Cambridge International Mathematics

270 Coordinate geometry (Chapter 12)

COLLINEAR POINTS

Three or more points arecollinearif they lie on the same straight line.

If three points A, B and C are collinear, the gradient of AB is equal

to the gradient of BC and also the gradient of AC.

Example 17 Self Tutor

Show that the following points are collinear: A(1,¡1),B(6,9),C(3,3).

EXERCISE 12E.2

1 Determine whether or not the following sets of three points are collinear:

a A(1,2),B(4,6) and C(¡ 4 ,¡4) b P(¡ 6 ,¡6),Q(¡ 1 ,0) and R(4,6)

c R(5,2),S(¡ 6 ,5) and T(0,¡4) d A(0,¡2),B(¡ 1 ,¡5) and C(3,7)

2 Findcgiven that:

a A(¡ 4 ,¡2),B(0,2) and C(c,5) are collinear

b P(3,¡2),Q(4,c) and R(¡ 1 ,10) are collinear.

Coordinate geometry is a powerful tool which can be used:

² tocheckthe truth of a geometrical fact

² toprovea geometrical fact by using general cases.

In these problems we find distances, midpoints, and gradients either from a sketch or by using the appropriate

formulae.

Example 18 Self Tutor

P(3,¡1),Q(1,7)and R(¡ 1 ,5)are the vertices of triangle PQR.

M is the midpoint of PQ and N is the midpoint of PR.

a Find the coordinates of M and N. b Find the gradients of MN and QR.

c What can be deduced fromb? d Find distances MN and QR.

e What can be deduced fromd?

F USING COORDINATE GEOMETRY [7.2 - 7.5]

A

B

C

gradient of AB =

9 ¡¡ 1

6 ¡ 1

=

10

5

=2

gradient of BC=

3 ¡ 9

3 ¡ 6

=

¡ 6

¡ 3

=2

) AB is parallel to BC, and as point B is common to both line segments,

A, B and C arecollinear.

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y:\HAESE\IGCSE01\IG01_12\270IGCSE01_12.CDR Friday, 3 October 2008 12:13:21 PM PETER