Calculate the equation of a perpendicular line. Enter the equation of the original line and the point it passes through to calculate the perpendicular line equation.

## Perpendicular Line Formula

Linear lines are almost always displayed in the form of

y = mx + b

Where m is the slope and b is the y-intercept. The first step in finding the equation of a line perpendicular to another is understanding the relationship of their slopes. The slope of a perpendicular line is always the inverse of the other. This means that the product of the two slopes is equal to -1.

With that in mind we can formulate the following equations.

a * m = -1

a = -1 / m

Where m is the original slope, and a is the slope of the perpendicular line. Now that we have the slope of our new line, all we need now is the y-intercept, b.

To calculate the y-intercept, we can use a similar formula to the one used to calculate the equation of a parallel line.

b = y₀ + 1 * x₀ / m

- Where b is the y-intercept
- y0 is the y coordinate the line passes through
- X0 is the x coordinate the line passes through
- m is the slope of the original line.

## How to calculate a perpendicular line

Let’s look at an example of how to use these equations. First, let’s assume you know the equation of the first line. It happens to be y=4x+5. Let’s also assume you know the x and y coordinates of a point that the perpendicular line passes through, say (4,5).

First, we need to calculate the slope. From the equation a = -1 / m we get a value of -1/4.

Next, we need to calculate the y-intercept of the new line using the equation b = y₀ + 1 * x₀ / m. From this, we get a value of 6.

Finally, we need to put this all together in the form: y=-1/4x + 6.

## Finding Parallel and Perpendicular Lines

in the 2-Dimension Cartesian Plane, all straight lines can be represented as an equation of the form y=mx+b, where m is the slope and x and y are points along the line. Since all lines can be described this way, it makes calculating parallel and perpendicular lines simple.

It’s just a matter of manipulating the equation. Let’s look at how the equation is manipulated in order to calculate the equation of a perpendicular line.

Conceptually, a perpendicular line is a line that crosses through the original line at any point and forms a 90-degree angle at intercept. Since there can be an infinite number of perpendicular lines, calculating any specific line requires a point.

The slope of a perpendicular line is the reciprocal of the slope of the original line. This simply means if the original slope is m, the reciprocal is 1/m. With this in mind, we can now manipulate the equation to determine the perpendicular line.

y=mx+b —> y2=(1/m)x2+b

The only thing left to do is solve for be using the point given along the perpendicular line.

## Steps to calculate a perpendicular line

We’ve gone over how to calculate the equation of a perpendicular line, and how manipulation of the original equation yields the results, but the following steps will outline the specific steps.

- Calculate the slope of the original line. This can be done through the use of two points along the original line. To learn more about calculating the slope of a line click here.
- Take the reciprocal of the original slope. If the slope is m, the reciprocal is 1/m.
- Calculate b, the y-intercept of the new line using the new slope and the point given along that line.
- Put all of the information together in point-slope form.

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