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Tangent to a Circle

 

The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.

 

 

 


 

 

 

Statement 1
The shortest segment that can be drawn from a given point to a given line is the perpendicular from the given point to the given line


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


Given
P is a point of contact of a tangent AB to the circle C (O, r).

 

To Prove
OP
AB

Construction  
Let Q be a point on AB, other than P. Join O to Q.

Proof    
Q is a point on the tangent AB other than the point of contact P cutting the circle at P'.
Now, OQ = OP' + P'Q
             = OP + P'Q              [OP = OP' = r]
OQ > OP, i.e. OP < OQ

From statement 1, OP is the shortest of all the distances thus we conclude

OP AB.





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