In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.
First, we will prove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. To do so, let's start by drawing a circle with center Q, a tangent line m with point of tangency P, and a radius from P.
Let's assume that the line m is not perpendicular to QP. Therefore, the perpendicular segment from point Q to line m must intersect line m at some other point R.
The perpendicular segment represents the shortest distance from point Q to line m. This means that QR must be less than QP.
QR < QP
If this is the case, and since P is on the circle, then point R must be inside the circle. Therefore, line m is a secant line. However, we have said before that line m is tangent to ⊙ Q. We have a contradiction! The contradiction came from supposing that m was not perpendicular to QP. Therefore, m must be perpendicular to QP.
b We will prove that, if line m is perpendicular to QP, then m is tangent to ⊙ Q. To do so, we will start by assuming that m is perpendicular to the radius QP at P.
We will suppose that line m is not tangent to ⊙ Q. In this case, line m intersects ⊙ Q at a second point R.
Since m is perpendicular to QP, then QP must be less than QR.
QP < QR
However, both QP and QR are radii of ⊙ Q, so they must have the same length.
QP = QR
We obtained two statements that contradict each other. The contradiction came from supposing that line m was not tangent to ⊙ Q. Therefore, m is a tangent line to the circle.
