Question

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

Answer 1

Given: A circle with centre O and a tangent AB at a point P of the circle.
To Prove: OP is perpendicular to AB
Construction: Take a point Q other then P, on AB. Join OQ
Proof: Q is a point on the tangent AB, other than the point of contact P.
Therefore, Q lies outside the circle.
Let OQ intersect the circle at R.
Then,OR is smaller than OQ(a part is less than the whole).......(1.)
But, OP=OR. (radii of the same circle)..........(2.)
Therefore, OP is greater than OQ {from (1.)&(2.)}
Thus, OP is shorter than any other line segment joining O to any point of AB, other than P.
In the other words, OP is the shortest distance between the point O and the line AB.
But, the shortest distance between a point and a line is the perpendicular distance.
Therefore OP is perpendicular to AB.
Hence proved.