Tangents

I used to get really mindfucked by tangents.  How can a line intersect with a point on a circle to form a ninety degree angle with the line only touching one point?  A circle is made of an infinite number of points.  Points have no width or height.  How can a line hit only one exact point on the circle without brushing into an infinite number of points right next to it that make up its curve?

It's a set of points where the supremum is in the set.