Prove that the tangent at any point of the circle is perpendicular to the radius through the point of contact.
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Prove that the tangent at any point of the circle is perpendicular to the radius through the point of contact.
Given: A circle with center O and radius r. Let P be a point on the circle where a tangent TP is drawn.
To Prove:The tangent TP is perpendicular to the radius OP, i.e.,
Proof: Consider a circle with center O and a tangent TP at point P.Since P lies on the circle, OP is the radius of the circle.
Assume another point Q on the tangent TP (other than P).Since Q lies outside the circle, the line segment OQ is longer than OP (as OP is the shortest distance from O to the circle).
In any circle, a line drawn from the center to a point outside the circle is always longer than the radius.That means:for all points Q on the tangent other than P.
This implies that OP is the shortest distance from O to the tangent.By the definition of perpendicularity, the shortest distance from a point to a line is along the perpendicular.
Thus, OP must be perpendicular to TP.
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