Lesson 3.2.6

Finding the Distance from a Point to a Line

The shortest path from a point to a line is always perpendicular. This lesson combines perpendicular slopes, systems of equations, and the distance formula into one powerful technique.

Introduction

This lesson is a capstone of Unit 3 coordinate geometry — it requires perpendicular slopes, writing equations, solving systems, and the distance formula all in one problem.

Past Knowledge

Perpendicular slopes (3.2.3). Writing equations (3.2.4-3.2.5). Distance formula (1.2.4).

Today's Goal

Find the shortest distance from a point to a line on the coordinate plane.

Future Success

Distance between parallel lines, altitudes of triangles, and perpendicular bisectors use this method.

Key Concepts

Method (Step by Step)

  1. Find the perpendicular slope from the given line's slope.
  2. Write the equation of the perpendicular line through the given point.
  3. Find the intersection of the original line and the perpendicular line (solve the system).
  4. Use the distance formula between the given point and the intersection point.

Shortcut Formula (for )

where is the point and is the line in standard form.

Interactive Diagram — Desmos Geometry

The shortest distance from a point to a line is always perpendicular. See the right angle marking where the perpendicular segment meets the line.

Worked Examples

Basic

Using the Formula

Find the distance from to .

units

Intermediate

Step-by-Step Method

Find the distance from to .

Step 1: Perp. slope =

Step 2: Perp. line: →

Step 3: Solve system: → ,

Step 4: Distance from to

units

Advanced

Distance Between Parallel Lines

Find the distance between and .

Pick any point on line 1: when , → point

Rewrite line 2:

units

Common Pitfalls

Forgetting Absolute Value in the Formula

The numerator uses — absolute value! Distance is always positive.

Wrong Standard Form

The formula requires . If the line is , rewrite it as first.

Real-Life Applications

GPS & Emergency Services

When an emergency is reported on a road, dispatchers calculate the perpendicular distance from the nearest ambulance station to the road to estimate response time. The point-to-line distance formula gives the shortest possible route.

Practice Quiz

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