Question 10 Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
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Solution
Consider a circle with centre O. Let P be an external point from which two tangents PA and PB are drawn to the circle which are touching the circle at point A and B respectively and AB is the line segment, joining point of contacts A and B together such that it subtends ∠AOB at center O of the circle. It can be observed that, OA ⊥ PA ∴∠OAP=90∘ Similarly, OB ⊥ PB ∴∠OBP=90∘ In quadrilateral OAPB, Sum of all interior angles = 360∘ ∠OAP+∠APB+∠PBO+∠BOA=360∘ ⇒90∘+∠APB+90∘+∠BOA=360∘ ⇒∠APB+∠BOA=180∘ ∴ The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.