Lesson 10.2.5

Segment Lengths: Secants & Tangents

When secants and tangents extend from an external point, specific product rules connect the segment lengths — all derived from similar triangles.

Introduction

This lesson completes the segment relationships for circles. The key idea: from an external point, whole × external = whole × external.

Past Knowledge

Chord products (10.2.4). Tangent properties (10.2.1). Similar triangles.

Today's Goal

Apply three external segment formulas: secant-secant, secant-tangent, tangent².

Future Success

Equation of a circle (10.3), power of a point, SAT problems.

Key Concepts

The Three External Formulas

1. Two Secants from External Point:

2. Secant-Tangent from External Point:

3. Two Tangents (review):

All-in-One: Power of a Point

All four segment theorems (chords, secant-secant, secant-tangent, tangent-tangent) are special cases of the Power of a Point theorem. The “power” of a point is constant for any line through it that meets the circle.

Worked Examples

Basic

Secant-Tangent

Tangent = 6, external part of secant = 4, whole secant = x. Find x.

→

Whole secant = 9 (internal chord part = 9 − 4 = 5)

Intermediate

Two Secants

From point P: secant 1 has whole = 12, external = 4. Secant 2 has whole = x, external = 3. Find x.

→ →

x = 16

Advanced

Finding the Tangent Length

From external point P: secant passes through circle with external part 5 and internal part 7 (whole = 12). Find the tangent from P.

Common Pitfalls

“Whole” vs. “External”

The formula uses whole × external, NOT chord × external. The "whole" includes the chord part + the external part.

Tangent² (Not Tangent × 2)

For secant-tangent: tangent is SQUARED (t²), not doubled. The tangent acts as both “whole” and “external” because it starts and ends at the circle instantly.

Real-Life Applications

Cell Tower Range

Engineers calculating signal line-of-sight to a circular coverage area use secant-tangent relationships to determine the farthest reachable point.

Pond Distance Estimation

To measure across a circular pond without crossing it, surveyors can use two secant measurements from an external point and the product formula to calculate the pond's chord length.