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Parallel and Perpendicular line calculator

Select the equation format, the parameter to be measured, and input the required entities. The calculator will determine the lines that may be parallel or perpendicular to the equation passing through a certain set of points, with the steps displayed.

Input Line as:

Ax + By + C = 0

Point A

Point B

Point C

Input Point

Enter 1

Enter 2

What to compute:

Related

The online Parallel and Perpendicular Line Calculator helps you find the equation of a line parallel or perpendicular to a given line that passes through a specified point. This free tool also provides step-by-step solutions for all calculations.

What are Perpendicular and Parallel Lines?

Perpendicular lines intersect at right angles (90°). Key properties:

They always meet at a 90° angle.
All perpendicular lines are intersecting lines, but not all intersecting lines are perpendicular.

Parallel lines never intersect and remain equidistant. Key properties:

They maintain a constant distance from each other.
They lie in the same plane.
They never meet at any point.

Example: Adjacent sides of a square are perpendicular. Two lines perpendicular to the same line are parallel. You can calculate these properties quickly using the online Parallel and Perpendicular Line Calculator.

Perpendicular Line Equation Examples

Example #1: Find the equation of a line passing through (5, 8) perpendicular to $y = 3x - 6$.

Solution:

Slope of given line: $m = 3$
Slope of perpendicular line: $a = -1/m = -1/3 \approx -0.333$
Equation: $y = ax + b \Rightarrow y = -0.333x + b$
Substitute point (5, 8): $8 = -0.333(5) + b \Rightarrow b = 9.665$
Perpendicular line equation: $y = -0.333x + 9.665$

Example #2: Determine if the lines through points (0, –3) & (–1, –7) and (3, 0) & (–5, 2) are parallel or perpendicular.

Solution:

Slope of first line: $m_1 = (-7 + 3)/(-1 - 0) = 4$
Slope of second line: $m_2 = (2 - 0)/(-5 - 3) = -1/4$
Check parallel: $m_1 \neq m_2 \Rightarrow$ not parallel
Check perpendicular: $m_1 \cdot m_2 = 4 \cdot (-1/4) = -1 \Rightarrow$ lines are perpendicular

You can verify these results with a Slope Calculator or the perpendicular/parallel line calculator.

Finding the Slope of a Perpendicular Line

If the original line is $y = mx + r$, the slope of a perpendicular line is:

$$a = -1/m$$

To find the y-intercept $b$, use a point $(x_j, y_j)$ on the line:

$$b = y_j - a x_j$$

Finding the Intersection Point of Perpendicular Lines

Example: Lines $y = 3x - 6$ and $y = -0.334x + 9.67$

Set equations equal: $3x - 6 = -0.334x + 9.67$
Combine terms: $3.334x = 15.67 \Rightarrow x \approx 4.7$
Substitute x into first equation: $y = 3(4.7) - 6 \approx 8.1$
Intersection point: $(4.7, 8.1)$

Finding the Slope of a Parallel Line

For parallel lines, slopes are equal: $a = m$. Using a point $(x_j, y_j)$, the equation is:

$$y_j = m x_j + b \Rightarrow b = y_j - m x_j$$

You can also use an Online Point Slope Form Calculator to find the line equation.

Distance Between Two Parallel Lines

Given parallel lines $y = mx + b_1$ and $y = mx + b_2$, distance is:

$$D = \frac{|b_2 - b_1|}{\sqrt{m^2 + 1}}$$

How the Calculator Works

Input:

Select the type of line.
Choose whether to compute a parallel or perpendicular line.
Enter the values in the input fields.
Click “Calculate”.

Output:

Displays entered values, slopes, and intersection points.
Provides the equation of the parallel or perpendicular line.

FAQ

Rule of Parallel Lines

Two lines are parallel if their slopes are equal but y-intercepts differ. Learn more.

Slope of Perpendicular Line

If a line has slope $m$, a perpendicular line has slope $-1/m$.

Is a Zero Slope Linear?

Yes. A zero slope means the line is horizontal, i.e., y remains constant while x changes.

Conclusion

This free online Parallel and Perpendicular Line Calculator efficiently computes slope-intercept forms, parallel and perpendicular line equations, and intersection points with accurate step-by-step solutions.

References

Wikipedia: Perpendicular lines

Splash Learn: Properties of perpendicular lines

Khan Academy: Equations of parallel and perpendicular lines

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