Calculate the equation of a perpendicular line. Enter the equation of the original line and a point that the perpendicular line passes through to calculate the perpendicular line equation.
- Perpendicular Bisector Calculator
- All Math and Numbers Calculators
- Slope Calculator
- Midpoint Calculator
Perpendicular Line Formula
Linear lines are almost always displayed in the form of
y = mx + b
Where m is the slope and b is the y-intercept. The first step in finding the equation of a line perpendicular to another is understanding the relationship of their slopes. For non-vertical lines, the slope of a perpendicular line is the negative reciprocal of the other slope. This means that (when both slopes are defined and nonzero) the product of the two slopes is equal to -1.
With that in mind we can formulate the following equations (for m ≠ 0):
a * m = -1
a = -1 / m
Where m is the original slope, and a is the slope of the perpendicular line. Now that we have the slope of our new line, all we need now is the y-intercept of the perpendicular line.
To calculate the perpendicular line’s y-intercept, use a point (x₀, y₀) that the perpendicular line passes through:
b = y₀ − a x₀
- Where a is the perpendicular slope
- b is the y-intercept of the perpendicular line
- y₀ is the y-coordinate of a point on the perpendicular line
- x₀ is the x-coordinate of a point on the perpendicular line
Special cases: if the original line is horizontal (m = 0), the perpendicular line is vertical and has the form x = x₀. If the original line is vertical (x = c), its slope is undefined, and the perpendicular line is horizontal (y = y₀).
What Is a Perpendicular Line?
A perpendicular line is a line that forms a 90-degree angle with another line. Such lines can be drawn in any plane.
On a grid, perpendicular lines can be positioned in any orientation (vertical, horizontal, or slanted) as long as they meet at a right angle.
They don’t have to be pointing upwards; they should only be at a 90-degree angle with respect to another line.
The Difference Between Parallel, Perpendicular, and Intersecting Lines
While it’s easy to confuse the lines with each other, they’re three very different concepts.
Parallel
Lines in a grid that are always the same spacing apart are known as parallel lines. Distinct parallel lines never intersect, and (in the Cartesian plane) they have the same slope (or they are both vertical, with undefined slope).
Perpendicular
Lines that intersect at a right 90-degree angle are known as perpendicular lines. They’re perpendicular if the slope of one (non-vertical) line is the negative reciprocal of the slope of the other.
The only thing that parallel lines and perpendicular lines have in common is that they’re both made up of straight lines.
Intersecting
Intersecting lines are formed when lines in a grid intersect with each other at a point of intersection. However, unlike perpendicular lines, they don’t have to form a right angle.
Do All Shapes Have Perpendicular Lines?
Perpendicular lines can be found in many shapes, but not all of them.
There will be no perpendicular sides on many polygons, but some might have perpendicular lines. Perpendicular lines will always exist in squares, right-angled triangles, and rectangles.
You can also find perpendicular lines on everyday objects such as decorations, fences, and doors.
What Is the Equation for Perpendicular Slope?
The perpendicular slope will be the negative reciprocal of the initial slope (for a non-vertical line with nonzero slope). For example, if the original slope is m, the perpendicular slope is -1/m.
To calculate the line equation, use the slope-intercept equation (y = mx + b) and substitute in the provided point and the new slope to solve for the new intercept.
Then, you can restore the equation to its standard form (ax + by = c), if desired.
Do Perpendicular Lines Have The Same Slope?
The slope of perpendicular lines is not the same. If two lines are perpendicular, one line’s slope is the negative reciprocal of the other line’s slope.
A number’s product and its reciprocal equals 1. When the slopes of two perpendicular (non-vertical) lines in the plane are multiplied, however, the result is -1.
This indicates the slopes of perpendicular lines are reciprocals in the opposite direction.
Do Perpendicular Lines Have to Touch?
Yes. In Euclidean geometry, two lines are perpendicular only if they intersect and form a 90-degree angle at their intersection point.
Sometimes people talk about line segments being perpendicular even if the segments do not meet, meaning that the lines containing the segments would intersect at 90 degrees if extended.
On 3D objects (like a cube), edges that meet at a corner can be perpendicular because they intersect at that corner and form a right angle.
Summary
Two intersecting lines that make a right angle are called perpendicular lines.
The slope of the lines differs because the slope of one (non-vertical) line is the negative reciprocal of the slope of the other.
Perpendicular lines aren’t present in all shapes, although they’re always found in squares, right-angled triangles, and rectangles.
Perpendicular lines must intersect (touch) at a right angle to be deemed perpendicular.
How to calculate a perpendicular line
Let’s look at an example of how to use these equations. First, let’s assume you know the equation of the first line. It happens to be y=4x+5. Let’s also assume you know the x and y coordinates of a point that the perpendicular line passes through, say (4,5).
First, we need to calculate the slope of the perpendicular line. From the equation a = -1 / m we get a value of -1/4.
Next, we need to calculate the y-intercept of the new line using the equation b = y₀ − a x₀. From this, we get a value of 6.
Finally, we need to put this all together in the form: y=-1/4x + 6.
Finding Parallel and Perpendicular Lines
In the 2-Dimension Cartesian Plane, any non-vertical straight line can be represented as an equation of the form y=mx+b, where m is the slope and b is the y-intercept. This makes calculating parallel and perpendicular lines simple for non-vertical lines.
It’s just a matter of manipulating the equation. Let’s look at how the equation is manipulated in order to calculate the equation of a perpendicular line.
Conceptually, a perpendicular line is a line that intersects the original line and forms a 90-degree angle at the point of intersection. Since there can be an infinite number of lines with the correct perpendicular slope, calculating a specific perpendicular line requires a point that the perpendicular line passes through.
The slope of a perpendicular (non-vertical) line is the negative reciprocal of the slope of the original line. This simply means if the original slope is m, then the perpendicular slope is -1/m.
y=mx+b —> y₂=(-1/m)x + b₂
The only thing left to do is solve for b₂ using the point given along the perpendicular line.
Steps to calculate a perpendicular line
- Calculate the slope of the original line. This can be done through the use of two points along the original line. To learn more about calculating the slope of a line click here.
- Take the negative reciprocal of the original slope. If the slope is m (and m ≠ 0), the perpendicular slope is -1/m.
- Calculate the y-intercept of the new line using the new slope and the point given along that line (for example, b = y₀ − a x₀).
- Write the equation in point-slope form or slope-intercept form.
For more math related calculators, click here.
