From an external point P, two tangents PA and PB are drawn to a circle with centre O. If angle APB = 70 degrees, find angle AOB.
Get the complete, step-by-step math solution for: "From an external point P, two tangents PA and PB are drawn to a circle with centre O. If angle APB = 70 degrees, find angle AOB.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify properties of tangents and radii
A fundamental property of circles states that the radius drawn to the point of tangency is perpendicular to the tangent line. Therefore, the angle between radius OA and tangent PA is 90∘, and the angle between radius OB and tangent PB is also 90∘.
Step 2: Find ∠ AOB in quadrilateral OAPB
The sum of the interior angles in any quadrilateral is 360∘. In quadrilateral OAPB, we know three angles: ∠OAP=90∘, ∠OBP=90∘, and the given ∠APB=70∘. By subtracting the sum of these three angles from 360∘, we can determine the measure of ∠AOB.