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

Here, we have to show that $∠ A P B + ∠ B O A = 180^{o}$ .

Let the centre of the circle be O. Tangents from PA and PB drawn to the circle from P meet the circle at A and B.

$A B$ is the line segment joining point of contacts $A$ and $B$ together such that it subtends $∠ A O B$ at center $O$ of the circle.

As the radius is perpendicular to the tangent at the point of contact. Hence, we can say

$∠ O A P = 90^{o}$ and $∠ OBP = 90^{o}$

As we know sum of interior angles in a quadrilateral is $360^{o}$ , hence in quadrilateral $O A P B$

$∠ O A P + ∠ A P B + ∠ P B O + ∠ BOA = 360^{o}$

$\Rightarrow 90^{o} + ∠ APB + 90^{o} + ∠ BOA = 360^{o}$

$\Rightarrow ∠ A P B + ∠ BOA = 180^{o}$

Hence, 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.

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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