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3.06 Parallel and perpendicular lines

Lesson

Parallel Lines

Parallel lines are lines that have the same gradient. 

Two parallel lines that never cross and don't have any points in common.

 

 

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=2x1

$y=4+2x$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+2y3=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.  

 

Here is the line $y=x$y=x

Here are two more lines in the same family of parallel lines: $y=x+3$y=x+3 and $y=x-6$y=x6.

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.

Horizontal Lines

All horizontal lines are parallel to the $x$x-axis and are of the form $y=c$y=c.

They have a gradient of $0$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.

 

Vertical lines

All vertical lines are parallel to the $y$y-axis and are of the form $x=c$x=c

Their gradient is undefined.  

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

Perpendicular lines
  • Two lines are perpendicular if their gradients are negative reciprocals of each other. 
  • To test if lines are perpendicular multiply the gradients together. If the result is $-1$1 then the lines are perpendicular. 

 

Examples

Question 1

Which lines are parallel to $y=-3x+2$y=3x+2?

Select all correct options.

  1. $y=3x$y=3x

    A

    $y=-\frac{2x}{3}+8$y=2x3+8

    B

    $-3y-x=5$3yx=5

    C

    $y=-10-3x$y=103x

    D

    $y+3x=7$y+3x=7

    E

    $y=3x$y=3x

    A

    $y=-\frac{2x}{3}+8$y=2x3+8

    B

    $-3y-x=5$3yx=5

    C

    $y=-10-3x$y=103x

    D

    $y+3x=7$y+3x=7

    E

Question 2

Write down the equation of a line that is parallel to the $x$x-axis and passes through $\left(-10,2\right)$(10,2).

Question 3

What is the gradient of the line perpendicular to $y=\frac{x}{8}+3$y=x8+3?

Question 4

Find the equation of a line that is perpendicular to $y=6x+10$y=6x+10, and has the same $y$y-intercept. Give your answer in the form $y=mx+c$y=mx+c.

Outcomes

VCMNA338

Solve problems involving gradients of parallel and perpendicular lines

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