Cambridge International AS and A Level Mathematics Pure Mathematics 1

The distance between two points

P1^

2

Lines which are parallel have the same slope and so m 1 = m 2. If the lines are

perpendicular, m 1 m 2 = −1. You can see why this is so in the activities below.

ACTIVITY 2.2 Draw the line L 1 joining (0, 2) to (4, 4), and draw another line L 2 perpendicular

to L 1. Find the gradients m 1 and m 2 of these two lines and show that m 1 m 2 = −1.

ACTIVITY 2.3 The lines AB and BC in figure 2.5 are equal in length and perpendicular. By

showing that triangles ABE and BCD are congruent prove that the gradients m 1

and m 2 must satisfy m 1 m 2 = −1.

!^ Lines for which m 1 m 2 =^ −1 will only look perpendicular if the same scale has been

used for both axes.

The distance between two points

When the co-ordinates of two points are known, the distance between them can

be calculated using Pythagoras’ theorem, as shown in figure 2.6.

y

x

JUadientm

JUadientm

$ (

' &

%

2

θ

θ

Figure 2.5

y

x

(2, 4)

A

B(6, 7)

O

Figure 2.6

C

AC = 6 − 2 = 4

BC = 7 − 4 = 3

AB^2 = 42 + 32

= 25

AB = 5