Name: Tangent as Perpendicular to the Radius
Uploaded: 2019-07-19
Duration: 4 min 9 s
Description: When a wheel rolls on the ground, the ground acts as a tangent to it. If the centre of the wheel is joined to the point where it touches the ground, what ang...

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 lines segments joining the points of contact at the centre.

Accepted Answer

Given: $P A$ and $P B$ are the tangents drawn from a point $P$ to a circle with centre $O .$ Also, the line segments $O A$ and $O B$ are shown.

To prove: $∠ A P B + ∠ A O B = 180 °$ .

Proof:

We know that the tangent is perpendicular to the radius through the point of contact.

$∴ P A ⊥ O A \Rightarrow ∠ O A P = 90 °$ .

$∴ P B ⊥ O B \Rightarrow ∠ O B P = 90 °$ .

$∴ ∠ O A P + ∠ O B P = 90 ° + 90 ° = 180 ° . . . (i)$ .

But, we know that the sum of all the angles of a quadrilateral is $360 °$ .

$∴ ∠ O A P + ∠ O B P + ∠ A P B + ∠ A O B = 360 ° . . . (i i)$

From (i) and (ii), we get

$∠ A P B + ∠ A O B = 180 °$ .

Hence proved.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the lines segments joining the points of contact at the centre.

Important Questions on Circles

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