Creating Equations for Parallel and Perpendicular Lines

• Find the equation of a line parallel or perpendicular to a given line that passes through a given point

Now that we've established the slope criteria for parallel and perpendicular lines, we can create our own equations.

Example 1

Give the equation of a line that passes through C and is parallel to line AB

So, we need to make an equation for a line. Well, we know that the equation for a line is

y = mx + b

So for our new line, we need to figure out m and b

m is pretty easy. We know know that for parallel lines, the slopes are equal, so we just need to figure out the slope of line AB

We can choose any two points on line AB that we want. So it's worth taking a moment to think about what would be easiest. We could use A and B, but it would be even easier to use the x-intercept and the y-intercept. So, I'm going to say

• We'll let point 1 be the x-intercept, which occurs at (4, 0).

  • This means x₁ = 4 and y₁ = 0

• We'll let point 2 be the y-intercept, which occurs at (0, 8).

  • This means x₂ = 0 and y₂ = 8

Plugging into our slope equation, we get

Since the slope of line AB is -2, the slope of our parallel line will be -2 as well. Let's plug it into our equation and see what it looks like now

y = mx + b

y = -2x + b

All we need now is b. To find this, remember that our line needs to pass through point C. Since C is at (2, 2), that means at when the line passes through C, x will be 2 and y will be 2. Let's pretend we are at that point right now and see what happens

y = -2x + b

(2) = -2(2) + b

2 = -4 + b

6 = b

So, the only way we will ever pass through point C is if b = 6.

y = -2x + b

y = -2x + 6

And we've got our answer. This picture shows our new line in blue, and by looking at the slope and y-intercept you can see it matches our equation.

Example 2

Give the equation of a line that passes through C and is perpendicular to line AB

Almost the same question, except this time the line must be perpendicular. So, from example 1 we already know that the slope of line AB is -2. Our criterion for the slopes of perpendicular lines is that their slopes must multiply to give -1. Written as an equation, this means

slope of AB X slope of line C = -1

Plugging in our slope for line AB

-2 X slope of line C = -1

Divide both sides by -2

So we know our slope will be 1/2

y = mx + b

y = (1/2)x + b

Now just like last time, at some point our line needs to pass through C. So let's see what happens when the line is at point C, whose coordinates are (2, 2).

y = (1/2)x + b

(2) = (1/2)(2) + b

2 = 1 + b

1 = b

So, the only way we will ever pass through point C is if b = 1.

y = (1/2)x + b

y = (1/2)x + 1

And we have our answer. This picture shows our new line in blue, and by looking at the slope and y-intercept you can see it matches our equation.