 iGCSE (2021 Edition)

# 2.05 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

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

## 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​ $m_2$m2​ $m_1\times m_2$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.

#### Practice questions

##### 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

Consider the following points on the number plane:

$A$A $\left(2,-1\right)$(2,1)

$B$B $\left(4,-7\right)$(4,7)

$C$C $\left(-3,1\right)$(3,1)

$D$D $\left(-6,10\right)$(6,10)

1. First, calculate the gradient of the line $AB$AB.

2. Now, find the gradient of the line $CD$CD.

3. Is the line $CD$CD parallel to $AB$AB?

Yes

A

No

B

Yes

A

No

B

##### Question 4

Assess whether the points $A$A, $B$B and $C$C are collinear.

1. If $A$A and $B$B have the coordinates $\left(-4,3\right)$(4,3) and $\left(-2,7\right)$(2,7) respectively, evaluate the gradient of $AB$AB.

2. If $C$C has the coordinates $\left(-7,-3\right)$(7,3), evaluate the gradient of $BC$BC.

3. Based on these two gradients, are $A$A, $B$B and $C$C collinear?

Yes

A

No

B

Yes

A

No

B

##### Question 5

A line goes through A$\left(3,2\right)$(3,2) and B$\left(-2,4\right)$(2,4):

1. Find the gradient of the given line.

2. Find the equation of the line that has a $y$y-intercept of $1$1 and is parallel to the line that goes through $A$A$\left(3,2\right)$(3,2) and $B$B$\left(-2,4\right)$(2,4).

##### Question 6

Find the equation of a line that is perpendicular to $y=-\frac{x}{2}+5$y=x2+5, and goes through the point $\left(0,6\right)$(0,6).

### Outcomes

#### 0606C8.4A

Know and use the condition for two lines to be parallel or perpendicular.