Distance Formula
In short: The distance formula gives the straight-line length between two points \((x_1, y_1)\) and \((x_2, y_2)\): the square root of the squared horizontal gap plus the squared vertical gap.
8th GradeGeometry8.G.B.8
What is Distance Formula?
Any two points on a grid can be joined by a slanted segment, and the distance formula measures its length without a ruler. Draw a horizontal and a vertical path between the points and you get a right triangle whose longest side is the segment.
In symbols, \(d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}\). It is the Pythagorean theorem with the legs taken from the coordinates, so the order of the points does not matter because the differences are squared.
How to spot it
- Two ordered pairs are given and the question asks how far apart the points are.
- A map uses a coordinate grid, with towns or checkpoints placed at labeled points.
- You must show a triangle is isosceles or a quadrilateral is a rectangle by comparing side lengths.
- The segment between two points is neither horizontal nor vertical, so counting squares fails.
The method
- Label the points \((x_1, y_1)\) and \((x_2, y_2)\) and subtract the x-values, then the y-values.
- Square each difference; negative signs disappear after squaring.
- Add the two squares.
- Take the square root and include the unit, simplifying the radical or rounding as asked.
Distance Formula: solved example
Problem
On a hiking map where 1 unit equals 1 mile, Fairview sits at \((-6, 2)\) and Linden at \((5, -3)\). How far apart are the towns, to the nearest hundredth of a mile?
Solution
Horizontal gap: \(5 – (-6) = 11\). Vertical gap: \(-3 – 2 = -5\).
Then \(d = \sqrt{11^2 + (-5)^2} = \sqrt{121 + 25} = \sqrt{146} \approx 12.08\). The towns are about 12.08 miles apart.
Try it
Question 1
Find the distance from \((0, 0)\) to \((-9, 40)\).
Show the answer
\(\sqrt{81 + 1{,}600} = \sqrt{1{,}681} = 41\).
Question 2
What is the distance between \((2, 7)\) and \((2, -5)\)?
Show the answer
The x-values match, so the segment is vertical: \(|7 – (-5)| = 12\).
Distance Formula vs. Midpoint Formula
Both formulas start from two points, but they return different things. The midpoint formula averages the coordinates and gives a point halfway between them. The distance formula squares and adds the gaps and gives one number, the length of the segment.
Keep going
Lessons
Worksheets and quizzes
- Distance on the Coordinate Plane – Worksheet
- Distance on the Coordinate Plane – Quiz
- Coordinate Geometry and Proofs – Worksheet
- Coordinate Geometry and Proofs – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry