Question

prove that the tangent at any point of a circle is perpendicular to the radius

Answer 1

prove that tangent at any point on the circle is perpendicular to the point of contact
given:- a circle with centre O .XPY is a tangent to a circle p
to prove that:-OP perpendicular to XY
proof:-Take a point Q on XY other than P and join OQ. point lie out side the circle if the point lies inside the circle then it will be secant not a tangent
soo u get the proof
OP perpendicular to XY



i hope this will make u to know the therom

Answer 2

given : A circle with centre. O and a tangent. PQ to the circle at a point X.
To prove: OX is perpendicular to PQ
proof : Let assume that OX is not perpendicular to PQ
Take a point Y must lie outside the circle
[if Y lies inside the circle, PQ becomes a secant. and not a tangent to the circle]
therefore,
OY is longer than the radius OX of the circle
OY> OX
this must be happen for all points on the line PQ.
therefore
OX is perpendicular to PQ
so our assumption thatbOX is not perpendicular to PQ is false
therefore OX is perpendicular to PQ.
hence proved.