Tangents and Points of Tangency

A tangent line is always perpendicular (makes a 90° angle) to the radius at the point where it touches the circle.

A tangent line touches the circle at one point and makes a 90° angle with the radius.

How It Works

What is a tangent?

A tangent is a straight line that just barely touches the outside of a circle. It touches at exactly ONE point. Think of it like a ball sitting on the floor — the floor is the tangent line.

The 90° rule

Draw a line from the center of the circle to where the tangent touches. That line (the radius) makes a perfect 90° angle with the tangent. Always. This is the most important rule.

Two tangents from the same point

If you draw two tangent lines from the same point outside the circle, those two lines are always the same length. This is super useful for solving problems.

Tangent-radius triangles

Since the radius meets the tangent at 90°, you often get right triangles. You can use the Pythagorean theorem (a² + b² = c²) to find missing lengths.

Key Formulas

Tangent-Radius Rule

tangent ⊥ radius (90° angle)

The tangent line is perpendicular to the radius at the point of tangency.

Two-Tangent Rule

If PA and PB are tangents from point P, then PA = PB

Two tangent segments from the same external point are equal in length.

Tangent-Secant Angle

angle = ½ × (far arc − near arc)

When a tangent and a secant meet outside the circle, the angle equals half the difference of the intercepted arcs.