Tangent test (perpendicular to a radius at the endpoint ⇒ tangent)

Depends on: Perpendicular segment is shortest (Perpendicular segment is shortest).

Statement

Let $\odot O$ be the circle with centre $O$ and radius $R$, and let $T$ be a point on $\odot O$. If a line $\ell$ passes through $T$ and is perpendicular to the radius $OT$, i.e.

then $\ell$ is a tangent to $\odot O$ (with point of tangency $T$).

In other words: at a point on the circle, drawing a line perpendicular to the radius produces a line tangent to the circle at that point — this is the converse of tangent ⊥ radius (Tangent is perpendicular to the radius at the point of tangency).