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Prove

A tangent to a circle is perpendicular to the radius through the point of contact.

1
by divyakag

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A tangent to a circle is perpendicular to the radius through the point of contact.

by divyakag

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given: a circle with center o and a tangent

to prove ; op ⊥ XY

construction plot a point q outside the circle but on tangent other than p

proof :xy is a tangent

p lies on the tangent xy

q lie on xy other than p outside the circle

∴ op is shorter than oq

is q lies inside the circle then xy will be secant

p is point on the tangent xy

∴ op is the shortest of all other points except point p

∴ op is shorter than oq

∴ op⊥ xy