MATHEMATICS

Discussion point
➜ Does it matter which
point you call (x 1
, y 1
)
and which (x 2
, y 2
)?
The gradient of a line
y
A
(2, 4)
O
Figure 5.4
θ
7 – 4 = 3
B (6, 7)
C
6 – 2 = 4
x
7 − 4 3
Gradient m = =
6 − 2 4
θ (theta) is the Greek letter
‘th’. α (alpha) and β (beta)
are also used for angles.
When you know the coordinates of any two points on a straight line, then you
can draw that line. The slope of a line is given by its gradient. The gradient is
often denoted by the letter m.
In Figure 5.4, A and B are two points on the line. The gradient of the line AB is
given by the increase in the y coordinate from A to B divided by the increase in
the x coordinate from A to B.
In general, when A is the point ( x , y ) and B is the point ( x , y ) 2 2
, the gradient is
y
m =
x
− y
− x
2 1
2 1
1 1
Gradient =
change in y
change in x
When the same scale is used on both axes, m = tanθ (see Figure 5.4).
5
Chapter 5 Coordinate geometry
Parallel and perpendicular lines
ACTIVITY 5.1
It is best to use squared paper for this activity.
Draw the line L 1
joining (0, 2) to (4, 4).
Draw another line L 2
perpendicular to L 1
from (4, 4) to (6, 0).
Find the gradients m 1
and m 2
of these two lines.
What is the relationship between the gradients?
Is this true for other pairs of perpendicular lines?
When you know the gradients m 1
and m 2
, of two lines, you can tell at once if
they are either parallel or perpendicular – see Figure 5.5.
Lines for which
m m = − 1 2
1 will
only look perpendicular
if the same scale has
been used for both axes.
Figure 5.5
m 1
m 2
parallel lines: m 1
= m 2
m 1
m 2
perpendicular lines: m 1
m 2
= –1
parallel lines: m1 = m2
perpendicular lines: m1m 2
= −1
So for perpendicular lines:
m
1
1
=− and likewise, m
1
So m 1 and m 2 are the negative
2
=−
m
m
reciprocal of each other.
2
1
5