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 tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Accepted Answer

Let the circle with centre $O$ and chord $P Q$ . Let tangents at $P$ and $Q$ meet at the point $A$ as shown below:

We have to prove that $∠ A P Q = ∠ A Q P$

Consider $Δ O P Q$

$\Rightarrow O P = O Q$ (radius of circle)

Hence $Δ O P Q$ is an isosceles triangle.

$\Rightarrow ∠ O P Q = ∠ O Q P$ (base angles of isosceles triangle) ...........(i)

As radius $O P$ is perpendicular to tangent $A P$ at point of contact $P$ .

$\Rightarrow ∠ A P O = 90 °$

From figure $∠ A P O = ∠ A P Q + ∠ O P Q$

$\Rightarrow 90 ° = ∠ A P Q + ∠ O P Q$

$\Rightarrow ∠ A P Q = 90 ° - ∠ O P Q$ ..........(ii)

As radius $O Q$ is perpendicular to tangent $A Q$ at point of contact $Q$ .

$\Rightarrow ∠ A Q O = 90 °$

From figure $∠ A Q O = ∠ A Q P + ∠ O Q P$

$\Rightarrow 90 ° = ∠ A Q P + ∠ O Q P$

$\Rightarrow ∠ A Q P = 90 ° - ∠ O Q P$

Using (i), we get

$\Rightarrow ∠ A Q P = 90 ° - ∠ O P Q$ ..........(iii)

Using (ii) and (iii), we can say that

$\Rightarrow ∠ A P Q = ∠ A Q P$

Hence, the tangents at the extremities of any chord of a circle make equal angles with the chord.

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Important Questions on Circles

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