Tangent to a Circle
A tangent to a circle is a line that touches the circle at only one point.
I. There is only one tangent at a point of the circle.
II. The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.
The tangent at any point of a circle is perpendicular to the radius, through the point of contact.
Proof: We have the centre O of the given circle and XY is the tangent to the circle at a point P.
Let us take a point Q on XY other than P. Join OQ.
Obviously, Q lies outside the circle. i.e., OQ > OP
Since, all the points on XY, except P lies outside the circle.
i.e., OP is smaller than all the distance of the point O from XY.
i.e., OP is the smallest distance of O from XY.
i.e., OP ⊥ XY