Circles are full of interesting relationships between lines and points. Tangents touch circles at one point, while secants cut through them twice. These interactions create special properties we can use to solve geometry problems.

Understanding tangents and secants helps us tackle real-world scenarios involving circular objects. We'll explore how to find lengths of segments and write equations for tangent lines, building our problem-solving toolkit for circular geometry.

Tangents and Secants of Circles

Tangents and secants of circles

• Tangent intersects a circle at exactly one point called the point of tangency
• Secant intersects a circle at exactly two points creates a chord, a line segment with endpoints on the circle (diameter is the longest chord)

Properties of circle tangents

• Tangent line is perpendicular to the radius at the point of tangency radius is the shortest distance from the center to the tangent line
• Two tangent segments from a common external point are congruent common external point is equidistant from the points of tangency (tangent to a circle from a point)

Lengths in tangent-secant problems

• Tangent segments theorem if two tangent segments are drawn to a circle from an external point, then the segments are congruent (common external tangent)
• Secant segments theorem if two secant segments are drawn to a circle from an external point, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment $\overline{PA} \cdot \overline{PB} = \overline{PC} \cdot \overline{PD}$
• Tangent-secant segments theorem if a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the length of the tangent segment equals the product of the lengths of the secant segment and its external segment $\overline{PA}^2 = \overline{PB} \cdot \overline{PC}$ (tangent meets a secant at an external point)

Equations of circle tangent lines

1. Find the slope of the radius at the point of tangency using the center and the point of tangency
2. The slope of the tangent line is the negative reciprocal of the slope of the radius
3. Use the point-slope form of a line $(y - y_1) = m(x - x_1)$ to write the equation of the tangent line, where $(x_1, y_1)$ is the point of tangency and $m$ is the slope of the tangent line (slope is perpendicular to radius)

Applying Tangent and Secant Properties

Solving problems with tangents and secants

• Use the tangent segments theorem, secant segments theorem, or tangent-secant segments theorem to set up an equation substitute the given values and solve for the unknown length (similar to proportions)
• Recognize when to apply the Pythagorean theorem in conjunction with the segment theorems when a right triangle is formed, the Pythagorean theorem can be used to solve for unknown lengths ($a^2+b^2=c^2$ for right triangles)