Given : A circle C(0 ,r) and a tangent L at point A
To Prove : OA is perpendicular to L
CONSTRUCTION : Take a point B , other than A ,on the tangent L .Join OB .
Suppose OB meets the circle in C.
PROOF : We know that , among all line segment joining the point O to a point on L , the perpendicular is shortest to L .
OA = OC ( Radius of same circle)
Now OB = OC + BC
Therefore OB greater than OC
⇒ OB greater than OA
⇒ OA is shorter than OB
B is an arbitrary point on the tangent L. Thus OA is shorter than any other line segment joining O to any point on L . Hence here OA is perpendicular to L.