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: