Perpendicular Line Calculator

What this calculator does

So, you have a line and a point, and you want the line that passes through that point at a right angle to the first. This tool builds it. You give it the point, plus the slope and y-intercept of the existing line, and it returns the equation of the perpendicular line and the point where the two lines cross.

Four inputs, the point's coordinates and the original line's slope and intercept, and out come the perpendicular equation and the intersection.

How to use it

1. Enter the point the perpendicular line should pass through.
2. Enter the slope and y-intercept of the existing line.
3. Press Calculate.

What perpendicular means for lines

Two lines are perpendicular when they meet at a right angle, a perfect ninety degrees, like the corner of a square or the cross of a plus sign. Perpendicularity is one of the most important relationships in geometry, underlying everything from the axes on a graph to the walls of a building. Given any line and a point, there is exactly one line through that point that stands at a right angle to it, and this tool finds that unique perpendicular.

The negative reciprocal slope

The heart of the calculation is a neat rule about slopes. If a line has a certain slope, the slope of any line perpendicular to it is the negative reciprocal: you flip the slope upside down and change its sign. So a slope of 2 becomes minus one half, and a slope of minus three becomes one third. The reason is that turning a line by a right angle swaps its rise and run and reverses the direction, which is exactly what taking the negative reciprocal does to the number. A handy check is that a slope multiplied by its perpendicular partner always gives minus one.

Pinning the line to a point

Knowing the perpendicular slope tells you the direction of the new line, but not which of the many parallel lines with that slope you want. That is where the point comes in. There is only one line with the perpendicular slope that also passes through your chosen point, and the tool locks onto it by using the point to work out the line's own y-intercept. The result is a complete equation, giving both the slope and the intercept of the perpendicular line, ready to use or graph.

Where the two lines meet

The tool also finds the point where the perpendicular line crosses the original one. Since perpendicular lines meet at a right angle, they share exactly one point, and that crossing is often the whole reason for drawing the perpendicular. It is the foot of the perpendicular from your point down to the original line, which is the closest point on that line to yours. Finding it is a matter of setting the two line equations equal and solving, which the tool does for you, reporting the intersection as a coordinate pair.

A worked example

Take the point 2 comma 3 and an existing line with slope 2 and y-intercept 1, which is y equals 2x plus 1. The perpendicular slope is the negative reciprocal of 2, which is minus one half. The perpendicular line through 2 comma 3 works out to y equals minus one half x plus 4. Setting the two equations equal to find where they cross gives the point 1.2 comma 3.4, the foot of the perpendicular on the original line.

Questions people ask

What does this tool find?

The line through a given point at a right angle to an existing line, plus the point where the two cross.

How is the perpendicular slope found?

It is the negative reciprocal of the original slope: flip the number over and change its sign. A slope of 2 becomes minus one half.

Is there a quick check on the slopes?

Yes. A slope and its perpendicular partner multiply to give minus one.

Why do I need to give a point?

Because many lines share the perpendicular slope. The point selects the single one that passes through it.

What is the intersection point?

The single place where the two lines cross, which is the closest point on the original line to your given point.

References

On perpendicular lines. Perpendicular lines meet at a right angle, and their slopes are negative reciprocals of each other.

1. Eric W. Weisstein, "Perpendicular," from MathWorld, a Wolfram resource, on perpendicular lines and the negative reciprocal relationship between their slopes.
2. Eric W. Weisstein, "Slope-Intercept Form," from MathWorld, a Wolfram resource, on the slope and y-intercept form in which the perpendicular line is given.