Coordinate geometry

Coordinate Geometry
Coordinate geometry is the study of the relationships between points on the Cartesian plane
What we will explore in this tutorial
(a) Explore gradient
I. Identify the gradient of a straight line
II. Calculate the gradient of a straight line
III. Determine the gradient of straight lines that are parallel to or perpendicular to a
given line
(b) Calculate the midpoint of a line/line segment
(c) Calculate the length of a given line
(d) Determine the equation of
I. A straight line
II. The equation of a line parallel to a given line
III. The equation of a line perpendicular to a given line
(e) Interpret the x and y intercepts of a given straight line
Gradient
Gradient may be described as a “rate of change” that is we examine how one thing is changing
as the other thing is changing; for example we may heat water and compare the temperature
as time passes or we may compare the distance travelled by a car compared with time.
Identifying the gradient from the equation of a straight line
The general form of a straight line is 0ax by c   , however a more popular version of this is
what we call the slope intercept form of a straight line y mx c  . Much of our work here will
be concentrated on this form of the line
The letterm , the coefficient of x , represents our gradient. For straight lines the gradient is
always constant for the whole line. You should be able to look at a straight line and easily
identify its gradient; examine the equations below;

Examples of the slope intercept form are
5 4y x  ; Here the gradient is 5 [our c value or the y intercept is – 4]
8 3y x  Here the gradient is – 3 [our c value or the y intercept is 8]
2
4
3
y x  ; Here the gradient is
2
3
[our c value or the y intercept is – 4]
1
8
2
y x  ; Here the gradient is
1
2
 [our c value or the y intercept is 8]
In some cases however a question may give you the general form of a straight line and ask you
to determine its gradient for example
1. 2 7 5y x 
2. 5 3 4x y 
3. 10 2 3 0x y  
In each case to get our answer we need to rewrite it in the form y mx c  so that we can
easily see the value of our gradient.
Example 1.
State the gradient of the line 2 7 5y x 
Solution
2 7 5
7 5
2 2
7
2
y x
x
y y mx c
m
 
    

Example 2
Write down the gradient of 5 3 4x y 

Solution
5 3 4
3 4 5
4 5
3 3
4 5 5
3 3 3
x y
y x
x
y
x
y m
 
  
 
 
   

Example 3
Determine the gradient of 10 2 3 0x y  
Solution
10 2 3 0
3 2 10
2 10 2 10
,
3 3 3
2
3
x y
y x
x x
y y
m
  
 

   
 

Calculate the gradient of a straight line given two pairs of coordinates   1 1 2 2, ,x y x y
To determine the gradient of line AB we need to examine the ratio of the change in the y -
distance compared with the change in the x – distance. We call the change in y the rise and the
change in x the run. This can be written down as
2 1
2 1
y y
m
x x



Examples
Know this formula/concept.
This formula is used to calculate the gradient of a straight
line given two points   1 1 2 2, ,x y x y

Find the gradient of the line passing through the points given
1. (5,6), (0,4)A B
2. (6, 2), ( 2,3)W X 
3. (3,13), (4,18)M N
Solution to 1
Using the formula 2 1
2 1
y y
m
x x



we have
4 6 2 2
0 5 5 5
m
 
  
 
note that
  1 1 2 25, 6 0, 4x y x y    or if you choose to use them alternately then
  1 1 2 20, 4 5, 6x y x y   
Solution to 2
The gradient of WX is given as
 
2 1
2 1
3 2 5
2 6 8
y y
m
x x
m



 
 
  
Solution to 3
The gradient of MN is given as
2 1
2 1
18 13 5
5
4 3 1
y y
m
x x
m




  

Note.
It may help to label each coordinate
individually as 1x , 1y , 2x , 2y
And use it as a guide to substitute the
numbers correctly until you build up a
rhythm

Finding the gradient of a straight line given its graph
The process for determining the gradient of the graph is to
1. Draw a suitable right angled triangle on the line
2. Determine the rise and the run
3. Divide the rise by the run
So we have the gradient of the line as
9 3 6
4 0 4
Rise
m
Run

  


Parallel and perpendicular lines
Two lines are parallel if they have the same gradient
Two lines are perpendicular if the product of their gradients is – 1 [negative 1]
Examples
1. A line has the equation 5 3y x  , write down the gradient of the line that is
a. Parallel to 5 3y x 
b. Perpendicular to 5 3y x 
Solution: Note that the gradient of 5 3y x  is 5 and therefore
(a) The equation of any line parallel to 5 3y x  will have a gradient of 5
(b) If two lines are perpendicular the product of their gradients is negative ONE, therefore,
we can use a simple equation to find it such as
5 1
1
5
m
m
 


, Note that
1
5 1
5

   , so the
gradient we need is
1
5
m

 . Note that
5
5
1
 so we invert
5
1
and change its sign to get
1
5
m


We could have also found this number
1
5
m

 by inverting our gradient and changing its
sign.
2. A straight line PQ has the equation
2
4
3
y x  , determine
a. The gradient of any line that is parallel to PQ
b. The gradient of the any line perpendicular to PQ

Solution
Our gradient here is
2
3

so
(a) Any line parallel to PQ will have a gradient of
2
3

(b) And using the explanations given above Any line perpendicular to
2
3

will have a
gradient of
3
2
m  , we invert the
2
3

and change its sign
The midpoint and length of a line segment
There are two formulae that we need here, that is given any two points   1 1 2 2, ,x y x y
1 2 1 2
,
2 2
x x y y
midpt
  
  
 
   
2 2
2 1 2 1L x x y y   
Example
A straight line passes through the points (6, 2), ( 5,3)J K  determine
(a) The midpoint of JK
(b) The length of line segment JK
The midpoint is given as
   6 5 2 3 1 1
, ,
2 2 2 2
      
   
  
The length is given as
     
 
2 2
2 2
5 6 3 2
11 5
146 12.1
L
L
L units
     
  
 
To find midpoint
To find the length

Finding the equation of a straight line
Case 1; given two points   1 1 2 2, ,x y x y
A straight line LM passes through the points (4,6), (6,10)L M , find the equation of LM
First we need to find the gradient which here is
10 6 4
2
6 4 2
m

  

Now using the general form of the line  1 1y y m x x   and the point (4,6)L we have the equation
 
 
1 1
6 2 4
6 2 8
2 8 6
2 2
y y m x x
y x
y x
y x
y x
  
  
  
  
 
Case 2; given the gradient and a point
A straight line CD passes through the point (4,3)C and has a gradient of
3
4
m  , calculate the
equation of CD
Again using the point given and the general equation of the line we have the equation of CD as
 
3
4 4
4
3
4 3
4
3
3 4
4
3
1
4
y x
x
y
x
y
x
y
  
  
  
 

Case 3; find the equation of a line given its graph
Here we see that the gradient
4
5
rise
m
run
  and our y intercept is 5, using the slope intercept of
the line y mx c  we have by substitution the equation of the line
4
5
5
y x 

Interpreting the x and y intercepts of a straight line
Consider the graph below
We already understand that the y – intercept is the point at which the line
cute/intersects the y – axis
The x – intercept is the solution of the equation
4
5 0
5
x   which gives
4
5
5
25 1
6
4 4
x
x
 

  

Coordinate geometry

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Coordinate geometry