Question

prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact ​

Answer 1

Answer:

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Answer 2

Statement:-

• the tangent at any point of a circle is perpendicular to the radius at the point of contact

Diagram:-

• In the attachment

Given:-

• C(O,r)
• tangent XY with point of contact as P.

To prove:-

• OP is perpendicular to XY

Proof:-

• as P lies on the circle, OP = r
• Q and S are on the exterior of the circle

hence, OS > OP

OQ > OP

So, OP is shortest among all the line segments that can be drawn with this tangent.

hence, OP is perpendicular to XY (because, of all line segments from a point, the perpendicular is the shortest).