# Parallel And Perpendicular Lines

Parallel Lines

These are just lines that have the same gradient. Above you will see the lines for y = 1.5x +1 and y = 1.5x – 0.5.

It doesn’t matter what the value of c is, so long as as the m values are equal when expressed in the standard y = mx + c form then the lines are parallel.

Based on this you might be asked a question like: Line T has a gradient of 2 and is parallel to Line U. Line U passes through point (4,5). Find its equation.

As you know they are both parallel you know m = 2 and it goes through (4,5) so just plug these into the standard straight line graph equation to get c:

y = mx + c

5 = (2 x 4) + c = 8 + c

c = 5 – 8 = -3

Therefore the equation of the line is y = 2x -3.

Perpendicular Lines

These are lines that cross each other at right angles. There are two important rules to learn about such graphs:

• If the gradient of one line is m, then the gradient of the other is -1/m
• If you multiply both gradients together you get -1

The examples above show the lines y = 2.5x and y = -(1/2.5)x = -0.4x

Example: Find the perpendicular line to 4y – 3x = 8 through the point (0,2).

Rearrange the first equation into the form y = mx + c

So y = 3/4x + 2

Find the value of the perpendicular line’s gradient.

If the first one is 3/4 then:

3/4 x m = -1

Therefore m = -1 ÷ 3/4 = -4/3

Put this, and the coordinates, into the standard format equation to find c.

y = mx + c

so 2 = (-4/3 x 0) + c

Therefore c = 2.

You now have c and m.

So the equation is y = -4/3x + 2.