Parallel and Perpendicular Lines

How to use Algebra to find parallel and perpendicular lines

Parallel Lines

How do we know when two lines are parallel?

Their slopes are the same!

The slope is the value m in the equation of a line:

y = mx + b

Two parallel lines with slope 2 on a coordinate plane

Example:

Find the equation of the line that's:

parallel to y = 2x + 1
and passes though the point (5,4)

The slope of y = 2x + 1 is 2

The parallel line needs to have the same slope of 2.

We can solve it by using the "point-slope" equation of a line:

y − y₁ = 2(x − x₁)

And then put in the point (5,4):

y − 4 = 2(x − 5)

That's an answer!

But it might look better in y = mx + b form. Let's expand 2(x − 5) and then rearrange:

y − 4 = 2x − 10

y = 2x − 6

Vertical Lines

But this doesn't work for vertical lines ... I explain why at the end.

Not The Same Line

Be careful! They may be the same line (but with a different equation), and so are not parallel.

Parallel lines are always the same distance apart and never meet.

How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:

Example: is y = 3x + 2 parallel to y − 2 = 3x ?

For y = 3x + 2: the slope is 3, and y-intercept is 2

For y − 2 = 3x: the slope is 3, and y-intercept is 2

So they are the same line and so aren't parallel

Try it!

See parallel lines here (try the sliders):

images/function-graph.js?fn0=mx+3&fn1=mx+b&xmin=-5.294&xmax=5.259&ymin=-1.447&ymax=4.553&varm=1|-10|10&varb=1|-10|10

Perpendicular Lines

Two lines are perpendicular when they meet at a right angle (90°).

To find a perpendicular slope:

When one line has a slope of m, a perpendicular line has a slope of −1m

In other words the negative reciprocal

Perpendicular lines with slopes -4 and 1/4

Example:

Find the equation of the line that's

perpendicular to y = −4x + 10
and passes though the point (7,2)

The slope of y = −4x + 10 is −4

The negative reciprocal of that slope is:

m = −1−4 = 14

So the perpendicular line will have a slope of 1/4:

y − y₁ = (1/4)(x − x₁)

And now we put in the point (7,2):

y − 2 = (1/4)(x − 7)

That answer is OK, but let's also put it in "y=mx+b" form:

y − 2 = x/4 − 7/4

y = x/4 + 1/4

Try it!

See perpendicular lines here (try the sliders):

images/function-graph.js?fn0=mx+3&fn1=(-1/m)x+b&xmin=-5.294&xmax=5.259&ymin=-1.447&ymax=4.553&varm=1|-10|10&varb=1|-10|10

Quick Check of Perpendicular

When we multiply a slope m by its perpendicular slope −1m we get simply −1.

So to quickly check if two lines are perpendicular:

When we multiply their slopes, we get −1

Like this:

Perpendicular lines with slopes 2 and -0.5

Are these two lines perpendicular?

Line	Slope
y = 2x + 1	2
y = −0.5x + 4	−0.5

When we multiply the two slopes we get:

2 × (−0.5) = −1

Yes, we got −1, so they are perpendicular.

Vertical and Horizontal Lines

The previous methods work nicely except for a vertical line:

Vertical line passing through the point (2,0)

In this case the gradient is undefined (as we can't divide by 0):

m = y_A − y_Bx_A − x_B = 4 − 12 − 2 = 30 = undefined

And for a horizontal line the slope is 0, so the negative reciprocal rule
−1m doesn't work here because we would get −10 which is undefined

And our quick check also fails because 0 × undefined isn't −1.

So just rely on the fact that:

Vertical lines are parallel to each other
Horizontal lines are parallel to each other
Vertical is perpendicular to Horizontal (and vice versa)

Summary

Parallel lines: same slope (m)
Perpendicular lines: negative reciprocal slope (−1/m)

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