Tangent Theorem 2

In the previous post on tangent theorem, we have learned that if a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. In this post, we prove the converse of this statement. We prove that if a line is tangent to a circle, then it is perpendicular to the radius at the point of tangency. Note that the point of tangency is the point where the circle and the line intersect.

Given

Line m is tangent to circle O at point A

Prove

Line m is perpendicular to OA

Proof

We will use proof by contradiction to prove the statement above. 

tangent theorem

Let us assume the opposite of the conclusion. That is, assume that line m is not perpendicular to OA. Then, there exists a segment OB which is perpendicular to OA. If so, OB < OA (Why?).  But OB lies exterior to the circle since m is a tangent line. This means that OB > OA.

The statement OB < OA and OB > OA are contradictory. Therefore, the supposition is false. So, m is perpendicular to OA.

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