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.

What Is a Perpendicular Line?

A perpendicular line is a line that forms a 90-degree angle with another line. Such lines can be positioned in any plane.

On the grid, the perpendicular lines can be positioned crosswise, vertically and horizontally, or sideways.

They don’t have to be pointing upwards; they should only be at a 90-degree angle with respect to another line.

The Difference Between Parallel, Perpendicular, and Intersecting Lines

While it’s easy to confuse the lines with each other, they’re three very different concepts.

Parallel

Lines in a grid that are always the same spacing apart are known as parallel lines. Parallel lines never cross one other and they never have the same slope.

Perpendicular

Lines that connect at a right 90-degree angle are known as perpendicular lines. They’re perpendicular if the slope of one line is the negative reciprocal of the slope of the other.

The only thing that parallel lines and perpendicular lines have in common is that they’re both made up of straight lines.

Intersecting

Intersecting lines are formed when lines in a grid intersect with each other at a point of intersection. However, unlike perpendicular lines, they don’t form a right angle.

Do All Shapes Have Perpendicular Lines?

Perpendicular lines can be found in many shapes, but not all of them.

There will be no perpendicular sides on many polygons, but some might have perpendicular lines. Perpendicular lines will always exist in squares, right-angled triangles, and rectangles.

You can also find perpendicular lines on everyday objects such as decorations, fences, and doors.

What Is the Equation for Perpendicular Slope?

The perpendicular slope will be the reciprocal of the initial slope in the opposite direction.

To calculate the slope, use the slope-intercept equation (y = mx + b) and substitute in the provided point and the new slope.

Then, restore the equation to its standard form (ax + by = c).

Do Perpendicular Lines Have The Same Slope?

The slope of perpendicular lines is not the same. If two lines are perpendicular, one line’s slope is the negative reciprocal of the other line’s slope.

A number’s product and its reciprocal equals 1. When the slopes of two perpendicular lines in the plane are multiplied, however, the result is -1.

This indicates the slopes of perpendicular lines are reciprocals in the opposite direction.

Do Perpendicular Lines Have to Touch?

If two nonvertical lines in the same plane intersect at a right angle, then they’re considered to be perpendicular. If the lines aren’t in the same plane, the term perpendicular doesn’t apply to the shape.

Visualize a 3D square object if you’re having trouble understanding this information.

Even if it doesn’t appear to be the case, the lines on a 3D square are perpendicular. This is due to the lines existing on the same plane.

Summary

Two intersecting lines that make a right angle are called perpendicular lines.

The slope of the lines differs because the slope of one line is the negative reciprocal of the slope of the other.

Perpendicular lines aren’t present in all shapes, although they’re always found in squares, right-angled triangles, and rectangles.

Perpendicular lines don’t have to touch to be deemed perpendicular if they’re on the same plane.

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.

  1. 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.
  2. Take the reciprocal of the original slope. If the slope is m, the reciprocal is 1/m.
  3. Calculate b, the y-intercept of the new line using the new slope and the point given along that line.
  4. Put all of the information together in point-slope form.

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