Algebra: standard slopes
Base line y = 2x + 1 through point (3, 4).
- Perpendicular slope: a = -1/2
- Intercept: b = 4 - (-0.5)(3) = 5.5
- Intersection: Solve 2x + 1 = -0.5x + 5.5 → (1.8, 4.6)
Answer: y = -0.5x + 5.5; intersection (1.8, 4.6).
Coordinate geometry desk
Perpendicular Line Calculator
Calculate the perpendicular line through any point, find where it meets the original line, and check your algebra homework in seconds. Built for coordinate geometry, engineering sketches, and exam prep.
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Perpendicular Line Calculator
Enter the slope m and intercept r for y = mx + r, then the point (x, y). The perpendicular equation and intersection update instantly in your browser.
First line
y = mx + r
Second line passes through point
Perpendicular line
y = ax + b
-
Point of intersection
Using this calculator
- Type slope m and intercept r for the first line y = mx + r.
- Enter the x and y coordinates of the point the perpendicular must pass through.
- Read a, b, and the intersection coordinates as soon as all four inputs are valid numbers.
What Is a Perpendicular Line?
A perpendicular line crosses another line at a 90-degree angle. In the coordinate plane, that right-angle relationship appears as a slope rule: the product of the two slopes equals -1 when neither line is vertical.
Perpendicular lines show up in algebra, analytic geometry, construction layouts, and CAD drawings whenever a design must meet a wall, axis, or reference line at a square corner.
On this page you learn the definition, the formula, manual steps, and how the calculator above applies the same algebra automatically.
Definition
Two lines are perpendicular if they intersect and form a right angle.
Meaning in slope form
If one slope is m, a non-vertical perpendicular slope is -1/m.
Real-world use
Floor plans, road offsets, beam supports, and graphing problems all rely on perpendicular direction.
Perpendicular Line Formula
Core slope relationship: m1 · m2 = -1
Given base line: y = mx + r
Perpendicular slope: a = -1/m (when m ≠ 0)
Through point (x0, y0): b = y0 - a·x0
Perpendicular equation: y = ax + b
Intersection: mx + r = ax + b
x = (b - r) / (m - a), then y = mx + r
The negative reciprocal is the heart of perpendicular slope work. Multiply m by a and you should get -1 for non-vertical pairs.
If m = 0 the base line is horizontal, so the perpendicular through (x0, y0) is vertical: x = x0. The calculator handles that case without dividing by zero.
For a deeper walkthrough see our perpendicular line formula article.
How to Calculate a Perpendicular Line
Follow the same sequence as the calculator near the top of this page: identify the base slope, flip it with the negative reciprocal, build the line through your point, then solve for the intersection.
- Find the original slope: Read m from y = mx + r.
- Compute the negative reciprocal: Set a = -1/m when m is not zero.
- Use point-slope logic: Substitute (x0, y0) into y = ax + b to find b.
- Write the perpendicular equation: State y = ax + b or x = x0 for horizontal bases.
- Find the intersection: Set mx + r equal to ax + b and solve for x, then y.
- Check with the tool: Enter the same numbers in the calculator to confirm rounding.
Perpendicular Line Examples
Five worked cases cover algebra, graphing, and applied geometry. Plug each set of values into the calculator to verify.
Coordinate geometry: negative base slope
Base line y = -3x + 2 through point (1, 5).
- Perpendicular slope: a = 1/3
- Intercept: b = 14/3
- Intersection: (11/12, 13/12)
Answer: y = (1/3)x + 14/3; intersection (11/12, 13/12).
Horizontal and vertical case
Base line y = 4 (m = 0) through point (2, 7).
- Perpendicular: Vertical line x = 2
- Intersection: Meets y = 4 at (2, 4)
Answer: Perpendicular x = 2; intersection (2, 4).
Engineering offset
A path follows y = 0.25x - 1. A survey stake sits at (40, 8). Find the perpendicular through the stake.
- Slope: a = -4
- Line: b = 8 - (-4)(40) = 168 → y = -4x + 168
- Foot of perpendicular: Intersection with y = 0.25x - 1 gives the closest point on the path
Answer: Perpendicular y = -4x + 168; use the calculator for decimal intersection coordinates.
Graphing check
Plot y = x + 2 and the perpendicular through (0, 3).
- Slope: a = -1
- Equation: y = -x + 3
- Intersection: (0.5, 2.5) where lines cross at a right angle
Answer: Perpendicular y = -x + 3; intersection (0.5, 2.5).
Perpendicular Slope Calculator
The perpendicular slope is the negative reciprocal of the base slope. When the base slope is m, a finite perpendicular slope is a = -1/m, which keeps m · a = -1.
Undefined slope cases matter: a horizontal line (m = 0) pairs with a vertical perpendicular. A vertical base line pairs with a horizontal perpendicular. The calculator above switches to x = x0 when m = 0.
Use this section when you only need the slope relationship before building a full equation. For practice problems see perpendicular slope calculator on the blog.
Negative reciprocal
Flip the fraction and change sign: 3/4 becomes -4/3.
Product test
Multiply slopes; a result of -1 signals perpendicular (non-vertical cases).
Sign analysis
Opposite signs on slopes often indicate a right angle in sketch work.
Perpendicular Line Equation
Most students start with slope-intercept form y = ax + b after finding a = -1/m. Point-slope form y - y0 = a(x - x0) is equivalent and useful when the point is the focus.
Standard form Ax + By + C = 0 is common in textbooks and engineering sheets. You can rearrange y = ax + b into standard form with integer coefficients when required.
Read perpendicular line equation for conversions between forms and simplification tips.
Point-slope: y - y0 = a(x - x0)
Slope-intercept: y = ax + b, b = y0 - a·x0
Standard form: rearrange to Ax + By + C = 0
Perpendicular vs Parallel Lines
Parallel lines never meet (unless they are the same line) and share the same slope. Perpendicular lines meet once at a right angle and use slopes whose product is -1.
Graphically, parallel lines look like railroad tracks; perpendicular lines form an L or T shape with square corners.
Compare both ideas in perpendicular vs parallel lines when you study systems of linear equations.
| Feature | Perpendicular | Parallel |
|---|---|---|
| Slope relation | m1 · m2 = -1 | m1 = m2 |
| Intersection | One point (usually) | None or infinite |
| Angle | 90° | 0° |
| Typical use | Heights, offsets, normals | Lanes, rails, copies |
Graphing Perpendicular Lines
Plot the base line first, mark the given point, then draw the perpendicular with the negative reciprocal slope.
On graph paper, count slope as rise over run for the base line, then swap and flip for the perpendicular. A base slope of 2/3 becomes a perpendicular slope of -3/2.
The intersection point is where the two lines cross. If your graph is accurate, the angles should look square. Algebra gives the exact intersection; see graphing perpendicular lines for plotting tips.
- Plot the base line: Use m and r from y = mx + r.
- Mark the point: Place (x0, y0) on the grid.
- Draw the perpendicular: Use slope a = -1/m through the point.
- Label intersection: Solve algebraically or read from a careful sketch.
Perpendicular Line Calculator
The interactive panel above accepts the base line as y = mx + r plus a point (x, y). It returns the perpendicular equation y = ax + b (or x = x0 when the base is horizontal) and the intersection coordinates.
Original line equation input uses slope m and intercept r. Point coordinate input uses separate x and y fields. Instant output includes a, b, and the intersection without sending data to a server.
Example calculations on this page walk through linear, horizontal, and applied cases. Scroll to perpendicular line examples for numbered practice, or open the blog article how the calculator works.
Common Perpendicular Line Mistakes
Avoid these algebra and graphing errors when slopes, signs, and forms mix together.
- Using the reciprocal without changing sign (3/4 is not perpendicular to 4/3; you need -4/3).
- Dividing by zero when m = 0 instead of recognizing a vertical perpendicular.
- Mixing up which point belongs to which line in word problems.
- Forgetting to solve for intersection after finding the perpendicular equation.
- Plotting rise/run on the wrong axis when graphing by hand.
Perpendicular Lines in Coordinate Geometry
Coordinate geometry connects algebra and diagrams. Perpendicular lines tie together slope formula, distance to a line, and intersection points in the plane.
You will see perpendiculars in proofs about rectangles, in finding shortest distance from a point to a line, and in checking whether corners in a polygon are square.
Strengthen foundations with perpendicular lines in coordinate geometry and related slope topics linked throughout this site.
Slope formula
m = (y2 - y1) / (x2 - x1) for non-vertical segments.
Distance
Perpendicular segments often represent shortest distance to a line.
Systems
Intersection solves a linear system of two equations.
FAQs About Perpendicular Lines
What is the perpendicular slope formula?
For non-vertical lines, multiply slopes to get -1: m1 · m2 = -1. So a = -1/m when m is the base slope.
How do I find the equation of a perpendicular line?
Compute a = -1/m, then use y = ax + b with b = y0 - a·x0 for the point (x0, y0).
Where do perpendicular lines intersect?
Set the two equations equal and solve. The calculator returns that point automatically.
What if the line is horizontal?
A horizontal base y = r has perpendicular x = x0 through (x0, y0), meeting the base at (x0, r).
How is perpendicular different from parallel?
Perpendicular slopes multiply to -1; parallel slopes are equal. Perpendicular lines meet at 90°.
Can I use this for engineering or construction?
Yes for layout sketches and quick checks, but verify critical builds with licensed plans and field measurements.
Does the point have to sit on the first line?
No. The tool builds the perpendicular through your point, then finds where it crosses the base line.