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2nd year Co-ordinate Geometry of the Line
When plotting a point always put the x 1st and y 2nd. The x value is on horizontal axis and y is
on the vertical axis.
Plot the point A= (3,2) and B =(-1,-2)
x, y
x, y
1.Distance/length between 2 points
A=(3,2) B =(-1,-2)
x1,y1 x2, y2
Sub into formula below
( x 2  x1) 2  ( y 2  y1) 2
Distance =
(1  3) 2  ( 2  2) 2
(4) 2  (4) 2
16  16
│ab│ =length/distance from a to b
│ab│= 32
x1  x 2 y1  y 2
,
)
2
2
3 1 2  2
,
)
= (
2
2
midpo int  (
2.
A=(3,2) B =(-1,-2)
x1,y1
x2, y2
=
(1, 0)
3. The slope/gradient of line when given 2 points
Slope/m =
y 2  y1
x 2  x1
A=(3,2) B =(-1,-2)
x1,y1 x2, y2
= 3–1
5-3
= 2 =1
2
slope/m
Finding the slope from a graph
Slope is rise
Run
m= 3
3
3 boxes up
3 boxes across
M=1
line goes up from left to right so plus
Slope is rise
Run
M= -4
2
4 boxes down
2 boxes across
M= -2
line going down so negative
Rules with slopes

Parallel lines means the slopes are the same

Perpendicular lines (∟) means the lines are at 90° angle and to get the slope you
must
Turn the other slope upside down and change the sign.
-1
2
Parallel lines both slopes = -1
2
-1
2
3
2
N
∟
Perpendicular lines M ∟ N
∟ = 90°
-2
3
M
IF N = 3 then M slope is -2
2
3
To prove two lines are perpendicular you multiply the slopes and you should get -1.
m1*m2 =-1
3 X
2
-2
3
= -1
4. Equation of a line
You need 2 things Slope and a point
y – y1 = m (x – x1)
Example
If m = 2 and point is (3,4) find the equation of the line?
X1, y1
Y - 4 = 2(x-3)
Y -4 =2x-6
put all on same side where x is
positive
2x-y-2=0
Using slope and point sub into equation of the line y – y1 = m (x – x1)
m = -3
(5,7)
4
x1, y1
y – y1 = m (x – x1)
y–7=
3
(x – 5)
4
4(y-7)=-3(x-5)
4y – 28= -3x + 15
3x+4y-28-15=0

3x + 4y – 43 = 0
Cross multiply
5. Slope of a line when given the equation
M = -a
b
Example:
a b c
3x + y -5=0
ax +by + c =0
whats with x is a and whats with the y is b
a=3, b=1
m= -3
1
= -3
Or use y=mx+c
You must write the line in this format with the y on the left and all the other
terms on the other side.
5x+2y=1
2y =-5x+1
y= -5x+1 m is the value with the x on top and bottom so m = -5
2
2
6. To prove a point is on a line
You sub it into the line and get 0 = 0 if its on the line
Show (-3,1) is on the line 2x+4y+2 =0
2(-3)+4(1)+2 =0
-6 +4+2 = 0
0 =0
point on line
7. Graphing lines
If cuts x-axis ----y=0
When given an equation and asked to draw it.
 Let x=0 find y
 Let y=0 find x
 Plot the 2 points
 Draw the line through these points
If cuts y-axis----- x=0
x- 2y -6 = 0
-6 = 0
x- 2y
x =0
y=0
0 – 2y -6 =0
2(0) -6 =0
x-
-6 = 2y
=0
-3 = y
x -6
(0,-3)
(6,0)
x=6
8. Lines Parallel to the axis
If the line is x = 2 you draw line at 2 down through x axis
If line is y = -1 you draw a line at -1 through y axis