Question

Prove that the line segment joining the points of contact of two parallel tangents
of a circle passes through its centre.​

Answer 1

Answer:

Solution :

Note : See the diagram in the attachment.

Given :

• A circle C (O, r) in which
• L and m are two parallel tangents at the point A and B.
• A ans B are joined, AB is a diameter of the circle.

To Prove :

• AB passes through the O.

Construction :

I) Join O, A and O, B

ii) Through O, draw OC // l ( Or m) .

Proof :

$\implies$l // m { Given }

Hence,

AO is a trasversal to them.

$\implies$ ∠PAO + ∠COA = 180°

{ Sum of interior angles }

$\implies$90° + ∠COA = 180°

[ OA ⊥ PQ ] By Tangent - Radius Theorem

∠COA = 180° - 90°

$\implies$∠COA = 90° ...... Eq (1)

Similarly, ∠COB = 90° ..... Eq(2)

Then,

$\implies$∠COA + ∠COB = 90° + 90° = 180°

[Adding (1) and (2) eqn ]

Hence,

AOB is a straight line.

AB is a line - segment that passes through O.

________________________

Answer 2

Answer:

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