Name: Tangents From an Exterior Point: Illustration
Uploaded: 2019-07-19
Duration: 4 min 47 s
Description: If you lift a circular object with a pair of tongs, then the tongs are two tangents to the circular object. Let us study the properties of tangents drawn fro...

Question

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.

Accepted Answer

Given: $O$ is the centre of circle, tangents $P A$ and $P B$ are drawn from the external point $P .$
To prove: $∠ A P B + ∠ A O B = 180^{o}$
Proof: $P A$ is a tangent
$\Rightarrow ∠ O A P = 90^{o}$ [radius is perpendicular to the tangent at the point of contact].
Also, $P B$ is a tangent,
$\Rightarrow ∠ O B P = 90^{o}$ [radius is perpendicular to the tangent at the point of contact]

In quadrilateral $O A P B,$ we have, $∠ O A P + ∠ O B P + ∠ A O B + ∠ A P B = 360^{o}$ [Sum of interior angles of a quadrilateral is $360 °$ ]
$\Rightarrow 90^{o} + 90^{o} + ∠ A O B + ∠ A P B = 360^{o}$
$\Rightarrow ∠ A O B + ∠ A P B = 360^{o} - 180^{o}$
$\Rightarrow ∠ A O B + ∠ A P B = 180^{o}$
Hence, the angle between two tangents drawn from an external point is supplementary to the angle subtended by the line joining the point of contact at the centre.

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.

Important Questions on Circles

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