Cambridge International Mathematics

Coordinate geometry (Chapter 12) 271

a Mis

μ

3+1

2

,

¡1+7

2

¶

which is (2,3).Nis

μ

3+¡ 1

2

,

¡1+5

2

¶

which is (1,2).

b gradient of MN=

2 ¡ 3

1 ¡ 2

=1

gradient of QR=

5 ¡ 7

¡ 1 ¡ 1

=1

c Equal gradients implies that MNkQR.

d MN =

p

(1¡2)^2 +(2¡3)^2

=

p

1+1

=

p

2

¼ 1 : 41 units

QR=

p

(¡ 1 ¡1)^2 +(5¡7)^2

=

p

4+4

=

p

8

=2

p

2

¼ 2 : 83 units

e Fromd, QR is twice as long as MN.

Q,()1 ¡7

M

N

R,()-1 ¡5

P,()3 -1

For figures named

ABCD, etc. the

labelling is in

cyclic order.

or

EXERCISE 12F

1 Given A(0,4),B(5,6) and C(4,1), where M is the midpoint of AB and N is the midpoint of BC:

a Illustrate the points A, B, C, M and N on a set of axes.

b Show that MN is parallel to AC, using gradients.

c Show that MN is half the length of AC.

2 Given K(2,5),L(6,7),M(4,1):

a Illustrate the points on a set of axes.

b Show that triangle KLM is isosceles.

c Find the midpoint P of LM.

d Use gradients to verify that KP is perpendicular to LM.

e Illustrate what you have found inb,canddon your sketch.

3 Given A(3,4),B(5,8),C(13,5) and D(11,1):

a Plot A, B, C and D on a set of axes.

b Use gradients to show that:

i AB is parallel to DC ii BC is parallel to AD.

c What kind of figure is ABCD?

d Check that AB=DC and BC=AD using the distance formula.

e Find the midpoints of diagonals: i AC ii BD.

f What property of parallelograms has been checked ine?

4 Given A(3,5),B(8,5),C(5,1) and D(0,1):

a Plot A, B, C and D on a set of axes. b Show that ABCD is a rhombus.

c Find the midpoints of AC and BD. d Show that AC and BD are perpendicular.

f 2

p

2 compared with

p

2 g

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Y:\HAESE\IGCSE01\IG01_12\271IGCSE01_12.CDR Thursday, 6 November 2008 9:44:44 AM PETER