Enter Line and Point Values
Formula Used
The starting line is written as y = mx + b.
The perpendicular slope is m₂ = -1 / m, when m is not zero.
The new intercept is b₂ = y₁ - m₂x₁.
The perpendicular equation is y = m₂x + b₂.
If the original slope is zero, the original line is horizontal. The perpendicular line is vertical, so the equation is x = x₁.
The foot point uses x = (x₁ + m(y₁ - b)) / (m² + 1) and y = mx + b.
The distance from the point to the original line is |mx₁ - y₁ + b| / √(m² + 1).
How to Use This Calculator
- Enter the original slope from y = mx + b.
- Enter the original y intercept.
- Enter the point that the perpendicular line must pass through.
- Add an x value for evaluation, if needed.
- Choose decimal places and fraction display.
- Press the calculate button.
- Review the final equation, point slope form, foot point, and distance.
- Download CSV or PDF output for records.
Example Data Table
| Original Line | Point | Perpendicular Slope | Final Perpendicular Line |
|---|---|---|---|
| y = 2x + 3 | (4, 1) | -1/2 | y = -0.5x + 3 |
| y = -3x + 5 | (2, -1) | 1/3 | y = 0.3333x - 1.6667 |
| y = 0x + 7 | (5, 2) | undefined | x = 5 |
| y = 0.25x - 4 | (8, 3) | -4 | y = -4x + 35 |
Article: Perpendicular Lines in Slope Intercept Form
Why Perpendicular Lines Matter
A perpendicular line is more than a rotated graph. It shows the direction that meets another line at a square corner. This calculator starts with the slope intercept form y = mx + b. You enter the original slope, the original intercept, and a point that the new line must pass through. The tool then creates the perpendicular equation.
The Main Slope Rule
The key idea is the negative reciprocal slope. When the original slope is m, the perpendicular slope is -1 divided by m. This rule works for every nonzero slope. If the original line is horizontal, its slope is zero. The perpendicular line becomes vertical, so the result is written as x = value.
Why the Point Is Required
The point matters because many lines can be perpendicular to the same original line. Only one of them passes through your chosen point. After finding the new slope, the calculator solves the intercept by using b = y - mx. It also shows the equation in slope intercept form and point slope form.
Advanced Checks
Advanced results help you check the answer. The dot product slope test confirms perpendicularity. The foot of the perpendicular shows where the new line meets the original line. Distance from the point to the original line is also shown. These values are useful in geometry, drafting, coordinate proofs, and layout checks.
Input and Export Tips
Use exact input when possible. Fractions, integers, and decimals are supported. A fraction view can make classwork easier to read. Decimal rounding helps when measurements come from real projects. The evaluation field also finds a y value on the perpendicular line for any selected x.
Careful Review
The calculator is designed for careful work. It separates givens, formulas, and final results. You can export the result as a CSV file or a PDF note. That makes it simple to save calculations, compare cases, or attach work to an assignment. Always review vertical and horizontal cases. They are valid answers, but they do not always fit y = mx + b.
Accuracy Practice
For best accuracy, match units before you start. Use the same scale for both coordinates. A map point in meters should not be mixed with inches. Record the chosen point clearly. Then compare the plotted crossing point with the displayed foot. If both lines meet there, the setup is consistent and visually ready.
FAQs
What is a perpendicular line?
A perpendicular line meets another line at a right angle. On a coordinate plane, its slope is the negative reciprocal of the original slope, except when a horizontal or vertical case appears.
What does y = mx + b mean?
It is slope intercept form. The letter m is the slope. The letter b is the y intercept. This form shows how the line rises, falls, and crosses the y axis.
How do I find the perpendicular slope?
Use the negative reciprocal. Change the slope into a fraction, flip it, and change the sign. For example, a slope of 2 becomes -1/2.
Why does the calculator need a point?
A slope alone gives direction, not location. Many perpendicular lines can share the same slope. The point tells the calculator exactly where the required perpendicular line must pass.
What happens when the original slope is zero?
The original line is horizontal. Its perpendicular line is vertical. A vertical line cannot be written as y = mx + b, so the calculator returns x = selected point x.
Can I enter fractions?
Yes. You can enter values like 1/2, -3/4, decimals, or whole numbers. The fraction display option also shows friendly fraction approximations beside decimal results.
What is the foot of the perpendicular?
It is the point where the perpendicular line meets the original line. It is useful for distance problems, projection work, coordinate geometry, and graph checking.
What do the download buttons do?
The CSV button saves the result as spreadsheet-ready data. The PDF button creates a simple printable note with the main equation, checks, distance, and related values.