Tangent to a Circle — Properties & Theorems

A tangent to a circle is a line that touches the circle at exactly one point. The key theorem is that the radius to the point of tangency is perpendicular to the tangent line, so tangent problems often turn into right-triangle or angle problems very quickly.

If a line only looks like it touches the circle, do not use tangent theorems until that condition is clear. Most mistakes happen when a secant or an ordinary line is treated like a tangent.

Tangent to a Circle: Main Property

If line $l$ is tangent to a circle at point $T$, then

$$OT \perp l$$

where $O$ is the center of the circle.

This is often called the radius-tangent theorem. The condition matters: the radius must end at the point where the tangent touches the circle.

Tangent vs. Secant

A tangent meets the circle once. A secant crosses the circle and meets it at two points.

That difference is small in a picture but important in a proof. Tangent theorems do not automatically apply to secants or chords.

Equal Tangents From One External Point

If two tangents are drawn from the same external point $P$ to a circle, and they touch the circle at $A$ and $B$, then their lengths are equal:

$$PA = PB$$

This is useful when one tangent length is known and the other is missing. The condition is important: both tangents must come from the same external point.

Worked Example: Find the Length of a Tangent

Suppose a circle has center $O$. From an external point $P$, a tangent touches the circle at $T$. Let

$$OT = 5$$

and

$$OP = 13.$$

Find the tangent length $PT$.

Because $OT$ is a radius to the point of tangency, it is perpendicular to the tangent line. So triangle $OPT$ is a right triangle with right angle at $T$.

Use the Pythagorean theorem:

$$OP^2 = OT^2 + PT^2$$

Substitute the values:

$$13^2 = 5^2 + PT^2$$ $$169 = 25 + PT^2$$ $$PT^2 = 144$$ $$PT = 12$$

So the tangent segment has length $12$.

This is the standard tangent-length pattern: find the point of tangency, mark the right angle, then solve the right triangle.

Common Mistakes in Tangent Problems

Perpendicular does not always mean tangent

A line perpendicular to a radius is tangent only when it passes through the radius endpoint on the circle. Perpendicular somewhere else is not enough.

A secant is not a tangent

If a line cuts through the circle at two points, it is a secant, not a tangent. Using tangent rules there will give the wrong result.

Equal tangents need the same outside point

The rule $PA = PB$ only applies when both tangent segments come from the same outside point to the same circle.

The right angle has a specific location

The right angle is between the tangent and the radius to the point of tangency. It is not automatically between the tangent and every segment from the center or external point.

When Tangents to Circles Are Used

Tangents to circles appear in school geometry, coordinate geometry, and diagram proofs about angles and lengths. They also lead into related ideas such as tangent-chord angles, circle constructions, and power-of-a-point problems.

Quick Check Before You Solve

When you see a tangent, ask:

1. Where is the point of tangency?
2. Which radius goes to that point?
3. Does that create a right triangle or an equal-tangents setup?

Those three checks catch most setup errors before you start calculating.

Try a Similar Problem

Try your own version with the same setup but different numbers, such as $OT = 7$ and $OP = 25$. Solve for $PT$, then check whether your answer makes geometric sense. If you want another case right away, explore a similar circle-geometry problem in GPAI Solver.