Tangent Line to a Circle — Definition, Formula & Examples

A tangent line to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. At that point, the tangent line is always perpendicular to the radius drawn to it.

Given a circle with center $O$ and radius $r$ , a line $\ell$ is tangent to the circle if and only if the perpendicular distance from $O$ to $\ell$ equals $r$ . Equivalently, $\ell$ intersects the circle in precisely one point $P$ , and $\overline{OP} \perp \ell$ .

Key Formula

d(O, \ell) = r

Where:

$O$ = Center of the circle
$\ell$ = The line being tested for tangency
$r$ = Radius of the circle
$d(O, \ell)$ = Perpendicular distance from center O to line ℓ

How It Works

To determine whether a line is tangent to a circle, measure the distance from the circle's center to the line. If that distance equals the radius, the line is tangent. Because the radius meets the tangent at a right angle, you can use the Pythagorean theorem in problems involving a tangent segment, a radius, and the line from the center to an external point.

Worked Example

Problem: A circle has center O and radius 5. A tangent line touches the circle at point P. An external point Q lies on the tangent line such that OQ = 13. Find the length of the tangent segment PQ.

Step 1: Recognize that OP is a radius and is perpendicular to the tangent line at P, so triangle OPQ is a right triangle with the right angle at P.

\angle OPQ = 90°

Step 2: Apply the Pythagorean theorem with hypotenuse OQ.

OQ^2 = OP^2 + PQ^2 \implies 13^2 = 5^2 + PQ^2

Step 3: Solve for PQ.

PQ^2 = 169 - 25 = 144 \implies PQ = 12

Answer: The tangent segment PQ is 12 units long.

Why It Matters

Tangent lines appear throughout geometry proofs, coordinate geometry problems, and circle theorems on standardized tests like the SAT. In engineering and physics, tangent lines model the direction of motion along a curved path at a single instant.

Common Mistakes

Mistake: Forgetting that the radius and tangent line are perpendicular at the point of tangency.

Correction: Always mark the right angle where the radius meets the tangent. This lets you set up right-triangle relationships and apply the Pythagorean theorem correctly.