Answer :

Let us consider a circle with center O, OP be a radius and XY be a tangent at point P.



To prove: OP XY


Proof: Let’s take a point Q on XY other than P.


Clearly, point Q lies outside the circle because if Q lies inside the circle then XY will be a secant (as it will intersect the circle at two points)


Point P is on circle and point Q is outside the circle


OP < OQ


This is true for all points on XY other than P


OP is the shortest distance between point P and line XY.


OP XY [Shortest side is perpendicular]


Hence, proved!


OR



Let us consider a circle with center O. PQ and PR be two tangents from an external point P to the circle.


To prove: QPR + QOR = 180°


Proof: In quadrilateral PROQ, we have


QPR + PQO + QOR + ORP = 360°


Now, OQP = ORP = 90° [Tangent at a point on a circle is perpendicular to the radius through point of contact]


⇒∠QPR + 90° + QOR + 90° = 360°


⇒∠QPR + QOR = 180°


Hence, proved!


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