Unit 5 Section 4 : Parallel and Perpendicular Lines

In this section we consider the particular relationship between the equations of parallel lines and perpendicular lines. The key to this is the gradient of lines that are parallel or perpendicular to each other.

Example 1

(a)
Draw the lines with equations
y = x y = x + 4 y = x – 2
(b)

What do the three equations have in common?

Note that the three lines are parallel, all with gradient 1. All the equations of the lines contain 'x'. This is because the gradient of each line is 1, and so the value of m in the equation y = mx + c is always 1.
Parallel lines will always have the same gradient, and so the equations of parallel lines will always have the same number in front of x (known as the coefficient of x).
For example, the lines with equations:
y = 4x – 2
y = 4x
y = 4x + 10
will all be parallel (the coefficient of x is 4 in each case).

Example 2

The equations of four lines are listed below:

A y = 3x + 2 B y = 4x + 2
C y = 3x – 8 D y = 4x + 12
(a)
Which line is parallel to A ?
C is parallel to A, because both equations contain 3x (the coefficient of x in both cases is 3).
(b)
Which line is parallel to B ?
D is parallel to B, because both equations contain 4x (the coefficient of x in both cases is 4).

Example 3

The graph shows two perpendicular lines, A and B:

(a)
Calculate the gradient of A and write down its equation.
Gradient of A =
=2
Intercept of A = –7
Equation of A is y = 2x – 7
(b)
Calculate the gradient of B and write down its equation.
Gradient of B =
=
Intercept of B = –2
Equation of B is y = x – 2
(c)
Describe how the gradients of the lines are related.
The gradients of the lines are 2 and .
So:
Gradient of B =
If two lines A and B are perpendicular,
Gradient of B =
OR
Gradient of A × Gradient of B = –1

Example 4

Line A has equation y = 3x + 2. Write down the gradient of line B that is perpendicular, and a possible equation for B.

Gradient of A = 3
Gradient of B =
=
Equation of B will be y = x + c.
This will be perpendicular to A for any value of c,
so a possible equation is y = x + 4.

Exercises

Question 1
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(a)

Draw the lines with the following equations on the same set of axes:

y = 2x + 5
y = 2x + 1
y = 2x – 3
(b)

Draw two other lines that are parallel to these lines.

Question 2
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(a)

Draw the line with equation y = 3x – 2.

(b)

Draw a line parallel to y = 3x – 2 that passes through the point with coordinates (0, 3)

(c)

Determine the equation of the second line.

Question 3
The equations of five lines are listed below.
Ay = 5x – 7
By = 2x + 8
Cy = 3x + 3
Dy = 3x – 8
Ey = 5x + 2
(a)Which line is parallel to A ?
(b)Which line is parallel to C ?
(c)Are there any lines parallel to B ?
There are no lines parallel to B, because B has gradient 2 whilst A and E have gradient 5, and C and D have gradient 3.
Question 4
The diagram shows the line with equation y = 3x + 2 and two other lines, A and B, parallel to it.
(a) What is the gradient of the line A ?
(b) What is the equation of the line A ?
(c) What is the equation of the line B ?
Question 5
The diagram shows the line with equation y = x + 5,
and three other parallel lines.
What is the equation of:
Note: Use slash (/) to write fractions. e.g.
(a) line A,
(b) line B,
(c) line C ?
Question 6
The graph shows two lines, A and B.
(a) Calculate the gradient of the line A.
(b) What is the equation of the line A?
(c) What is the equation of the line B ?
Question 7
The graph shows two lines, A and B.
(a) Calculate the gradient of A.
(b) Calculate the gradient of B.
(c) Are the lines perpendicular? Use your answers to (a) and (b).
The lines are perpendicular, because –3 × = –1
and the product of the gradient is –1 for perpendicular lines.
Question 8
The equations of five lines are given below:
Ay = 5x + 2
B
y = x + 4
Cy = 2x + 1
D
y = x + 6
Ey = –2x + 3
(a)Which line is perpendicular to A ?
(b)Which line is perpendicular to B ?
(c)Which line is not perpendicular to any of the other lines?
Question 9
The line A joins the points with coordinates (4, 2) and (6, 8).
The line B joins the points with coordinates (5, 5) and (11, 3).
The line C joins the points with coordinates (6, 8) and (11, 4).
(a)

Calculate the gradient of each line.

A: B: C:
(b)

Which two lines are perpendicular?

and
Question 10
A line has equation y = 4x + 3.
Note: Use slash (/) to write fractions. e.g.
(a)

Write down the equation of 2 lines that are parallel to y = 4x + 3.

y = and y =
Any two lines of the form y = 4x + c, with c ≠ 3 are correct.
(b)

Write down the equation of 2 lines that are perpendicular to y = 4x + 3.

y = and y =
Any two lines of the form y = x + c are correct.
Question 11
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The diagram shows the graph of the straight line y = 3x.
(a)On this diagram, draw the graph of the straight line y = 2x.
(b)Write the equation of another straight line which goes through the point (0, 0).
y =
(c)The straight line with the equation y = x – 1 goes through the point (4, 3). On your diagram, draw the graph of the straight line y = x – 1.
(d)Write the equation of the straight line which goes through the point (0, –1) and is parallel to the straight line y = 3x.
y =
Question 12
Lucy was investigating straight lines and their equations. She drew the following lines.
(a)
y = x is in each equation.
What does this tells you about all the lines?
(b)
The lines cross the y axis at (0, –3), (0, 0) and (0, 4). Which part of each equation helps you see where the line crosses the y axis?
(c)
Lucy decided to investigate more lines. She needed longer axes.
Where will the line y = x – 20 cross the y axis?
()
(d)
On a copy of the graph, draw another line which is parallel to y = x.
Write the equation of your line.

y =
Any line of the form y = x + c is correct, where c ≠ 4, 0, –3 or –20.