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how to prove algebraicaly that tangent creates 90 degree angle

how to prove algebraicaly that tangent creates 90 degree angle

Grade:11

2 Answers

deepak
84 Points
7 years ago
consider the tangent and the circle ,
now consider the distance between the center and the tangent since the distance between a pt and a line will be minimum
iff its measure along a line perpendicular to the tangent
and as the tanget touches the circle only at single pt the distance between that pt of interraction and the centre will obviously be the
shortest distance hence the line along which the distance is measured i.e the radius will be perpendicular to the tangent
 
hope it helps
Ritika Das
105 Points
7 years ago
Suppose a circle has a centre O with T on the boundary of the circle from where the tangent has been formed. Let P be the foot of the radius drawn from centre O of the circle. Suppose T and P to be different points at first.
Now, since OT is greater than OP, and angle OPT is equal to 90 degrees, therefore angle OTP is greater than angle OPT.
So angle OTP is greater than 90 degrees. 
Contradictingly, the angle sum of triangle OPT is seen to be greater than 180 degrees, while violates the angle sum property of a triangle. So we conclude that T and P are the same point, and moreover the tangent creates a 90 degrees angle along with the radius perpendicular on it.

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