Cambridge International Mathematics

268 Coordinate geometry (Chapter 12)

PERPENDICULAR LINES

Notice thatline 1andline 2are perpendicular.

Line 1has gradient^31 =3:

Line 2has gradient ¡ 31 =¡^13 :

We see that the gradients arenegative reciprocalsof each other

and their product is 3 £¡^13 =¡ 1.

For lines which are not horizontal or vertical:

² if the lines areperpendicularthen their gradients arenegative reciprocals

² if the gradients arenegative reciprocalsthen the lines areperpendicular.

Proof:

Example 14 Self Tutor

If a line has gradient^23 , find the gradient of:

a all lines parallel to the given line

b all lines perpendicular to the given line.

a Since the original line has gradient^23 , the gradient of all

parallel lines is also^23.

b The gradient of all perpendicular lines is¡^32 :

fthe negative reciprocalg

Example 15 Self Tutor

Findagiven that the line joining A(2,3) to B(a,¡1) is parallel to a line

with gradient¡ 2.

gradient of AB =¡ 2 fparallel lines have equal gradientg

)

¡ 1 ¡ 3

a¡ 2

=¡ 2

)

¡ 4

a¡ 2

=

¡ 2

1

3

3

1

• 1

line 1

line 2

y

x

(1)

(2)

A,()ab¡

A'()-¡ba,

O

Suppose the two perpendicular lines are translated so that they

intersect at the origin O. If A(a,b) lies on one line, then under

an anticlockwise rotation about O of 90 oit finishes on the

other line and its coordinates are A^0 (¡b,a).

The gradient of line (1) is

b¡ 0

a¡ 0

=

b

a

:

The gradient of line (2) is

a¡ 0

¡b¡ 0

=¡

a

b

:

b

a

and¡

a

b

are

negative

reciprocals of

each other.

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Y:\HAESE\IGCSE01\IG01_12\268IGCSE01_12.CDR Thursday, 2 October 2008 12:52:34 PM PETER