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Perpendicular linear graphs

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Lesson details

Learning outcome

I can prove the relationship between the gradients of perpendicular lines.

Key learning points

  1. Two lines are perpendicular if they meet at right angles.
  2. The product of the gradients of two perpendicular lines is -1
  3. This can be derived from Pythagoras' theorem.
  4. The product of the gradients of two lines is -1 iff the lines are perpendicular.

Keywords

  • Perpendicular - Two lines are perpendicular if they meet at right angles.

  • IFF - IFF (If and only if) is an implication that goes both ways.

Common misconception

Perpendicular lines have to be drawn as a horizontal and vertical pair.

Some pupils may struggle to see that two lines are perpendicular when they are not drawn horizontally and vertically. Drawing this on paper and then rotating the paper will help them see that it is the angle between the lines that matters.

Teacher tip

Consider printing Task 2 Q3 card to make sets of steps that pupils can cut and rearrange to form the proof. Different coloured card means the sets are colour coded for ease when collecting sets in at the end.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
 except where otherwise stated. See Oak's terms & conditions
 (Collection 2).

Lesson video

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Prior knowledge starter quiz

Download starter quiz questions (PDF)

6 Questions

Q1.
Which of these statements is true for parallel lines?

If the gradients of two lines are a zero pair, the lines are parallel.
Parallel lines are the same length.
Correct answer: Parallel lines have the same gradient.
Parallel lines have the same $$y$$-intercept.
Parallel lines intersect at right angles.

Q2.
The gradient of this line is .

An image in a quiz
Correct Answer: -1, negative 1, negative one

Q3.
The gradient of this line is .

An image in a quiz
$$-{4\over 3}$$
$$-1$$
Correct answer: $$-{3\over 4}$$
$$-{1\over 2}$$
$$-{1\over 4}$$

Q4.
The equation of the line drawn below is $$y =$$ $$x -1$$.

An image in a quiz
Correct Answer: 3

Q5.
The equation of the line drawn below is $$y = -{4\over 5}x +$$ .

An image in a quiz
Correct Answer: 4

Q6.
Using Pythagoras' theorem, the length of this line segment is units.

An image in a quiz
Correct Answer: 10, ten

Assessment exit quiz

Download exit quiz questions (PDF)

6 Questions

Q1.
Which of these statements best describes perpendicular lines?

Two lines which are always the same distance apart.
Two lines which are reflections of each other in the $$y$$-axis.
Two lines which are the same length.
Correct answer: Two lines which intersect at right-angles.
Two lines which never intersect.

Q2.
Which of these statements is true for perpendicular lines?

Their gradients are a zero pair.
Their gradients are reciprocals of each other.
Their gradients are the same.
Correct answer: Their gradients have a product of -1
Their gradients have a product of 1.

Q3.
Which of these diagrams show two lines which are perpendicular?

An image in a quiz
Correct Answer: An image in a quiz
An image in a quiz
An image in a quiz
Correct Answer: An image in a quiz
An image in a quiz

Q4.
Which of these statements is a correct use of the phrase 'if and only if' (iff)?

A shape is a rectangle iff it is a square.
Correct answer: A triangle has a right-angle iff Pythagoras' theorem holds.
A number is an integer iff it is positive.
Correct answer: Two lines are perpendicular iff their gradients have a product of -1.

Q5.
Which of these is an expression for the length of the line AB?

An image in a quiz
$$\sqrt{m^2} $$
$$1^2+m^2 $$
Correct answer: $$\sqrt{1^2+m^2} $$
$$(\sqrt{1^2+m^2})^2 $$
$$1 + m$$

Q6.
Which of these is equivalent to the equation $$(m_1 )^2 + (m_2 )^2 + 2 = (m_1 )^2 + (m_2)^2− 2m_1m_2$$?

$$m_1m_2=1$$
Correct answer: $$2 =− 2m_1m_2$$
$$2(m_1 )^2 + 2(m_2 )^2 = 2− 2m_1m_2$$
$$(m_1 ) + (m_2 ) + 1 = (m_1 ) + (m_2)− m_1m_2$$

To help you plan your 10 maths lesson on: Perpendicular linear graphs, download all teaching resources for free and adapt to suit your pupils' needs...