Given: A circle with center 'O' and a tangent XY to the circle at a point P and OP radius

To prove: OP is perpendicular to XY

Proof:

Take a point Q on Line XY other than P and join O and Q. The point Q must lie outside the circle(note that if Q lies inside the circle,XY become a secant and not a tangent to the circle)

Therefore, OQ is longer than the radius of the circle.

OQ > OP

This must happen for all points on the line XY. It is therefore true that OP is the shortest of all the distance of the point O to the

As a perpendicular is the shortest in length among line segments drawn from a point to the line.Therefore is perpendicular to XY

OP ⟂ XY