Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

This method can be generalised to find the distance between any two points,

A(x 1 , y 1 ) and B(x 2 , y 2 ), as in figure 2.7.

The length of the line AB is ()xx 21 −+^2 ()yy 21 −^2.

The mid-point of a line joining two points

Look at the line joining the points A(2, 1) and B(8, 5) in figure 2.8. The point

M(5, 3) is the mid-point of AB.

Notice that the co-ordinates of M are the means of the co-ordinates of A and B.

52 =+^12 () 83 ;(=+ 2115 ).

This result can be generalised as follows. For any two points A(x 1 , y 1 ) and

B(x 2 , y 2 ), the co-ordinates of the mid-point of AB are the means of the

co-ordinates of A and B so the mid-point is

xx 12 yy 12

22

 ++





, .

y

x

A C

B(x 2 , y 2 )

(x 1 , y 1 )

O

Figure 2.7

BC = y 2 − y 1

The co-ordinates

of this point must

be (x 2 , y 1 ).

AC = x 2 − x 1

y

0 x

1 A

M

P

Q

B(8, 5)

(2, 1)

3

3

2

3

2

1 2

2

3

4

5

4 5 6 7 8

Figure 2.8