Perpendicular Line Calculator

The perpendicular line calculator is a beneficial tool for calculating the equation of a line from the perpendicular line quickly.

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Introduction to Perpendicular Line Calculator:

Perpendicular line Calculator is a tool designed that is used to find the equation of a line from the perpendicular line that passes through a specific point. The calculator is a valuable tool in geometry as it provides a way to determine and simplify the process of finding perpendicular lines.

Perpendicular Line Calculator with Steps

The perpendicular slope calculator is a useful tool for students, professionals, and anyone who is working with geometry problems and is not familiar with perpendicular concepts, it gives you a complete procedure while you use it for finding different examples.

What is a Perpendicular Line?

A perpendicular line is a line that makes a 90-degree angle with another line in geometry and it can be expressed with ⊥ a symbol. When two lines m1 and m2 are perpendicular, they intersect each other and form a right angle. The slopes of perpendicular lines are the negative reciprocals of the equation of a line or a point.

Perpendicular lines are fundamental elements that play a crucial role in various mathematical and practical applications. As it provides a basis for understanding angles, shapes, and spatial relationships between them.

What is the Rule for Perpendicular Lines?

There are two distinct lines l and q in a plane, it makes a right angle (90c) when they intersect. The slope of perpendicular lines l and q are the negative reciprocal of the given equation. The perpendicular line calculator uses the following perpendicular slope formula for evaluation,

$$ y \;=\; mx + c $$

$$ml\; mq \;=\; -1 $$

$$ m_l \;=\; -\frac{1}{m_q} $$

$$ m_q \;=\; -\frac{1}{m_l} $$

How to Solve Perpendicular Line?

To find the slope of a perpendicular line, the perpendicular line equation calculator finds out the slope of the equation of a line and the point at which a line passes through it. Let's see how to solve the perpendicular line from the given equation.

Method 1: Use the Slope and a Point,

If the slope of the original line is given in question then, let us suppose the slope of the line is m1.
To find m2 you need to take the negative reciprocal of m1 as:

$$ m_2 \;=\; \frac{−1}{m_1} $$

If you have a point is a perpendicular line that passes through it, denote the coordinates as (x1,y1)

$$ y - y_1 \;=\; m(x - x_1) $$

where m is the slope

Method 2: Use the Equations of Lines

If the equation of the line, let us the y=mx + b, where m is the slope and b is the y-intercept.
To find the slope of the perpendicular line from m1, the m2 is the negative reciprocal of m1.
If you have a specific point, you can choose any point on the original line. Then use the point-slope that forms with the slope of 𝑚2.

$$ m_2 \;=\; \frac{-1}{m_1} $$

If you have the equation of the original line, you can use its slope directly to find the slope of the perpendicular line.

The perpendicular equation calculator uses both methods to solve the perpendicular line according to the given questions.

Solved Example of a Perpendicular Line:

Let's observe an example that explains the calculation process of the perpendicular line Calculator and find out how to calculate perpendicular line problems manually. For that, the examples are given below,

Example:

The line l passes through points (0,1) and (1,3) and the line q passes through points (-1, 4) then Find out if the given lines are perpendicular.

Solution:

To find the perpendicular line first find the slope of the equation

$$ ml \;=\; \frac{y_2 - y_1}{x_2 - x_1} $$

$$ mq \;=\; \frac{y_2 - y_1}{x_2 - x_1} $$

$$ ml \;=\; \frac{3 - 1}{1 - 0} $$

$$ =\; \frac{2}{1} \;=\; 2 $$

The slope of line q, mq passes through the points (-1, 4) and (5,1) is,

$$ m_q \;=\; \frac{1 - 4}{5 - (-1)} $$

$$ =\; \frac{-3}{6} $$

$$ =\; \frac{-1}{2} $$

Now lines l and q are perpendicular if and only if,

$$ m_l \;=\; -\frac{1}{m_q} \;and\; m_q \;=\; -\frac{1}{m_l} $$

$$ m_l \;=\; 2 \;and\; m_q \;=\; -\frac{1}{m_l} \;=\; -\frac{1}{2} $$

As, the slopes of the lines are negative reciprocals so that lines l and q are perpendicular lines.

How to Use Perpendicular Line Calculator?

The perpendicular slope calculator has a simple layout that enables you to calculate the perpendicular line from the given equation. Follow our guidelines before using it to find the solution to the equation and get an amazing experience. These guidelines are:

Enter the equation in the form of “y=mx+c” to find the perpendicular line in the input field of the perpendicular equation calculator.
Review your equation before pressing the calculate button of the perpendicular line equation calculator to get the precise solution without any mistakes.
Click on the “Calculate” button to get the solution of the perpendicular line problem.
Click on the “Recalculate” button to get a new page for more evaluation of the perpendicular line questions.
If you want to check the accuracy of our perpendicular lines calculator then you should first try out the load example and get an accurate solution every time.

Final Result of Perpendicular Slope Calculator:

The calculator for perpendicular lines gives you a solution to your given orthogonal equation problem after you click on the calculated button. It may include the following:

Result Option:

The result option provides you with solutions for Perpendicular line problems.

Possible Steps:

Possible steps of perpendicular lines calculator provide you with solutions to Perpendicular line equation examples in a step-wise process.

Benefits of Using Perpendicular Line Equation Calculator:

The calculator for perpendicular lines provides you many benefits whenever you use it to calculate the equation question to find its solution. These benefits help you to get a better understanding of Perpendicular line concepts.

The perpendicular gradient calculator saves the time and energy that you consume for finding the orthogonal equation questions manually.
It is a handy tool, so you can easily use it to solve the orthogonal equation
The line perpendicular calculator has a user-friendly design so that you can solve different equations easily.
The perpendicular slope calculator provides you with an accurate solution as per your given input term for the perpendicular line question
Perpendicular line Calculator is an educational tool that helps you to become familiar with the concept of the perpendicular line through an online platform.

Related References

How would you define perpendicular lines?
Explain the solve perpendicular lines problem. Explain in detail.
Give some examples of perpendicular lines:
Give an overview of perpendicular lines?

Frequently Ask Questions

What line is perpendicular to 3y= 5x+1?

To determine a perpendicular line from the given equation 3y = 5x + 1, to find the negative reciprocal of its slope. First, convert it into a standard form of equation as y = mx + c. To put it in slope-intercept form (y = mx + b), divide both sides by 3, and the given equation becomes

$$ y \;=\; \frac{5}{3}x + \frac{1}{3} $$

The slope of the equation is,

$$ m_1 \;=\; \frac{5}{3} $$

The slope of a line perpendicular to m2 is the negative reciprocal of m1 the perpendicular line of the slope is,

$$ m_2 \;=\; \frac{−1}{m_1} $$

$$ m_2 \;=\; \frac{−1}{\frac{5}{3}} $$

$$ =\; \frac{−3}{5} $$

Equation of the Perpendicular Line:

$$ m_2 \;=\; −\frac{3}{5} $$

To find the equation of the perpendicular line using the point-slope form:

$$ y − y_1 \;=\; m_2(x − x_1) $$

$$ y − y_1 \;=\; −\frac{3}{5}(x − x_1) $$

What is the slope of a line perpendicular to -10/3?

To find the slope of a line perpendicular to a given slope, you need to take the negative reciprocal of that slope. The given slope is:

$$ m_1 \;=\; −\frac{10}{3} $$

To find the slope of a line, you need to know m2, which is the negative reciprocal of m1.

$$ m_2 \;=\; −\frac{1}{m_1} $$

$$ m_1 \;=\; -\frac{10}{3} $$

$$ m_2 \;=\; −\frac{1}{−\frac{10}{3}} $$

For simplification multiply the numerator and denominator by −3:

$$ m_2 \;=\; -(\frac{-3}{10}) $$
$$ =\; \frac{3}{10} $$

So, the slope of a line perpendicular to m1 = −10/3 is m2 = 3/10.

What is the difference between intersecting and perpendicular lines?

Intersecting lines and perpendicular lines are two different concepts in geometry, but they are related to each other.

Intersecting Lines:

Intersecting lines are lines that meet or intersect at a common point. They have no angle between them but merely form right angles. Intersecting lines can be any angle other than 0 or 180 degrees but not have a negative reciprocal.

Perpendicular Lines:

Perpendicular lines are a special case in which these lines meet at a right angle (90 degrees). The angle between perpendicular lines is 90 degrees. Perpendicular lines have slopes that have negative reciprocals of each other.

What is the difference between parallel and perpendicular lines?

Parallel lines and perpendicular lines are both important concepts in mathematics, but they have distinct characteristics that make them different from each other.

Parallel lines never intersect, while perpendicular lines always intersect at a right angle. Parallel lines do not form any type of angles between them, while perpendicular lines form right angles.

On the other hand, perpendicular lines have negative reciprocal slopes while parallel lines have equal slopes. For a perpendicular line, finding the equation of a perpendicular line is important and for a parallel line, finding the parallel line on the graph is important.

Do two perpendicular lines always intersect?

Yes, in geometry, two perpendicular lines always intersect, if they are not parallel. When two lines are perpendicular, it means they make a right angle (90 degrees) at the point of intersection.

Amanda Jepson

Last Updated February 21, 2024