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Here we will use the method of assumes that statement is wrong. so we will suppose it is not perpendicular.take a point Q on XY other than P & join OQ. the point Q lie outside the circle that if Q lies inside the circle XY becomes secant not a tangent. therefore OQ is longer than OP so it is true that OP is shortest of all points.so our assumption is wrong .

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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.