Perpendicular Line Equation Calculator

Calculator

Original line input type

Original slope m

Used in y = mx + b.

Original intercept b

Standard coefficient A

Standard coefficient B

Standard coefficient C

Used in Ax + By + C = 0.

First point x1

First point y1

Second point x2

Second point y2

Point x-coordinate

Point y-coordinate

Decimal precision

Example Data Table

Original Line	Point	Original Slope	Perpendicular Slope	Perpendicular Equation
y = 2x + 1	(3, -2)	2	-1/2	y = -0.5x - 0.5
3x + 4y - 8 = 0	(2, 5)	-3/4	4/3	y = 1.3333x + 2.3333
x = 6	(1, 4)	Undefined	0	y = 4
y = -3	(7, 1)	0	Undefined	x = 7

Formula Used

For a nonvertical original line, the perpendicular slope is found with this rule:

m_{perpendicular} = -1 / m_original

After the perpendicular slope is found, the point-slope equation is:

y - y₁ = m_{perpendicular}(x - x₁)

For standard form, use Ax + By + C = 0. Its slope is -A / B, when B is not zero.

The foot of perpendicular from a point uses projection on the original line. The distance is |Ax + By + C| / √(A² + B²).

How to Use This Calculator

Select the original line input type.
Enter the slope, standard coefficients, or two points.
Enter the point where the perpendicular line must pass.
Choose decimal precision for rounded output.
Press Calculate to view the result above the form.
Use CSV or PDF buttons when you need a saved copy.

Understanding Perpendicular Lines

A perpendicular line meets another line at a right angle. This angle measures ninety degrees. The idea is simple, yet it is very useful. It appears in geometry, coordinate graphs, surveying, design, construction, robotics, and many classroom problems.

The most important fact is the slope relationship. When two nonvertical lines are perpendicular, their slopes are negative reciprocals. If the first slope is m, the perpendicular slope is -1 divided by m. This rule makes equation building fast. It also helps students check answers without drawing a graph.

Flexible Input Methods

This calculator accepts several common starting formats. You can enter a slope and intercept. You can enter a standard form line. You can also enter two points from the original line. Then you choose the point through which the perpendicular line must pass. The tool converts the original line into a slope model when possible. It also handles horizontal and vertical cases.

Special Line Cases

Special cases matter. A vertical line has no finite slope. Its perpendicular line is horizontal. A horizontal line has slope zero. Its perpendicular line is vertical. These cases often cause mistakes in manual work. The calculator reports them directly and gives clean equation forms.

Advanced Geometry Outputs

The result includes point-slope form, slope-intercept form when available, and standard form. It also shows the original slope, perpendicular slope, angle data, foot of perpendicular, and distance from the chosen point to the original line. These extra values are useful for analytic geometry tasks.

Use the answer as a guide, not only as a final number. Review the steps beside the result. Compare the displayed formula with your textbook method. Change the precision setting when you need rounded decimals. Use exact input values when possible, because early rounding can change later results.

Practice and Export Use

For deeper study, test parallel lines too. Parallel lines keep the same slope. Perpendicular lines reverse and invert slope. Seeing both patterns together improves graph reading and equation fluency. It also strengthens checks during exams and projects with less doubt.

The export buttons help with records. CSV is useful for spreadsheets. PDF is useful for printed homework notes. The example table below shows how different original lines create different perpendicular equations. This makes the calculator useful for practice, teaching, and quick verification.

FAQs

What is a perpendicular line?

A perpendicular line crosses another line at a ninety degree angle. On a coordinate plane, its slope has a special relationship with the original line slope.

What is the perpendicular slope rule?

For nonvertical lines, multiply the original slope by the perpendicular slope. The product equals -1. So the perpendicular slope is the negative reciprocal.

Can this calculator handle vertical lines?

Yes. If the original line is vertical, the perpendicular line is horizontal. The result becomes y equals the chosen point’s y-coordinate.

Can this calculator handle horizontal lines?

Yes. If the original line is horizontal, the perpendicular line is vertical. The result becomes x equals the chosen point’s x-coordinate.

Which equation forms are shown?

The tool shows point-slope form, slope-intercept form when possible, and standard form. Vertical lines do not have slope-intercept form.

Why enter a point?

Many perpendicular lines can share one slope. The point fixes the exact location, so only one perpendicular line is produced.

What does the foot of perpendicular mean?

It is the point where the perpendicular line touches the original line. It is useful for shortest distance and projection problems.

What does decimal precision control?

It controls rounding in the displayed results. Higher precision gives more decimal places. Lower precision creates shorter classroom-friendly answers.