Secant of a Circle — Definition, Formula & Examples

A secant of a circle is a line that passes through the circle and intersects it at exactly two points. Unlike a chord, which is a line segment with both endpoints on the circle, a secant extends infinitely in both directions.

Given a circle with center $O$ and radius $r$ , a secant is a line $\ell$ such that $|\ell \cap \text{circle}| = 2$ . The portion of the secant that lies inside the circle forms a chord, while the line itself continues beyond the circle in both directions.

Key Formula

a \cdot b = c \cdot d

Where:

$a$ = External segment of the first secant (from the external point to the nearer intersection)
$b$ = Whole length of the first secant (from the external point to the farther intersection)
$c$ = External segment of the second secant
$d$ = Whole length of the second secant

How It Works

When two secants are drawn from an external point to a circle, they create segments whose lengths are related by the secant–secant power-of-a-point theorem. If secant one passes through the circle creating an external segment

a

and a whole secant length

b

, and secant two creates external segment

c

and whole length

d

, then

a \cdot b = c \cdot d

. This relationship is useful for finding unknown lengths when partial measurements are given.

Worked Example

Problem: From an external point P, two secants are drawn to a circle. The first secant has an external segment of 3 and passes through the circle so that its total length from P to the far intersection is 12. The second secant has an external segment of 4. Find the total length of the second secant from P to its far intersection.

Step 1: Apply the secant–secant theorem using the known values.

a \cdot b = c \cdot d \implies 3 \cdot 12 = 4 \cdot d

Step 2: Solve for d.

36 = 4d \implies d = 9

Answer: The total length of the second secant from P to the far intersection is 9.

Why It Matters

Secant lines appear frequently in SAT and ACT geometry problems where you must find missing lengths using the power-of-a-point theorem. They also arise in engineering and optics when analyzing how lines intersect curved surfaces.

Common Mistakes

Mistake: Confusing a secant with a chord. Students sometimes treat a secant as the segment between the two intersection points.

Correction: A chord is the segment between the two points on the circle. A secant is the full line that extends beyond the circle in both directions. In length problems, the secant measurement starts from the external point, not from the circle.