Calculator form
Choose one input mode for the original line, then provide the point through which the perpendicular line must pass.
Example data table
| Original Line | Given Point | Perpendicular Line | Distance to Original | Angle |
|---|---|---|---|---|
| y = 2x + 3 | (4, 1) | y = -0.5x + 3 | 1.341641 | 90° |
| x - 5 = 0 | (2, -1) | y = -1 | 3 | 90° |
| 3x + 2y - 8 = 0 | (1, 5) | 2x - 3y + 13 = 0 | 2.773501 | 90° |
Formula used
Slope relationship
For non-vertical and non-horizontal lines, perpendicular slopes satisfy m₁ × m₂ = -1. So the perpendicular slope becomes m₂ = -1 / m₁.
From standard form
If the original line is Ax + By + C = 0, then a perpendicular line through (x₀, y₀) is Bx - Ay + (Ay₀ - Bx₀) = 0.
Intersection point
The calculator solves the original line and the generated perpendicular line together. Their solution gives the exact intersection, which is the foot of the perpendicular.
Point-to-line distance
Distance from (x₀, y₀) to Ax + By + C = 0 is |Ax₀ + By₀ + C| / √(A² + B²).
How to use this calculator
- Choose how you want to describe the original line.
- Enter the needed values for that mode.
- Provide the point through which the perpendicular line must pass.
- Add an optional comparison line if you want a direct check.
- Press the calculate button to see the result above the form.
- Review slope values, equation forms, distance, intersection, and the graph.
- Use the CSV or PDF buttons to save the output.
FAQs
1) What makes two lines perpendicular?
Two lines are perpendicular when they meet at a right angle. For regular slopes, their slope product equals negative one. Vertical and horizontal lines are also perpendicular to each other.
2) Can I use coordinates instead of an equation?
Yes. Choose the two-point mode, enter two coordinates on the original line, then enter the point through which the perpendicular line should pass.
3) Does the calculator support vertical lines?
Yes. Vertical lines have undefined slope, so the perpendicular line becomes horizontal. The result panel shows equation forms that still make sense for that case.
4) Why is slope-intercept form sometimes unavailable?
A vertical line cannot be written as y = mx + b. In that situation, the calculator displays the standard form and a vertical equation such as x = 4.
5) What does the distance result mean?
It is the shortest distance from your given point to the original line. If the point already lies on the original line, the distance becomes zero.
6) What is the intersection point used for?
The intersection is where the original line meets the generated perpendicular line. It is useful for geometry work, projections, and shortest-path calculations.
7) Can I compare another line against the result?
Yes. Enter an optional comparison line in standard form. The calculator will test whether that line is perpendicular to the original and whether it matches the generated line.
8) Is this useful for school and exam practice?
Yes. It helps verify manual answers, understand line relationships, and review how slope, standard form, intersection, and distance connect in coordinate geometry.