If PA and PB are tangents to a circle with centre O such that angle APB = 50 degrees, what is the measure of angle OAB?

Introduction / Context:
This geometry question focuses on properties of tangents to a circle and the relationship between angles at an external point, the centre and points of tangency. Understanding these angle relations helps in many circle related questions in competitive exams.

Given Data / Assumptions:
• A circle has centre O. • From an external point P, two tangents PA and PB are drawn to the circle. • Angle APB, the angle between the tangents, is 50 degrees. • We are asked to find angle OAB, an angle at point A inside triangle OAB.

Concept / Approach:
Two main concepts are used here. First, the radii OA and OB are perpendicular to the tangents PA and PB at the points of contact, so OA ⟂ PA and OB ⟂ PB. Second, in quadrilateral AOPB, the sum of interior angles is 360 degrees. This gives a direct link between angle APB and angle AOB, the central angle between the radii to the tangency points. Finally, triangle AOB is isosceles with OA = OB, so its base angles at A and B are equal and can be found from its angle sum.

Step-by-Step Solution:
1. Consider quadrilateral AOPB formed by joining O to A and B. 2. At point A, OA is a radius and PA is a tangent, so angle OAP = 90 degrees. 3. Similarly, at point B, OB is a radius and PB is a tangent, so angle OBP = 90 degrees. 4. Let angle AOB at the centre be x degrees. 5. Sum of interior angles of quadrilateral AOPB is 360 degrees, so 90 + 90 + 50 + x = 360. 6. Simplify: 230 + x = 360, so x = 130 degrees. 7. Triangle AOB is isosceles with OA = OB, so its base angles at A and B are equal. 8. Sum of angles in triangle AOB is 180 degrees, so angle OAB + angle OBA + angle AOB = 180. 9. Let angle OAB = angle OBA = y degrees. Then y + y + 130 = 180, so 2y = 50, giving y = 25 degrees. 10. Therefore, angle OAB = 25 degrees.

Verification / Alternative check:
You can also think of this as a known relation: the angle between two tangents from an external point is supplementary to the central angle subtended by the points of tangency. That is, angle APB + angle AOB = 180, giving angle AOB = 130 degrees. Then the isosceles triangle reasoning leads again to angle OAB = (180 - 130) / 2 = 25 degrees. Both methods are consistent.

Why Other Options Are Wrong:
• 30°: This would make the angles at A and B sum to 60 degrees, leaving only 120 degrees for angle AOB, contradicting the 130 degree result. • 40°: Then triangle AOB would have angles 40, 40 and 100 degrees, and angle APB would not be 50 degrees. • 50°: This would leave no angle for the centre consistent with the tangent quadrilateral sum. • 35°: Again, this does not satisfy the relationship between central and base angles in triangle AOB.

Common Pitfalls:
A frequent error is to assume angle OAB equals half of angle APB directly, which is not a general rule. Others forget that radii are perpendicular to tangents and miss the 90 degree angles at A and B. Drawing a clear diagram and marking all right angles usually prevents such mistakes.

Final Answer:
The measure of angle OAB is 25°.