Circle Tangent: Definition, Radius Rule, Examples, Chart, and Practice

What is a circle tangent?

A circle tangent is a line that touches a circle at exactly one point.

That one touch point is called the point of tangency.

A tangent line does not cut through the circle. It just meets the circle at the edge.

Tangent and radius rule

The most important tangent rule uses the radius drawn to the point of tangency.

A radius to the point of tangency is perpendicular to the tangent line.

Perpendicular means they meet at a right angle, so the angle is exactly 90°.

Circle tangent chart

Use the chart to decide whether a line is tangent, secant, or outside the circle.

A tangent has one touch point. A secant cuts the circle at two points. A line outside the circle has no touch point.

Circle tangent vs central angle

A tangent line touches the circle from outside, while a central angle starts at the center of the circle.

The tangent rule talks about a 90° angle at the touch point. The central angle rule talks about the arc opened by two radii.

Worked example: radius to tangent

Problem: Line m is tangent to a circle at point T. Radius OT is drawn to point T. What is the angle between OT and line m?

Step 1: Notice that OT is a radius to the point of tangency.

Step 2: A radius to a tangent is perpendicular to the tangent line.

Answer: The angle is 90°.

Equal tangent segments

If two tangent segments are drawn from the same outside point to the same circle, the two tangent lengths are equal.

For example, if PA and PB are tangents from point P, then PA = PB.

This rule is useful when a diagram gives one tangent length and asks for the other.

Example: equal tangent lengths

Problem: Two tangent segments from point P touch the same circle. One segment is 13 cm. What is the other segment?

Step 1: Both segments start from the same outside point.

Step 2: Tangent segments from the same outside point are equal.

Answer: The other tangent segment is 13 cm.

Common mistakes

Do not call a line tangent if it crosses through the circle.

Do not place the 90° angle at the center. The right angle is at the point where the tangent touches the circle.

Do not confuse a circle tangent line with the tangent ratio used in trigonometry. In lessons like angle of elevation, tangent is a ratio, not a line.

Quick practice

1. A tangent line touches a circle at exactly one point.

2. A radius to a tangent point makes a 90° angle with the tangent line.

3. A line that cuts a circle in two places is not a tangent.

4. Two tangent segments from the same outside point have equal lengths.