Question

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Answer 1

Here is the answer to your question!

(Refer to the attached image )

Let AB be the tangent drawn at a point C on the circle with centre 'O'..

To prove :

perpendicular at point C passes through the centre O.

Construction:

If possible, let the perpendicular passing through some other point, say " O' "

Join OC and O'C

Proof:

Since ,tangents at any point of a circle is perpendicular to the radius through the point of contact,

OC perpendicular AB => Angle OCB = 90°

Also,

Angle O'CB = 90° ( as it is supposed that CO' perpendicular AB)

:- angle OCB = Angle O'CB

Which is possible only when points O and O' Coincide.

So our supposition is wrong.

Hence , the perpendicular at the point of contact to the tangent to a circle always passes through the centre.