Circle - Tangent at every point is perpendicular to the Radius

Another property we find related to circles is the tangent at any point on a circle is always perpendicular to the radius at that point. Remember that tangent to a circle is a line that intersects the circle at one and only one point. 

You can take any point on a circle and draw the tangent at that point, the corresponding radius at that point will always be at right angle with the tangent.

‘Proof by contradiction’ is usually the method used to prove this statement. In this method, we first assume that our statement is not true, i.e, we assume that in a circle the radius is never perpendicular to the tangent. 

We then use some mathematical results which will eventually lead us to contradiction. Contradiction means something like “night and day occurring at the same time”, or “x is both bigger and smaller than y(x > y & x < y)”. 

Arriving at some sort of contradiction would imply that our initial assumption that radius is not perpendicular to the tangent was false, and that they really must be perpendicular. 

Given a circle with center ‘O’, let ‘t’ be a tangent at an arbitrary point ‘P’ on the circle. The radius OP then should be perpendicular to the tangent ‘t’. But instead, assume that it is not perpendicular to the tangent. 

Let OR be the segment from the center of the circle and perpendicular to the tangent line ‘t’.

Here we are basically using the geometrical result that there always exists a (unique) line passing through a point and perpendicular to some other line. So if OP is not the perpendicular to line ‘t’, there must be some other segment(OR) passing through point O and perpendicular to line ‘t’.

We see that we get a right angled triangle ORP. And in a right angled triangle the hypotenuse(side opposite to 90 degree angle) is the longest side. So in our figure, the longest side is the radius OP in the right triangle ORP. So it should be bigger than OR(and PR).

OP > OR

But we also see that point R lies outside of the circle. So it must be farther away from the center of the circle than the point P which is on the circle. 

OR > OP

And so we have a contradiction in which we have OP less than and greater than OR at the same time. 

(Note that we will obtain the same two contradicting conclusions everywhere else on the circle as well where we draw the tangent. From the definition, tangent to a circle at a point on the boundary intersects the circle at that point only. Every other point on the tangent will be outside of the boundary of the circle. And so this assumed point R can never be inside or on the perimeter of the circle. Hence, we will always get OP > OR and OP < OR at the same time.)

Hence our initial assumption that the radius OP is not perpendicular to the tangent must be false. The radius must be perpendicular to the tangent.