The Distance Formula
Measure the straight-line distance between any two points on the coordinate plane — powered by the Pythagorean Theorem.
Introduction
The Ruler Postulate gives distance on a number line. But on a plane, two points aren't generally on the same horizontal or vertical line. The Distance Formula uses horizontal and vertical legs of a right triangle to compute the hypotenuse — the straight-line distance.
Past Knowledge
Ruler Postulate (1.2.1). Coordinate Plane basics.
Today's Goal
Apply the Distance Formula to find the length of a segment on the coordinate plane.
Future Success
Used throughout coordinate proofs (Unit 5, 9) and circle equations (Unit 10).
Key Concepts
The Distance Formula
Given and :
Connection to the Pythagorean Theorem
Draw a right triangle with legs (horizontal) and (vertical). The distance is the hypotenuse:
Squaring Eliminates Negatives
Since we square the differences, . Order doesn't matter!
Worked Examples
Distance Between Two Points
Find the distance between and .
Answer:
Irrational Distance
Find the distance between and .
Answer:
Is It a Right Triangle?
Determine if , , form a right triangle.
, ,
Check: — not a right triangle (isosceles but not right).
Answer: No — it is isosceles, not right.
Common Pitfalls
Forgetting to Square Root
The formula gives inside the radical. Don't forget the final step.
Real-Life Applications
GPS & Navigation
GPS systems calculate straight-line (“as the crow flies”) distances between two coordinates using a version of the Distance Formula adapted for Earth's surface (the Haversine formula is its spherical cousin).
Practice Quiz
Loading...