 # Find Perpendicular Lines

In this worksheet, students will investigate perpendicular lines and find their gradients or equations in the form of numbers or expressions. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra, Graphs of Equations and Functions

Curriculum subtopic:   Graphs, Straight Line Graphs

Difficulty level:   ### QUESTION 1 of 10

Perpendicular lines are lines which cross each other at a right angle.

If the gradient of a line is m, then the gradient of a perpendicular line is When we multiply the gradients of two perpendicular lines, we get an answer of -1.

Let's see this in action now.

e.g. Find the gradient of a line which is perpendicular to a line with gradient:

a) 3

b) -1/4

c) 1.5

a) 3 × -1/3 = -1, so the gradient is =1/3

b) -1/4 × 4 = -1, so the gradient is 4

c) 1.5 × -2/3 = -1, so the gradient is -2/3

e.g. Find the gradient of a line which is perpendicular to 2y = 6x - 1.

Here we need to rearrange the equation into the form y = mx + c where m will be the gradient:

2y = 6x - 1

y = 3x - 1/2

The gradient of this line is 3.

The gradient of the perpendicular line is -1/3, because 3 × -1/3 = -1.

In this activity, we will investigate perpendicular lines and find their gradients or equations in the form of numbers or expressions.

What is the gradient of a line which is perpendicular to a line with a gradient of -7?

-7

7

1/7

-1/7

Which of the statements below are true about a pair of perpendicular lines?

They cross at 90 degrees

They have gradients which multiply to give 1

They have gradients which multiply to give -1

Match the pairs of gradients of lines which are perpendicular to each other below.

## Column B

2/3
1/3
-3
-3/2
0.3
2/9
-4.5
-10/3

Tick all the equations of lines below which are perpendicular to the line y = -3x + 11.

y = 3x + 11

y = 1/3 x + 1

3y = x - 9

y = 5 - 1/3 x

Match the pairs of equations of lines which are perpendicular to each other below.

## Column B

y = 2x + 3
6y = x + 1
y = 5 - 6x
4x + 6y - 3 = 0
4y + 8x = 6
y = 5 - 1/2x
2y - 3x = 6
8y - 4x = 3

Which of the lines below are not perpendicular to y = 3x - 2?

y = 5 - 1/3x

3y + x = -1

y = 1/3x + 1

y + 1/3x = 1

What is the gradient of a line which is perpendicular to a line with an equation y = 5/7x - 5?

5

7

-5/7

-1.4

Consider the equations of two lines below:

y = 4x - 5

4y = 8 - x

Which statement below is true about these two lines?

They are parallel lines

They are perpendicular lines

They are neither parallel nor perpendicular

They are the same line

Select all the lines from the options below which are perpendicular to 2x - 3y = 7.

4y + 6x = 1

2y = 7 - 3x

1 - 4x = 6y

2y - 3x - 3 = 0

A line has a gradient of -4.

Which of the lines below is perpendicular to it?

y + 4x = 3

4y - x - 7 = 0

4y = 2x + 5

2x - 3y = 7

• Question 1

What is the gradient of a line which is perpendicular to a line with a gradient of -7?

1/7
EDDIE SAYS
If the gradient of a line is m, then the gradient of a perpendicular line is -(1/m). So if we substitute -7 for m in this equation, we get: -(1 / (-7)) = 1/7
• Question 2

Which of the statements below are true about a pair of perpendicular lines?

They cross at 90 degrees
They have gradients which multiply to give -1
EDDIE SAYS
Perpendicular means crossing at a right angle or 90 degrees. The gradients of perpendicular lines also multiply to give a total of -1 not +1. Remember these facts to support you in the rest of this activity.
• Question 3

Match the pairs of gradients of lines which are perpendicular to each other below.

## Column B

2/3
-3/2
-3
1/3
0.3
-10/3
-4.5
2/9
EDDIE SAYS
Did you remember that the gradients of two perpendicular lines multiply to give -1? It's always easiest to spot these relationships if we have our gradients in fraction form, so let's change our decimal: 0.3 = 3/10 To find a matching pair, flip the fraction upside down and give it an opposite sign. e.g. 3/10 becomes -10/3, so this is a matching pair of gradients of perpendicular lines. Can you flip the remaining pairs independently and find the matching pairs?
• Question 4

Tick all the equations of lines below which are perpendicular to the line y = -3x + 11.

y = 1/3 x + 1
3y = x - 9
EDDIE SAYS
The lines perpendicular to y = -3x + 11 will have a gradient of 1/3. So y = 1/3 x + 1 is definitely perpendicular. 3y = x - 9 is also perpendicular. If we rearrange it, we reach: y = 1/3x - 3 so we can see that gradient we are looking for of 1/3. Did you spot those two perpendicular lines?
• Question 5

Match the pairs of equations of lines which are perpendicular to each other below.

## Column B

y = 2x + 3
y = 5 - 1/2x
y = 5 - 6x
6y = x + 1
4y + 8x = 6
8y - 4x = 3
2y - 3x = 6
4x + 6y - 3 = 0
EDDIE SAYS
We need to rearrange some of these equations into the form y = mx + c so that we can see what the gradient (m) is. e.g. 4y + 8x = 6 4y = 6 - 8x y = 3/2 - 2x The gradient of this line is -2. So the gradient of the line which is perpendicular to this will be 1/2. Which equation has a gradient of 1/2? Again, we need to rearrange our equations to find this. 8y - 4x = 3 8y = 3 + 4x y = 3/8 + 1/2x So we have found our match! Can you rearrange the equations and find the remaining three matches, using this example to help you?
• Question 6

Which of the lines below are not perpendicular to y = 3x - 2?

y = 1/3x + 1
EDDIE SAYS
The gradient of the lines perpendicular to y = 3x - 2 must be -1/3. We find this gradient by converting m from y = mx + c into -(1/m). We need to rearrange all the options into the form y = mx + c to be able to compare them: 3y + x = -1 > 3y = -1 - x > y = -1/3 - 1/3x y + 1/3x = 1 > y = 1 - 1/3x The two equations above, plus the first option in the list, all have gradients of -1/3 so are perpendicular to the line y = 3x - 2. y = 1/3x + 1 has a gradient of +1/3 so this is the line which is not perpendicular.
• Question 7

What is the gradient of a line which is perpendicular to a line with an equation y = 5/7x - 5?

-1.4
EDDIE SAYS
To find the gradient of a perpendicular line, we need to calculate -(1/m). In our equation y = 5/7x - 5, m = 5/7. So lines which are perpendicular to this will have the gradient -7/5. -7/5 is equivalent to - 1.4 so this is the correct answer.
• Question 8

Consider the equations of two lines below:

y = 4x - 5

4y = 8 - x

Which statement below is true about these two lines?

They are perpendicular lines
EDDIE SAYS
The first line has a gradient of 4. The second line has a gradient of -1/4. 4 × -1/4 = -1, so these lines are perpendicular.
• Question 9

Select all the lines from the options below which are perpendicular to 2x - 3y = 7.

4y + 6x = 1
2y = 7 - 3x
EDDIE SAYS
Let's first rearrange 2x - 3y = 7 into the form y = mx + c: 2x = 3y + 7 2x - 7 = 3y 2/3x - 7/3 = y The gradient of this line is 2/3. So the gradient of lines which are perpendicular to this will be -3/2. We will need to rearrange the options provided into the form y = mx + c to compare them. 4y + 6x = 1 > 4y = 1 - 6x > y = 1/4 - 6/4x > y = 1/4 - 3/2x which is the correct gradient we are looking for. 2y = 7 - 3x > y = 7/2 - 3/2x which is also the correct gradient. 1 - 4x = 6y > y = 1/6 - 4/6x > y = 1/6 - 2/3x which is not the correct gradient. 2y - 3x - 3 = 0 > 2y = 3x + 3 > y = 3/2x + 3/2 which is also not the correct gradient.
• Question 10

A line has a gradient of -4.

Which of the lines below is perpendicular to it?

4y - x - 7 = 0
EDDIE SAYS
If our starting line has a gradient of -4, any line perpendicular to it will have a gradient of 1/4. We need to rearrange the options provided into the form y = mx + c so that we can compare their gradients. y + 4x = 3 > y = 3 - 4x This line has a gradient of -4 so this is not a perpendicular line. 4y - x - 7 = 0 > 4y = x + 7 > y = 1/4x + 7/4x y + 4x = 3 > y = 3 - 4x This line has a gradient of 1/4 so this is perpendicular and, therefore, is our correct answer. 4y = 2x + 5 > y = 2/4x + 5/4 This line has a gradient of 1/2 so this is not a perpendicular line. 2x - 3y = 7 > 2x = 7 + 3y > 2x - 7 = 3y > 2/3x - 7/3 = y This line has a gradient of 2/3 so this is not a perpendicular line. Great job completing this activity! Hopefully you are feeling much more confident to recognise and work with perpendicular lines now.
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