Mathematics
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.
Circles
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Answer
Let us consider O as the centre point of the circle. Let P be a point outside the circle from which two tangents PA and PB are drawn to the circle which touches the circle at point A and B respectively.
Draw a line segment between points A and B such that it subtends ∠AOB at centre O of the circle.
We know that,
The tangent at any point of a circle is always perpendicular to the radius through the point of contact.
∴ ∠OAP = ∠OBP = 90° …………(1)
In a quadrilateral, the sum of interior angles is 360°.
In quadrilateral OAPB,
⇒ ∠OAP + ∠APB + ∠PBO + ∠BOA = 360°
Using Equation (1), we can write the above equation as
⇒ 90° + ∠APB + 90° + ∠BOA = 360°
⇒ ∠APB + ∠BOA = 360° - 180°
⇒ ∠APB + ∠BOA = 180°
Where,
∠APB = Angle between the two tangents PA and PB from external point P.
∠BOA = Angle subtended by the line segment AB joining the point of contacts at the centre.
Hence, proved 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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