Lesson 1.3 Perpendicular lines

This is a free lesson. We trust you enjoy it!

Gradients of perpendicular lines

Up to now you have learned that parallel lines have the same gradient. What about perpendicular lines? Do you think we can work out some precise numerical relationship between their gradients?

  • Can you say anything about the signs (negative/positive) about the gradients of perpendicular lines?
  • Can you say anything more than this?

 The following video explains how to derive a precise numerical relationship between the gradients of perpendicular lines. Once again, you don't need to be able to reproduce this derivation, but you must know the result and how to use it. You can also read the explanation given in the text book in section 1.9. (It is important to learn to read mathematical explanations in written form.)

For your notebook:

Gradients of perpendicular lines

Two lines are perpendicular if and only if

 $$ m_1 m_2 = -1 $$

where  \(m_1 \) and  \(m_2 \) are the gradients of the two lines. This is equivalent to \( m_1 =  - { 1 \over m_2 } \), or \(m_2 = - { 1 \over m_1 } \).

Note: we sometimes say that the one gradient is the negative reciprocal of the other. (The reciprocal of  \(x \) is  \( { 1 \over x } \).)

 This video gives an alternative explanation of Example 1.9.1 in your text book. The question asks you to prove that four points form a rhombus.

 Here is the answer to another problem involving gradients and perpendicular lines (see Exercise 1C, no. 3). (The answer is also given in the answer key.)

Weight lifting

Complete the following exercises from your text book:

Exercise 1C

  • Question 1 c,d,g,h,k,l.
  • Question 2 b,d,f,h,j,l.
  • Questions 4,5,6.

You can download the answer key from here, or from your download area on the My Account menu. Do so, and mark your work as explained in the Preliminaries.