Cambridge International Mathematics

Coordinate geometry (Chapter 12) 259

EXERCISE 12B.1

1

a A and B b A and D c C and A

d F and C e G and F f C and G

g E and C h E and D i B and G.

2

a A(3,5)and B(2,6) b P(2,4)and Q(¡ 3 ,2) c R(0,6)and S(3,0)

d L(2,¡7)and M(1,¡2) e C(0,5)and D(¡ 4 ,0) f A(5,1)and B(¡ 1 ,¡1)

g P(¡ 2 ,3)and Q(3,¡2) h R(3,¡4)and S(¡ 1 ,¡3) i X(4,¡1)and Y(3,¡3)

THE DISTANCE FORMULA

To avoid drawing a diagram each time we wish to find a distance, a

distance formulacan be developed.

In going from A to B, the x-step =x 2 ¡x 1 , and

the y-step =y 2 ¡y 1.

Now, using Pythagoras’ theorem,

(AB)^2 =(x-step)^2 +(y-step)^2

) AB=

p

(x-step)^2 +(y-step)^2

) d=

q

(x 2 ¡x 1 )^2 +(y 2 ¡y 1 )^2.

Example 4 Self Tutor

Find the distance between A(¡ 2 ,1)and B(3,4).

A(¡ 2 ,1) B(3,4)

x 1 y 1 x 2 y 2

AB=

p

(3¡¡2)^2 +(4¡1)^2

=

p

52 +3^2

=

p

25 + 9

=

p

34 units

The distance formula saves

us having to graph the points

each time we want to find a

distance. However, you can

still use a sketch and

Pythagoras if you need.

y

x

d

x 1

y 1

x 2

y 2

A( , )xy 11

B( , )xy 22

x-step

y-step

O

If A(x 1 ,y 1 ) and B(x 2 ,y 2 ) are two points in a plane, then the

AB=

p

(x 2 ¡x 1 )^2 +(y 2 ¡y 1 )^2

or d=

p

(x-step)^2 +(y-step)^2.

distance between these points is given by:

y

x

G

F

E

A

C

B

D

O

If necessary, use Pythagoras’ theorem to find the

distance between:

Plot the following pairs of points and use Pythagoras’ theorem to find the distances between them.

Give your answers correct to 3 significant figures:

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Y:\HAESE\IGCSE01\IG01_12\259IGCSE01_12.CDR Thursday, 2 October 2008 12:44:32 PM PETER