**Math Calculators** ▶ Parallel and Perpendicular line calculator

An online parallel and perpendicular line calculator will help you to determine the equation of the perpendicular and parallel to the given line that passes through the entered points. Also, this free parallel line calculator provides a step-by-step solution for particular equations. Let’s take a look to get information about parallel and perpendicular lines.

In mathematics, the perpendicular lines intersect each other at right angles. In simple words, a line is a perpendicular, when one line meets another at 90 degrees.

Conversely, when two lines will never intersect to each other is called parallel lines. The lines maintain to the same distance and never touch all the time.

For instance, the adjacent sides of the square remain perpendicular to each other. However, if two lines are perpendicular to the same line, then these two lines will are parallel and never intersect to each other.

Suppose a line passes through the point (5, 8) and perpendicular to the line \(y = 3x – 6\). We can find the perpendicular equation by following these steps:

Identify the slope (j) and the y-intercept (k) of given line. So, j = 3 and k = -6.

Compute the slope of the line which, is equal to\( a = -1/j = -1/3 = -0.334\)

Substitute the value of a in line equation\( y = ax + b : y = -0.334x + b\)

Plug the coordinates (5, 8) for x and y:

$$8 = -0.334 * 5 + b$$

$$8 = – 1.67 + b$$

$$b = 9.67$$

So, put the value of b in line equation:

$$y = – 0.334x + 9.67$$

When you put the same values in the perpendicular line calculator it’ll provide the same values.

However, an Online Slope Calculator helps to find the slope (m) or gradient between two points in the Cartesian coordinate plane.

By a simple line equation:

$$y = ax + b$$

Where x is the x-coordinate, “a” and “b” are coefficients, and y is the y-coordinate.

Suppose that you already know the following information:

The equation of the line is \(y = mx + r\) which is used by perpendicular line calculator. And the coordinates of the point of the line are \(x_j and y_j\).

If two lines are perpendicular, then, the product of slopes equals -1. Hence,

$$a * m = -1$$

$$a = -1 / m$$

To find the coefficient b also known as the y-intercept, substitute the coordinates (x, y) and the value of “a” into the equation of your line:

$$y = ax + b$$

$$y = -1 . x / m + b$$

$$b = y+ 1 . x / m$$

To find the intersection points between two lines, first identify a point with coordinates \((x_j,y_j)\) which, lies on each of the two lines.

When we found two perpendicular lines: y = 3x – 6 and y = – 0.334x + 9.67. Then, these two equations form the equations with two unknowns.

Let’s solve these equations:

$$Y_j = 3x_j – 6$$

$$Y_j = -0.334x_j + 9.67$$

By multiplying the 9 with the second equation

$$Y_j = 3x_j – 6$$

$$9y_j = -3x_j + 87$$

Sum these two equations,

$$10y_j = 87$$

So,

$$y_j = 8.7$$

$$x_j = 0.5y_j + 1 = 6.96 + 1 = 7.96$$

Hence, the coordinates of the point of intersection are \((7.96, 8.7)\).

Take a simple line equation:

$$y = ax + b$$

Where x is the x-coordinate, y is the y-coordinate, “a” and “b” are coefficients.

And the coordinates of the point of the line are x and y.

If two lines are parallel with the same slop. Then,

$$a = m$$

When you plug in the coordinates \((x_j,y_j)\) and also the value of a into the equation line.

$$y = ax + b$$

$$y_j = m . x_j + b$$

Furthermore, an Online Point Slope Form Calculator will find the equation of a line by using two coordinate points and the slope of the line.

First know the equation of your line, which, you can use to find the distance between it and the other line. However, an online parallel line calculator determines the distance between two parallel lines automatically. Otherwise, you can use this distance formula:

$$D = \frac{|b – r|} { \sqrt(m^2 + 1)}$$

This parallel line equation calculator provides the slope of parallel lines and intercepts points of the slope of a perpendicular line with the following steps:

- First, select the line type from a drop-down list.
- Then, choose the parallel or perpendicular line which, your desire to compute.
- Now, plug in the values in a specific box.
- Click the “Calculate.”

- The perpendicular calculator first displays your entered values, answer, and interception points.
- Then it provides a parallel line equation and the equation of the line.

Perpendicular slopes are –ve reciprocals of each other. In simple words, two lines are parallel when their slopes are equal and they have different y-intercept.

If a line is perpendicular to a line that has slop j, then the slope is -1/j. For instance, the slope of line x = (1/2)y + 3 is 1/2.

A zero (0) slope means there is no change in the x-coordinates as the y-coordinates change, so the line is horizontal.

Use this free online perpendicular line calculator that allows you to find the slope-intercept of the equation of a line and the equation of the parallel line. Additionally, this perpendicular line equation calculator provides the interception points where two lines meet with each other.

From the source of Wikipedia: Foot of a perpendicular, Construction of the perpendicular, In relationship to parallel lines, Graph of functions.

From the source of Splash Learn: What is Perpendicular, Properties of perpendicular lines, perpendicular lines.

From the source of Khan Academy: Equations of parallel and perpendicular lines, Types of Problems, Strategies, Real-life Applications.