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PA and PB are two tangents to a circle ...

PA and PB are two tangents to a circle with centre O, from a point P outside the circle . A and B are point P outside the circle . A and B are point on the circle . If `angle PAB = 47^(@)` , then `angle OAB ` is equal to :

A

`43^(@)`

B

`53^(@)`

C

`50^(@)`

D

`20^(@)`

Text Solution

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The correct Answer is:
To solve the problem, we will follow these steps: ### Step 1: Understand the Geometry We have a circle with center O and two tangents PA and PB drawn from an external point P to points A and B on the circle. The angle ∠PAB is given as 47°. ### Step 2: Identify Key Angles Since PA and PB are tangents to the circle, we know that the radius OA and OB are perpendicular to the tangents at points A and B respectively. Therefore, we have: - ∠OAP = 90° (because OA is perpendicular to PA) - ∠OBP = 90° (because OB is perpendicular to PB) ### Step 3: Use the Angle Relationships We need to find the angle ∠OAB. We can use the following relationship: - ∠OAB = ∠OAP - ∠BAP ### Step 4: Substitute Known Values From our previous steps: - ∠OAP = 90° - ∠BAP = ∠PAB = 47° Now, substituting these values into the equation: - ∠OAB = 90° - 47° ### Step 5: Calculate the Angle Now we perform the subtraction: - ∠OAB = 90° - 47° = 43° ### Conclusion Thus, the angle ∠OAB is equal to 43°. ### Final Answer The angle OAB is equal to 43°. ---
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