Chords, Secants, and Tangents in Circles

Understanding the lines that intersect and touch circles is a fundamental part of geometry. The three main types of lines we look at are chords, secants, and tangents.

• Chord: A line segment whose endpoints lie on the circle.
• Secant: A line that intersects a circle at exactly two points (an extended chord).
• Tangent: A line that touches the circle at exactly one point (the point of tangency).

The Tangent-Radius Theorem

A critical rule to remember is that a tangent is always perpendicular to the radius drawn to the point of tangency.

If a line is tangent to a circle at point $P$, and $O$ is the center of the circle, then the radius $OP$ forms a $90^\circ$ angle with the tangent line.

Segment Length Relationships

When these lines intersect, they create specific, predictable relationships between the lengths of their segments.

1. Intersecting Chords Theorem

If two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.

$a \cdot b = c \cdot d$

(Where $a$ and $b$ are segments of the first chord, and $c$ and $d$ are segments of the second chord.)

2. Secant-Secant Theorem

If two secants intersect at a point outside the circle, the product of the external segment and the whole segment of one secant equals the product of the external segment and the whole segment of the other.

$\text{External}_1 \cdot \text{Whole}_1 = \text{External}_2 \cdot \text{Whole}_2$

3. Tangent-Secant Theorem

If a tangent segment and a secant segment intersect at a point outside the circle, the square of the tangent's length is equal to the product of the external secant segment and the whole secant segment.

$\text{Tangent}^2 = \text{External} \cdot \text{Whole}$

Example Problems

Example 1: Intersecting Chords Two chords intersect inside a circle. One chord is divided into segments of $3$ and $8$. The other chord has one segment of $4$. Find the length of the unknown segment, $x$.

Using the Intersecting Chords Theorem: $3 \cdot 8 = 4 \cdot x$ $24 = 4x$ $x = 6$ The unknown segment is $6$.

Example 2: Tangent and Secant A tangent and a secant are drawn to a circle from the same external point. The tangent segment has a length of $6$, and the external part of the secant has a length of $3$. Find the length of the whole secant segment, $w$.

Using the Tangent-Secant Theorem: $6^2 = 3 \cdot w$ $36 = 3w$ $w = 12$ The whole secant segment has a length of $12$.