Question

prove that perpendicular at the point of contact to the tangent to a circle passes through a circle
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Answer 1

Answer:

Step-by-step explanation:

Due to network issues I can't post the picture

Let ,

O is the centre of the given circle.

A tangent PR has been drawn touching the circle at point P.

Draw QP ⊥ RP at point P, such that point Q lies on the circle.

∠OPR = 90°  (radius ⊥ tangent)

Also, ∠QPR = 90°  (Given)

∴ ∠OPR = ∠QPR

Now, above case is possible only when centre O lies on the line QP.

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

Answer 2

Answer:

Figure in attachment .

OB is the radius of the circle .

A is any other point on the circle .

We know that ∠OBC = 90°

[ Tangent joining the centre is perpendicular to the point of contact ]

We draw any point A such that ∠ABC = 90° .

This means that ∠ OBC = ∠ ABC which is only possible if :

O and A are the same points .

Hence we proved that perpendicular passes through the centre of the circle.

Step-by-step explanation:

Here we have assumed a point A so that ∠ ABC = 90 and hence we proved by using one of the tangent property .

The line drawn from the point of contact of the tangent to the centre makes 90° .