10.4 Tangents and secants

Tangents and Secants of Circles

Tangent lines touch a circle at exactly one point, while secant lines cut through a circle at two points. The relationships between these lines and the circle produce predictable geometric properties, especially when the lines originate from the same external point. This section covers those properties and the segment-length theorems you'll use to solve problems.

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Tangents and Secants: Definitions

A tangent intersects a circle at exactly one point, called the point of tangency.

A secant intersects a circle at exactly two points. The segment of the secant that lies inside the circle is a chord (a segment with both endpoints on the circle). The longest possible chord in any circle is the diameter.

Properties of Circle Tangents

Perpendicularity: A tangent line is always perpendicular to the radius drawn to the point of tangency. This means the radius is the shortest distance from the center to the tangent line. Whenever you see a tangent meeting a radius, mark a right angle at that intersection.

Congruent tangent segments: Two tangent segments drawn from the same external point are congruent. If point $P$ is outside a circle and you draw tangent segments $\overline{PA}$ and $\overline{PB}$ to the circle, then $PA = PB$. This also means the external point is equidistant from both points of tangency.

Segment-Length Theorems

These three theorems all deal with segments drawn to a circle from an external point $P$. Each one sets up an equation you can solve for an unknown length. The key to using them is correctly identifying which segment is the whole length (from $P$ all the way through the circle) and which is the external part (from $P$ to the nearer intersection point).

Tangent Segments Theorem: If two tangent segments are drawn from an external point, the segments are congruent.

$PA = PB$

Secant-Secant Theorem: If two secants are drawn from an external point $P$, where one secant passes through $A$ and $B$ (with $B$ closer to $P$) and the other passes through $C$ and $D$ (with $D$ closer to $P$), then:

$PA \cdot PB = PC \cdot PD$

Each side multiplies the full secant length (the whole) by its external segment length (the outside). The shorthand to remember: whole × outside = whole × outside.

For example, if one secant has a whole length of 12 and an external segment of 4, and the other secant has an unknown whole length $x$ with an external segment of 3, you'd write $12 \cdot 4 = x \cdot 3$ and solve to get $x = 16$.

Tangent-Secant Theorem: If a tangent segment and a secant segment are drawn from an external point $P$, where the tangent touches at $A$ and the secant passes through $B$ and $C$ (with $C$ closer to $P$), then:

$PA^2 = PB \cdot PC$

The tangent segment gets squared because it serves as both the "whole" and the "outside" at the same time. For example, if the tangent length is 6 and the secant's external segment is 3 with a whole length of $x$, you'd write $36 = x \cdot 3$, giving $x = 12$.

Equations of Circle Tangent Lines

To write the equation of a tangent line at a given point on a circle:

1. Find the slope of the radius. Use the center of the circle and the point of tangency. If the center is $(h, k)$ and the point of tangency is $(x_1, y_1)$, the slope of the radius is $m_r = \frac{y_1 - k}{x_1 - h}$.

2. Take the negative reciprocal. Since the tangent is perpendicular to the radius, the tangent's slope is $m_t = -\frac{1}{m_r}$. (If the radius is vertical, the tangent is horizontal, and vice versa.)

3. Write the equation using point-slope form. Plug the tangent point and the tangent slope into $y - y_1 = m_t(x - x_1)$.

For example, if a circle has center $(2, 3)$ and the point of tangency is $(5, 7)$, the radius slope is $\frac{7-3}{5-2} = \frac{4}{3}$. The tangent slope is $-\frac{3}{4}$, and the tangent line equation is $y - 7 = -\frac{3}{4}(x - 5)$.

Applying Tangent and Secant Properties

Solving Problems with Tangents and Secants

Most problems follow the same general approach:

1. Identify what's given. Determine whether you're working with two tangents, two secants, or a tangent and a secant from the same external point.
2. Choose the right theorem. Match the configuration to the correct segment-length theorem.
3. Set up the equation. Substitute the known lengths and use a variable for the unknown. Be careful to distinguish the whole secant length from just the external part.
4. Solve. These equations are usually linear or quadratic after expanding. If you get a quadratic, check both solutions and reject any negative lengths.

Watch for problems where a tangent meets a radius. That right angle means you can also apply the Pythagorean theorem alongside the segment theorems. If you know the radius $r$ and the distance from the center to an external point $d$, you can find the tangent segment length $t$ using:

$t^2 + r^2 = d^2$

For instance, if the radius is 5 and the center-to-external-point distance is 13, the tangent length is $t = \sqrt{169 - 25} = \sqrt{144} = 12$.