3.06 Parallel and perpendicular lines

Parallel lines

Parallel lines are lines that have the same gradient. 

 

Equations of parallel lines

Let's look at how we can identify parallel lines given their equations.

Equation Form	Characteristic of parallel lines	Examples
$y=mx+c$y=mx+c	Parallel lines have the same $m$m value.	$y=2x-1$y=2x−1 $y=4+2x$y=4+2x
$ax+by+c=0$ax+by+c=0	Parallel lines have the same value of $\frac{-a}{b}$−ab​.	$x+2y-3=0$x+2y−3=0 $2x+4y+1=0$2x+4y+1=0

For every straight line $y=mx+c$y=mx+c, there exist infinitely many lines parallel to it.  

 

Notice that they have the same gradient ($m$m-value) but different $x$x and $y$y-intercepts ($c$c-values).

Lines parallel to the axes

Horizontal lines

Horizontal lines are lines where the $y$y-value is always the same.

Let's look at the coordinates for $A$A, $B$B and $C$C on this line.  

$A=\left(-8,4\right)$A=(−8,4)

$B=\left(-2,4\right)$B=(−2,4)

$C=\left(7,4\right)$C=(7,4)

All the $y$y-coordinates are the same, $y=4$y=4. 

This means that regardless of the $x$x-value the $y$y value is always $4$4.

The equation of this line is $y=4$y=4

So if the equation of a straight line is $y=c$y=c, then it will be a horizontal line passing through the point  $\left(0,c\right)$(0,c).  

The $x$x-axis itself is a horizontal line.  The equation of the $x$x-axis is $y=0$y=0.

Vertical lines

Vertical lines are lines where the $x$x-value is always the same.

Let's look at the coordinates for $A$A, $B$B and $C$C on this line.  

$A=\left(-3,8\right)$A=(−3,8)

$B=\left(-3,3\right)$B=(−3,3)

$C=\left(-3,-3\right)$C=(−3,−3)

All the $x$x-coordinates are the same, $x=-3$x=−3. 

This means that regardless of the $y$y-value the $x$x-value is always $-3$−3.

The equation of this line is $x=-3$x=−3

So if an equation of a straight line is $x=c$x=c, then  it will be a vertical line passing through the point $\left(c,0\right)$(c,0).  

The $y$y-axis itself is a vertical line.  The equation of the $y$y-axis is $x=0$x=0.

 

Perpendicular lines

Lines that meet at right angles ($90^\circ$90°) are called perpendicular lines. 

Play with this applet creating pairs of perpendicular lines.  

Fill in this table as you go.

Gradient of line 1	$m_1$m1​
Gradient of line 2	$m_2$m2​
Product of line 1 and line 2	$m_1\times m_2$m1​×m2​

What do you notice about the product of the gradients of lines $1$1 and $2$2?  (The pair of perpendicular lines)

You will have discovered the perpendicular lines have gradients whose product is equal to $-1$−1.

We say that $m_1$m1​ is the negative reciprocal of $m_2$m2​.  

Negative reciprocal is a complex sounding term, but it just means two numbers that have opposite signs and are reciprocals of each other. 

Here are some examples of negative reciprocals:

$2$2 and $-\frac{1}{2}$−12​

$\frac{3}{4}$34​ and $-\frac{4}{3}$−43​

$-10$−10 and $\frac{1}{10}$110​

 

Practice questions

Question 1

Question 2

Question 3

Question 4