Perpendicular Lines

¹ How to find a perpendicular line?

Every straight line in two-dimensional space can be described by a simple line equation:

$y=ax+b$

where $a$ and $b$ are coefficients, $x$ is the x-coordinate, and $y$ is the y-coordinate. Every line is uniquely defined if the values of $a$ and $b$ are known.

Let’s assume that you know the following information:

1. The equation of the given line is $y=mx+r$. You know the values of $m$ and $r$ and are looking for a line perpendicular to this one.
2. You also know the coordinates of the point your line is supposed to pass through. They are $x_0$ and $y_0$.

The slope of any line is equal to the value of the a coefficient. If two lines are perpendicular, the product of their slopes equals -1. Hence,

$a\times m=-1$

$a=-1/m$

To find the $b$ coefficient (also known as the y-intercept), you have to substitute in the coordinates $(x_o,y_o)$ and the value of $a$ into the equation of your line:

$y=ax+b$

$y_O=-1\times x_O/m+b$

$b=y_b+1\times x_b/m$

Use Slopes if you are in need of $M$

Perpendicular line equation: an example

How do these calculations look in practice? Let’s assume that you want your line to pass through the point (3, 5) and be perpendicular to the line $y=2x-2$. You can find the perpendicular line equation when following these steps:

1. Identify the slope (m) and the y-intercept (r) of the given line. In this case, $m=2$ and $r=-2$.

2. Calculate the slope of your line. It is equal to

   a = -1 / m = -1/2 = -0.5

3. Input this value into the line equation $y=ax+b$:

   $y=-0.5x+b$

4. Substitute the coordinates (3,5) for the values of $x$ and $y$:

   5 = -0.5 × 3 + b

   5 = -1.5 + b

   b = 6.5

5. As the last step, input the $b$ coefficient into the line equation:

   y = -0.5x + 6.5

Don’t believe it? Check the result with this perpendicular line calculator!

Finding the intersection point

Once you know the equation of the new line, finding the intersection point between it and the first (given) line is a straightforward task. All you have to do is find a point with coordinates (xₐ,yₐ) such that it lies on each of the two lines.

Consider the example we’ve just analyzed. We found two perpendicular lines: y = 2x - 2 and y = -0.5x + 6.5. These two equations form a system of equations with two unknowns - the coordinates of the point of intersection.

Let’s solve this system of equations:

yₐ = 2xₐ - 2

yₐ = -0.5xₐ + 6.5

Multiplying the second equation by 4, you get

yₐ = 2xₐ - 2

4yₐ = -2xₐ + 26

Adding the two equations together,

5yₐ = 24

From there,

yₐ = 4.8

xₐ = 0.5yₐ + 1 = 2.4 + 1 = 3.4

The coordinates of the point of intersection are (3.4, 4.8).

Of course, you don’t have to carry out these tedious calculations all by yourself – our perpendicular line calculator can do the same in just a few seconds! And don’t forget to check our other coordinate geometry calculators like the average rate of change calculator.

FAQ

What are perpendicular lines?

We say that two lines are perpendicular if they intersect each other at a right angle (90°). You can find perpendicular lines in many geometric figures, such as squares, rectangles, and triangles. We can also use them to describe the orientation of lines, planes, and surfaces in three-dimensional space.

How do I verify if two lines are perpendicular?

To check if two lines are perpendicular, follow these steps:

1. Write down the slopes of these lines. (Recall the slope is the coefficient a standing next to the variable x in the formula y = ax + b.)
2. Multiply the two slopes together.
3. If the product is equal to -1, your lines are perpendicular. If not, they are not perpendicular.

Is the line y = 5x perpendicular to the line y = 0.2x - 1?

No, these two lines are not perpendicular. The product of their slopes is equal to 1. Recall that the product of slopes must be equal to -1 for the lines to be perpendicular. The sign matters!

Do two perpendicular lines always intersect?

Yes, if some two lines are perpendicular, there exists a point where they intersect. However, non-perpendicular lines may also intersect! It is only parallel lines that do not intersect (in standard geometry, at least).

Footnotes

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