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
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.
: )
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.
: )
Attachments:
Similar questions