Tangent to a Circle — Properties & Theorems

A tangent to a circle touches it at exactly one point, and the radius to that point is perpendicular to the tangent — which is why tangent problems collapse into right-triangle or angle problems so quickly. The first job in any tangent problem is to confirm the line really is tangent, because most errors come from treating a secant or an ordinary line as a tangent.

Tangent vs. secant at a glance

                 Tangent                      Secant
Meets circle     at exactly one point         at two points
Key relation     radius at the point of       no special perpendicular
                 tangency is perpendicular    relation from this alone
Theorems apply   radius-tangent theorem,      tangent theorems do NOT
                 equal tangents from a point   apply

The difference looks tiny in a picture but is decisive in a proof: tangent theorems do not automatically apply to secants or chords. Always confirm the line touches the circle only once first.

The properties you will use

Radius-tangent theorem. If line $l$ is tangent to a circle at point $T$, then

where $O$ is the center. The condition matters: the radius must end at the point where the tangent touches the circle.

Equal tangents from one external point. If two tangents are drawn from the same external point $P$, touching the circle at $A$ and $B$, then

This is useful when one tangent length is known and the other is missing — but only when both segments come from the same external point.

When to use which

If you have one tangent line and a radius, reach for the radius-tangent theorem and build a right triangle. If you have two tangents from a single outside point, reach for the equal-tangents rule. A quick three-question scan tells you which setup you are in:

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?

Worked example: length of a tangent

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

Find the tangent length $PT$.

Because $OT$ is a radius to the point of tangency, it is perpendicular to the tangent, so triangle $OPT$ is right-angled at $T$. Apply the Pythagorean theorem:

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.

Frequently confused points

• 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 elsewhere is not enough.
• A secant is not a tangent. A line cutting the circle at two points is a secant; tangent rules give the wrong result there.
• Equal tangents need the same outside point. $PA = PB$ holds only when both segments come from the same external point to the same circle.
• The right angle has a specific location. It is between the tangent and the radius to the point of tangency — not between the tangent and every segment from the center or external point.

Apply it yourself

Try the same setup with $OT = 7$ and $OP = 25$. Solve for $PT$, then check the answer makes geometric sense (it should be shorter than $OP$). Tangents to circles run through school geometry, coordinate geometry, and diagram proofs, and lead into tangent-chord angles, circle constructions, and power-of-a-point problems.