Name: Circle Passing Through One, Two & Three Points
Uploaded: 2019-07-19
Duration: 8 min 20 s
Description: Do you know the number of circles that pass through one, two or three given points? Well, watch this amazing video to get answers to questions like this!

Question

Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

Accepted Answer

Given: $X Y$ and $X' Y'$ are two parallel tangents to the circle with centre $O$ and $A B$ is the tangent at the point $C$ ,which intersects $X Y$ at $A$ and $X' Y'$ at $B$ .

To Prove: $∠ A O B = 90 °$

Construction: Join $O C$ .

Proof: In $△ O P A$ and $△ O C A$ ,

$O P = O C$ (Radii)

$A P = A C$ (Tangents from point $A$ )

$A O = A O$ (Common side)

$△ O P A ≅ △ O C A$ (By $SSS$ criterion)

Therefore, $∠ P O A = ∠ C O A$ __________ $(1)$ $[C . P . C . T]$

In $△ O Q B$ and $△ O C B$

$O Q = O C$ (radii)

$B Q = B C$ (Tangents from point $A$ )

$B O = B O$ (Common side)

$△ O Q B ≅ △ O C B$

$∠ Q O B = ∠ C O B$ _________ $(2)$

$P O Q$ is a diameter of the circle.

Hence, it is a straight line.

$∴ ∠ P O A + ∠ C O A + ∠ C O B + ∠ Q O B = 180 °$

$\Rightarrow ∠ C O A + ∠ C O B + ∠ C O A + ∠ C O B = 180 °$ $[$ From $(1)$ and $(2)]$

$\Rightarrow 2 ∠ C O A + 2 ∠ C O B = 180 °$

$\Rightarrow ∠ C O A + ∠ C O B = 90 °$

$\Rightarrow ∠ A O B = 90 °$

Hence, proved.

Radius	Chord	Diameter	Arc	Sector	Segment

Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

Important Questions on Circle and Tangents

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