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Question

# The angle between two tangents drawn from an external point to a circle is _______ to the angle subtended by the line segments joining the points of contact at the centre.

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Solution

## PA and PB are two tangents drawn from P to circle with centre O. Now, ∠OAP = 90º (Radius is perpendicular to the tangent at the point of contact) Also, ∠OBP = 90º (Radius is perpendicular to the tangent at the point of contact) In quadrilateral OAPB, ∠AOB + ∠OAP + ∠OBP + ∠APB = 360º (Angle sum property of quadrilateral) ⇒ ∠AOB + 90º + 90º + ∠APB = 360º ⇒ ∠AOB + ∠APB = 360º − 180º = 180º Thus, the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre. The angle between two tangents drawn from an external point to a circle is __supplementary__ to the angle subtended by the line segments joining the points of contact at the centre.

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