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Find Perpendicular Lines

In this worksheet, students will find equations of perpendicular lines.

'Find Perpendicular Lines' worksheet

Key stage:  KS 4

Curriculum topic:  

Curriculum subtopic:  

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Perpendicular lines are lines which cross 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.

 

Example 1

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

a) 3

b) -1/4

c) 1.5

Answers

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

 

Example 2

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

Answer

You need to rearrange the equation into the form y = mx + c. 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.

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

-7

7

1/7

-1/7

Perpendicular lines:

have the same gradient.

cross at 90 degrees.

have gradients which multiply to give 1.

have gradients which multiply to give -1.

Match the pairs of gradients of perpendicular lines.

Column A

Column B

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

Tick all the equations of lines 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 following equations into pairs of perpendicular lines.

Column A

Column B

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

Which of these lines is 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 perpendicular to the line with an equation y = 5/7x - 5?

5

7

-5/7

-1.4

y = 4x - 5

4y = 8 - x

These two lines are:

parallel

perpendicular

neither parallel nor perpendicular

the same line

Choose all the lines 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 perpendicular to a line with gradient -7?

CORRECT ANSWER
1/7
EDDIE SAYS
The correct answer is 1/7. Gradients of perpendicular lines multiply to give -1 and -7 × 1/7 = -1.
  • Question 2

Perpendicular lines:

CORRECT ANSWER
cross at 90 degrees.
have gradients which multiply to give -1.
EDDIE SAYS
Perpendicular means crossing at a right angle (90 degrees). The gradients of perpendicular lines multiply to give -1.
  • Question 3

Match the pairs of gradients of perpendicular lines.

CORRECT ANSWER

Column A

Column B

2/3
-3/2
-3
1/3
0.3
-10/3
-4.5
2/9
EDDIE SAYS
Did you remember that gradients of perpendicular lines multiply to give -1? It's always easiest to see this if you have your gradients in a fraction form. 0.3 = 3/10 To find a pair, flip the fraction upside down and give it an opposite sign. In this case, 3/10 becomes -10/3, so this the perpendicular gradient.
  • Question 4

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

CORRECT ANSWER
y = 1/3 x + 1
3y = x - 9
EDDIE SAYS
The lines perpendicular to y = -3x + 11 have a gradient of 1/3. y = 1/3 x + 1 is one of them. 3y = x - 9 is also perpendicular. If you rearrange it you get y = 1/3x - 3.
  • Question 5

Match the following equations into pairs of perpendicular lines.

CORRECT ANSWER

Column A

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
Some of these equations need to be rearranged to the form y = mx + c so that you can see what the gradient is (m). 4y + 8x = 6 4y = 6 - 8x y = 3/2 - 2x The gradient of this line is -2. Can you find the line with the perpendicular gradient (1/2)?
  • Question 6

Which of these lines is not perpendicular to y = 3x - 2?

CORRECT ANSWER
y = 1/3x + 1
EDDIE SAYS
y = 1/3x + 1 is not perpendicular to y = 3x - 2. The gradient of the lines perpendicular to y = 3x - 2 must be -1/3 and the gradient of this line is 1/3.
  • Question 7

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

CORRECT ANSWER
-1.4
EDDIE SAYS
The answer is -1.4. The perpendicular gradient is -7/5 which is equivalent to - 1.4.
  • Question 8

y = 4x - 5

4y = 8 - x

These two lines are:

CORRECT ANSWER
perpendicular
EDDIE SAYS
These two lines are parallel. The first line has a gradient of 4. The second line has a gradient of -1/4. 4 × -1/4 = -1, so the lines are parallel.
  • Question 9

Choose all the lines perpendicular to 2x - 3y = 7.

CORRECT ANSWER
4y - 6x = 1
2y = 7 - 3x
EDDIE SAYS
Let's first rearrange 2x - 3y = 7. 2x = 3y + 7 2x - 7 = 3y 2/3x - 7/3 = 3 The gradient of this line is 2/3. The gradient of any line that is perpendicular is -3/2. 4y - 6x = 1 and 2y = 7 - 3x have a gradient of -3/2.
  • Question 10

A line has a gradient of -4.

Which of the lines below is perpendicular to it?

CORRECT ANSWER
4y - x - 7 = 0
EDDIE SAYS
A line perpendicular to a line with a gradient of -4 must have a gradient of 1/4. 4y - x - 7 = 0 is perpendicular to a line with a gradient of -4. If we rearrange, we get: 4y = 7 + x y = 7/4 + 1/4x The gradient is 1/4 so this line is perpendicular.
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