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.

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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