Click here to return to the A-level revision guides main page

 

Core 1: Geometry and straight lines

Contents

1)           Gradients

2)          Equation of a straight line and its properties

3)          Perpendicular gradients

4)          Parallel lines and Perpendicular lines

 


 

Gradients

Beneath the line I have drawn a right-angled triangle

The height:

The base:   

The gradient is defined by:

 

 

In this case:

The gradient is defined by:

 

 

In this case:

The gradient is defined by:

Observe it is negative

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Observe in each case how the gradient equals the x-coefficient in the equation.

Also observe how the y-intercept equals the remainder added onto the equation.

Equation of a line

We are therefore in a position to generalise.

Generalisation 1:

An equation of the form:

 has a gradient=m and a y-intercept=c

Generalisation 2:

There is a second, slightly more powerful version of the equation of a straight line which does not rely so heavily on knowing the y-intercept.

If you know the gradient and a particular coordinate (

Then:

Example 1: Find the equation of the line with a gradient of 4, through the coordinate (2,-3)

Example 2: Gradient  through the coordinate (5,1)

 multiply through by 5

      so,


 

Perpendicular Gradients

You need to be aware of a particular property about perpendicular lines

Observe the following

These lines are at right-angles

This is called perpendicular

The gradients of the lines:

Line 1:

Line 2:

Observe how

 

Does this generalisation always hold?

These lines are also perpendicular

Line 1:

Line 2:

Observe how

 

 

 

Generalisation:

For two lines:  implies that they are perpendicular

This can be recognised when  

This generalisation only fails when one gradient is 0. In this case the perpendicular line has an “infinite gradient”

Finding parallel and perpendicular lines

Find the equation of the line parallel to y=4x+2, passing through the point (1,5)

 

 

 

 

 

Because it is parallel we know that m=4 since parallel means same gradient.

We also know the coordinate

(

We can therefore use the equation:

 

Find the equation of the line perpendicular to y=4x+2, passing through the point (1,5)

 

 

 

 

This is often written as:

 

Because it is perpendicular we know that  

(since )

We also know the coordinate

(

We can therefore use the equation: