Question

PROOF Theorm 10.1 The Tangent at any point of a circle is perpendicular to the radius through the point of contact

Answer 1

consider a circle with centre o and tangent by at p

now join o and p(given)

to show :opis perpendicular to xy

now mark an another point q somewhere else on xy (other than on p)

now join q and o(construction)

you will observe that op is lesser than oq

we know that out of all lines drawn to join a line and a point perpendicular will be the shortest

this shows po is perpendicular to XP
this radius is perpendicular to tangent

Answer 2

Hi mate your answer is
Let, a circle with Centre O MN is a tangent line to the circle at the point P.
To prove - OP PERPENDICULAR MN
Construction- let Q be any point other than P MN. Join OQ.
Proof- since, Q lies in the exterior of circle OQ>OP, Thus of all the segments that can be drawn from the centre to any point on the line PQ ,OP is shortest. We know that the shortest
segment that can be drawn from a given point to given line is perpendicular from the given point to the given line.
Hence ,OP PERPENDICULAR MN
I hope it will help you
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