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Prove that angle between two tangents …

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Prove that angle between 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

Posted by Faisal Pasha 6 years, 6 months ago

CBSE > Class 10 > Mathematics

1 answers

Gaurav Seth 6 years, 6 months ago

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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Given : PA and PB arc two tangents drawn from an external point P to a circle with centre O.
To prove : ∠AOB + ∠APB = 180°
Const : Join OA and OB.
Proof : ∵ The tangent at any point of circle is perpendicular to the radius through the point of contact.
∴ ∠OAP = 90°    .....(i)
and    ∠OBP = 90°    .....(ii)
Adding (i) and (ii), we get
∠OAP + ∠OBP = 180°
Now in quadrilateral AOBP,
∠OAP + ∠OBP + ∠APB + ∠AOB = 360°
⇒    180° + ∠APB + ∠AOB = 360°
∴ ∠APB + ∠AOB = 180°.

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