **Solve for α**:
- Rearranging gives:
\[ 2α = 180^\circ - 54^\circ \]
\[ 2α = 126^\circ \]
\[ α = \frac{126^\circ}{2} = 63^\circ \]

6. **Use the Tangent-Radius Property**:
- The tangent at point A and the radius OA form a right angle:
\[ \angle OAP + \angle OAB = 90^\circ \]
- Substituting α for angle OAP:
\[ 63^\circ + \angle OAB = 90^\circ \]

7. **Solve for Angle OAB**:
- Rearranging gives:
\[ \angle OAB = 90^\circ - 63^\circ = 27^\circ \]

### Final Answer:
- Therefore, angle OAB = 27 degrees.

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