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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 rightangled 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 xcoefficient in the equation.
Also observe how the yintercept
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
yintercept=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 yintercept.
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 rightangles 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: 