In the figure, PA and PB are two tangents to the circle with centre O such that ∠APB=50∘. Then, the measure of ∠OAB is :
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Step-by-Step Solution
Step 1: Identify properties of tangents and radii
We know that the radius drawn to the tangent at the point of contact is perpendicular to the tangent. Therefore, the angle between the radius OA and tangent PA is 90∘, and similarly, the angle between radius OB and tangent PB is 90∘.
Step 2: Find ∠ AOB in quadrilateral OAPB
The sum of angles in a quadrilateral is 360∘. In quadrilateral OAPB, we have ∠OAP=90∘, ∠OBP=90∘, and ∠APB=50∘. We can find ∠AOB by subtracting the sum of these three angles from 360∘.
Step 3: Identify properties of triangle OAB
In triangle OAB, OA and OB are both radii of the same circle. Therefore, triangle OAB is an isosceles triangle, which means the angles opposite to the equal sides are also equal.
Step 4: Calculate ∠ OAB
Since △OAB is an isosceles triangle with OA=OB, the base angles ∠OAB and ∠OBA are equal. The sum of angles in a triangle is 180∘. So, we can find ∠OAB by subtracting ∠AOB from 180∘ and dividing the result by 2.