Math, asked by ilovevirat2336, 1 year ago

How can you prove the converse of the below theorem?
“If a line in the plane of a circle is perpendicular to the radius at its endpoint on the circle, then the line is tangent to the circle”.

Answers

Answered by mysticd
27
Given ;

A radius OP of a Circle C( O , r ) and

a line APB , perpendicular to OP.

RTP : AB is a tangent to the circle at

the point P

proof :

Take a point Q , different from P , on

the line AB .

OP perpendicular to AB

Among all the line segments joining O

to a point on AB , OP is the shortest .

=> OP < OQ

=> OQ => OP

=> Q lies outside the circle .

Therefore ,

every point on AB , other than P lies

outside the circle .

This shows that AB meets the circle

only at the point P.

Hence , AB is a tangent to the circle

at P .

I hope this helps you.

: )
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