How to prove.
consider a few points on the tangent.
by the definition of tangent there's only one point which is common to the tangent and the circle.
so all the other points have to be outside the circle.
so distance of all these points from the center has to be more than the radius.
so the point of contact is the only point whose dist from the cener is equal to radius.
so of all the points on the tangent all points other than the pont of contact are at a distance >radius
so as far as the tangent is concerned , the distance of the center from this line is shortest at the point of contact.
since the shortest dist of a point from a line is the perpendicular distance, hence the result.
this was the explanation.
now the geom proof.
consider that the point of contact is P and the center is O.
consider any other representative point on the tangent , say, Q.
now, join O to P and O to Q.
So we have a triangle OPQ.
OP So for all points on the tangent other than P, their distance from O is > OP. So OP is the shortest distance so it has to be perpendicular distance. hence the answer.
So for all points on the tangent other than P, their distance from O is > OP.
So OP is the shortest distance so it has to be perpendicular distance.
hence the answer.
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