What Is a Tangent?

What tangency tells you

Tangents are important because they turn circle vocabulary into theorem language. They appear in construction work, in tangent-chord angle problems, and in arguments about equal tangent segments from an external point.

This page keeps the radius and the touch point on the screen together so the defining geometry of tangency stays visible instead of being inferred loosely.

A tangent may touch at the top, side, or any other point of the circle. Its location does not matter; the one-contact condition does. If you tilt the line until it crosses the boundary twice, the object has changed into a secant and the perpendicular-radius fact no longer applies.

At the touch point, the radius and tangent make a right angle. This gives a quick construction check: if the line touches the circle but the radius to contact is not perpendicular, either the drawing or the claimed tangent is wrong.

• A tangent touches a circle at exactly one point.
• A tangent and the radius to the point of tangency meet at a right angle.
• A tangent touches the circle at one point, whereas a secant cuts through at two points.
• Tangency is a local contact condition, not simply a line that passes near the circle's edge.

Where tangents help

• Use tangents in compass-and-straightedge constructions and formal geometry proofs.
• Use them in tangent-chord angle problems and in right-angle relationships at the circle boundary.
• Use them when modelling contact paths such as wheels touching a surface or external support lines.

Tangent mistakes to avoid

• Do not label a line as tangent if it intersects the circle in two places, even if the second cut looks small.
• Do not forget the perpendicular radius fact at the point of tangency.
• Do not place the point of tangency away from the actual contact point on the boundary.