PA and PB are tangents from an external point P to the circle with centre O. If ∠OPB = 30°
then find ∠AOB.
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∠BPO = 30° …given From P we have two tangents PA and PB We know that if we join point P and centre of circle O then the line PO divides the angle between tangents
⇒ ∠APO = ∠OPB = 30° …(i)
∠OAP = ∠OBP = 90° …radius is perpendicular to tangent …
(ii) Consider quadrilateral OAPB
⇒ ∠OAP + ∠APB + ∠PBO + ∠AOB = 360°…sum of angles of quadrilateral
From figure ∠APB = ∠APO + ∠OPB ⇒ ∠OAP + ∠APO + ∠OPB + ∠PBO + ∠AOB = 360° Using (i) and (ii)
⇒ 90° + 30° + 30° + 90° + ∠AOB = 360°
⇒ 240° + ∠AOB = 360°
⇒ ∠AOB = 120°
Hence ∠AOB is 120°
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