# How to Use the Distance Formula on a Coordinate Plane

By Kathleen Cantor, 03 Apr 2021

Let's say you want to measure the length of any distance or line sitting on horizontal or vertical axes on a coordinate plane. You can count the coordinates. The tricky part is using the same formula for a diagonal line—that's where the distance formula comes in.

Based on the Pythagorean Theorem, the distance formula sets your line as a section of the hypotenuse of a right-angled triangle. The theorem suggests that `a² + b² = c²`

when a and b are adjacent to the right angle of the triangle and c is the hypotenuse’s length.

## Calculating Length Using the Pythagorean Theorem and Distance Formula on a Coordinate Plane

Using the distance formula, you can determine the length of a line between any given two coordinates. This distance (d) can be expressed in the following way:

`d=√ ((x₁-x₂)² + (y₁-y₂)²)`

X and y are coordinates of a line on a Cartesian plane, and this line can be considered as the hypotenuse of a right-angled triangle. The Pythagorean Theorem is used to find the length of this hypotenuse using the distance formula. The coordinates x₂ and x₁ are one side of this triangle.

The third side is composed of y₂ and y₁, so the line from x₂, x₁ to y₂, y₁ will be the hypotenuse or distance between these two given points. It shouldn’t matter which coordinate points are called as long as they can be denoted as x, y coordinates. This is because the distance is always a positive number. For instance, finding the distance or length between coordinates 4, -2 and -10, 3 using the distance formula will involve;

Plugging in (x₁, y₁) as (4, -2) and (x₂, y₂) for (-10, 3) to simplify the equation as `d= √ (−10−4)² + (3+2)²`

`= √ (−14)² + (5²)`

`= √ 196+25`

`= √ 221`

`= 14.87 units`

## Applying the Distance Formula to Calculate For Line Length on a Coordinate Plane

After all these points are numbered, finding coordinates on a Cartesian plane is easy. And all you'll have to do is count them. Count along the x and y-axis to find these coordinates, the y axis being the vertical while the x is the horizontal plane. Coordinates are written as both numbers for each axis, ie: (4, 2), where four represents the x while 2 is the point where it crosses the y axis.

Taking a line segment as your distance between two points, assume that one endpoint is at (2, 1) while the other is at (6, 4). Now take the distance formula and plug in the coordinates, taking care to make the correct substitutions for both axes on the Cartesian plane. Derived from your (2, 1) and (6, 4) coordinates, your distance calculations should resemble these expressions:

`d = √ (6 -2)² + (4 - 1)²`

`√ (4)² + (3)²`

When calculating these subtractions, ensure you use the order of operations, meaning whatever is in the parentheses or exponents should be addressed first.

`√ 16 + 9`

`√25`

`Distance = 5 units`

The answer is rounded off since you are looking to find a square root, and as you're working on a coordinate plane it'll be in units as opposed to inches, meters, miles, or other conventional units.

## Finding the Coordinates of a Point When Given Its Distance from another Cartesian Plane Point

The distance formula can be used to also find any of the two coordinates on a Cartesian plane when you are given one coordinate and the length between them. You can assume that the second coordinate point is made up of integers, when you're given one of the points as 1, -6, and the distance between them is 4 units.

Using the distance formula when the length is `(d = 4)`

, and one of the coordinates is (x₁, y₁) is (1, -6), solving for the other point, (x₂, y₂) is made easy. Let's say you're given the following info:

`d= √ ((x₁-x₂)² + (y₁-y₂)²)`

`4= √ (1−x₂)² + (−6−y₂)²`

`16 = (1−x₂)² + (−6−y₂)²`

This point requires that you figure out what square numbers will provide the root for 16, and you'll find that the only numbers that qualify are 16 and 0.

`16 = ((1−x₂)² / 4²) + ((-6−y₂)² / 0²)`

Side A

`1 - x₂ = ± 4`

`-x₂ = -5 or 3`

`-x₂ = -5 or 3`

Side B

`-6 - y₂ = 0`

`-y₂ = 6`

`-y₂ = - 6`

Or

`16 = ((1−x₂)² / 0²) + ((-6−y₂)² / 4²)`

Side A

`1 - x₂ = 0`

`-x₂ = -1`

`-x₂ = 1`

Side B

`-6 - y₂ = ± 4`

`-y₂ = 10 or 2`

`y₂ = - 10 or - 2`

So the second coordinate point can be any of four possibilities, including (-3, -6), (5, -6), (1, -2), or (1, -10).

## Example of Distance Formula on a Coordinate Plane

Find the distance between a given place in China and another in India when the longitudinal and latitudinal coordinates are 35.8617, 104.1954, and 20.5937, 78.9629 degrees, respectively. Immediately plug in the distance formula LaTeX: syntax error (Unable to compile LaTeX formula,

{d}=√{\left({\left({x}₁-{x}₂\right)}²+{\left({y}₁-{y}₂\right)}²\right)}

Check syntax.), and insert the coordinates accordingly:

`d = √ ((104.1954 - 78.9629)² + (35.8617 - 20.5937)²)`

`d = 29.49222`

This geometric formula simplifies the task of getting the length or measuring the distance between two points on a coordinate path.

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