# Parallel and Perpendicular Lines

## Parallel and Perpendicular Lines Revision

**Parallel and Perpendicular Lines**

**Parallel Lines** – are always the same distance apart and never meet (they have the same gradient).

**Perpendicular Lines** – meet at a right angle (the gradients of perpendicular lines multiply to -1).

Before continuing this topic, recap the following:

**Parallel Lines – Same Gradient**

**Parallel** lines have the **same gradient.**

Equation of a straight line: y=\textcolor{#10a6f3}{m}x+c

\textcolor{#10a6f3}{m} \rightarrow gradient

c \rightarrow y- intercept

For lines to be parallel, the value of \textcolor{#10a6f3}{m} must be the same.

Let’s take a look at the following lines:

y=\textcolor{#10a6f3}{2}x+1 (*)

y\textcolor{#10a6f3}{-2}x=5 (**)

y=\textcolor{#10a6f3}{2}x -4 (***)

The second equation (**) must be rearranged into the form

y=\textcolor{#10a6f3}{m}x+c , before comparing the gradients:

y=\textcolor{#10a6f3}{2}x+5

These lines all have a gradient of \textcolor{#10a6f3}{2} \, (m=2) . These straight line equations are shown on the graph to the right.

**Perpendicular Lines – Right Angles**

**Perpendicular lines** meet at a **right angle**. When we multiply the gradients of a pair of perpendicular lines they **multiply** to \textcolor{#00d865}{-1}.

Another way of looking at this concept is to think of the perpendicular gradient as the **‘opposite sign and reciprocal’**.

For instance, let’s say we have a straight line with a gradient: \textcolor{#00d865}{m=2}

Then the perpendicular gradient would be: \textcolor{#00d865}{m=-\dfrac{1}{2}}

\textcolor{#00d865}{2}\times\textcolor{#00d865}{-\dfrac{1}{2}}=\textcolor{#00d865}{-1}Let’s take a look at the following lines:

y=\textcolor{#00d865}{3}x+1

y=\textcolor{#00d865}{-\dfrac{1}{3}}x+2

These lines are perpendicular as: \textcolor{#00d865}{3}\times\textcolor{#00d865}{-\dfrac{1}{3}}=\textcolor{#00d865}{-1}. These straight line equations are shown on the graph to the right.

**Example 1: Parallel Lines**

Find the equation of a straight line that is parallel to \textcolor{#f95d27}{y=3x+5} and passes through (1,-2)

Equation of the parallel line: y=3x+c

**[3 marks]**

c is the y \,intercept and can take any value. However, because we know the line passes through the coordinate (1,-2), we can find the exact value of c

To do this, we substitute x=1 and y=-2

-2=3(1)+c \rightarrow c=-5 \rightarrow \textcolor{#f21cc2}{y=3x-5}**Example 2: Perpendicular Lines**

Find the equation of a straight line that is perpendicular to

\textcolor{#f95d27}{y=\dfrac{2}{3}x+1} and passes through (2,1)

**[3 marks]**

Equation of the perpendicular line: y=-\dfrac{3}{2}x+c

c is the y \,intercept and can take any value. However, because we know the line passes through the coordinate (2,1), we can find the exact value of c

To do this, we substitute x=2 and y=1

1=-\dfrac{3}{2}(2)+c \rightarrow c=4 \rightarrow \textcolor{#f21cc2}{y=-\dfrac{3}{2}x+4}## Parallel and Perpendicular Lines Example Questions

**Question 1:** From the following equations of a straight line, state which pair are parallel and another pair which are perpendicular.

**[2 marks]**

a) y=2x+5

b) \dfrac{y+3x}{2}=3

c) 2y+6=4x

d) y=\dfrac{1}{3}x+9

After some rearrangement, to get the equations in the form of y=mx+c, we get the following:

a) y=2x+5

b) y=-3x+6

c) y=2x-3

d) y=\dfrac{1}{3}x+9

a) and c) are parallel, as the gradients are the same.

b) and d) are perpendicular, as when you multiply the gradients together you get -1.

**Question 2:** Find the equation of a straight line that is parallel to y-8x=11 and passes through (2,5)

**[2 marks]**

We first need to rearrange the equation into the form of y=mx+c:

y=8x+11Equation of parallel line:

y=8x+cSubstituting x=2 and y=5

5=8(2)+c \rightarrow c=-11 \rightarrow y=8x-11**Question 3:** Find the equation of a straight line that is perpendicular to 5y+x=50 and passes through (2,12)

**[2 marks]**

We first need to rearrange the equation into the form of y=mx+c:

y=-\dfrac{1}{5}x+10Equation of perpendicular line:

y=5x+cSubstituting x=2 and y=12

12=5(2)+c \rightarrow c=2 \rightarrow y=5x+2