Buzz
An algebraic equation known as the distance formula calculates the shortest path between two points in a 2D plane. Dream01/Shutterstock/MytourImagine sitting in math class, nervously facing a surprise quiz. As you glance at the question, "Determine the distance between these points," beads of sweat form on your brow.
The distance formula is simple yet powerful, rooted in one of mathematics' most renowned principles: the Pythagorean theorem.
What Is the Distance Formula?
The distance formula is a mathematical expression designed to calculate the length of a line segment connecting two points on a Cartesian coordinate system, also referred to as the coordinate plane.
This plane consists of two perpendicular axes, typically labeled the x-axis and y-axis, which intersect at a central point known as the origin. Here's the breakdown:
For two points P (x₁, y₁) and Q(x₂, y₂) in a two-dimensional space, the distance (d) between them is calculated using the formula: d = √ (x₂ - x₁)² + (y₂ - y₁)²
In three-dimensional space, for points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), the distance (d) is determined by the formula: d = √ (x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²
Next, we’ll explore the coordinate plane, a tool that pinpoints exact locations using horizontal and vertical positions. This is crucial for tasks ranging from solving math problems to GPS navigation.
Distance Formula Calculator
Understanding the Point Coordinate Plane
For many, the term "graph" conjures an image of a chart featuring two intersecting lines — one vertical and one horizontal — meeting at a right angle.
The vertical line is known as the y-axis, while the horizontal one is the x-axis. Together, they create a framework for interpreting and visualizing data.
To determine the position of a point on a graph, measure its location along the two axes: the x-axis and the y-axis. These measurements are referred to as the point's coordinates.
Before calculating the distance between two points, you must first identify their coordinates. The distance formula is then used to measure the straight-line segment that connects these two points.
Next, let’s delve into the fascinating connection between the Pythagorean theorem and the distance formula.
The Pythagorean Theorem and the Distance Formula
The Pythagorean theorem is named after the Greek philosopher Pythagoras, but the geometric principle it describes was understood by the ancient Babylonians over a thousand years before his time.
At its core, the Pythagorean theorem explains how to determine the longest side of a right triangle when the other two sides are known. The distance formula applies this concept by treating two points on a graph as the corners of a right triangle to measure the separation between them.
For a quick reminder, the Pythagorean theorem states: The area of the square constructed on the hypotenuse of a right triangle equals the combined areas of the squares built on the other two sides.
A few essential concepts are worth noting. A right triangle, also called a right-angled triangle, features one 90-degree angle. The side opposite this angle, known as the hypotenuse, is the longest side of the triangle.
While a triangle has three sides, a square has four. Picture transforming the hypotenuse of a right triangle into one side of a new square. Repeat this process for the other two sides, and you’ll create three distinct squares.
The Pythagorean theorem asserts that the area of the square based on the hypotenuse equals the sum of the areas of the squares based on the other two sides. If the hypotenuse is labeled "c" and the other sides "a" and "b," this relationship can be expressed as:
According to the Pythagorean theorem, the sum of the squares of the two shorter sides of a right triangle equals the square of the hypotenuse, expressed as a² + b² = c². This principle is foundational for deriving the distance formula.How to Find the Distance Between Two Points
Each point on a graph is defined by its x and y coordinates. By applying the Euclidean distance formula, an algebraic expression rooted in the Pythagorean theorem, you can determine the shortest path between any two points.
In this context, D represents the term 'distance.' The variables x₂ and x₁ denote the x-coordinates of Point 2 and Point 1, respectively, while y₂ and y₁ correspond to their y-coordinates.
To determine the distance, begin by subtracting x₁ from x₂. Next, square the result of this subtraction to prepare for the next steps in the calculation.
Following this, subtract y₁ from y₂ and square the outcome. This process yields two squared values that need to be summed together to proceed further.
Finally, compute the square root of the sum obtained in the previous step. This square root value, my friends, is the precise distance between the two points.
Distance Formula Example
Let’s consider Point A with coordinates (2, 5), where 2 is the x-coordinate and 5 is the y-coordinate. Similarly, Point B has coordinates (9, 13), with 9 as the x-coordinate and 13 as the y-coordinate. Insert these values into the formula, and you’ll arrive at the following:
Subtracting 2 from 9 gives 7, and subtracting 5 from 13 results in 8. Simple arithmetic confirms these values.
After performing the subtractions, the equation simplifies to:
When you square 7—meaning you multiply the number by itself—the result is 49. Similarly, squaring 8 gives you 64. Let’s substitute these values back into the equation.
Moving forward, adding 49 and 64 together gives a total of 113.
The square root of 113 is approximately 10.63, leading us to the final result.
Go ahead and excel in your upcoming quiz!
Pythagoras followed a vegetarian diet. As Tristam Stuart explains in his 2008 book, "The Bloodless Revolution: A Cultural History of Vegetarianism: From 1600 to Modern Times," the philosopher believed in "the idea that all living beings are connected, and therefore, causing harm to animals was morally wrong."
