Cambridge International AS and A Level Mathematics Pure Mathematics 1

The mid-point of a line joining two points

P1^

2

EXAMPLE 2.1 A and B are the points (2, 5) and (6, 3) respectively (see figure 2.9). Find:

(i) the gradient of AB

(ii) the length of AB

(iii) the mid-point of AB

(iv) the gradient of a line perpendicular to AB.

SOLUTION

Taking A(2, 5) as the point (x 1 , y 1 ), and B(6, 3) as the point (x 2 , y 2 ) gives x 1 = 2,

y 1 = 5, x 2 = 6,y 2 = 3.

(i) Gradient =

yy

xx

21

21

35

62

1

2

–

–

–

= – =–

=

yy

xx

21

21

35

62

1

2

–

–

–

= – =–

(ii) Length AB

(iii) Mid-point =

 ++







=()++ =

xx 12 yy 12

22

26

2

53

2 44

,

,(,)

(iv) Gradient of AB = m 1 =–.^12

If m 2 is the gradient of a line perpendicular to AB, then m 1 m 2 = − 1

⇒ ––^12 m 2 =^1

m 2 = 2.

EXAMPLE 2.2 Using two different methods, show that the lines
joining P(2, 7), Q(3, 2) and R(0, 5) form a
right-angled triangle (see figure 2.10).
SOLUTION
Method 1
Gradient of RP =^7520 – – =^1

Gradient of RQ =^2530 – – =–^1

⇒ Product of gradients = 1 × (−1) = − 1

⇒ Sides RP and RQ are at right angles.

y

x

B(6, 3)

A(2, 5)

O

Figure 2.9

=− +−

=− +−

=+ =

() ()

() ()

xx 212 yy 212

622 352

16 420

y

x

P(2, 7)

R(0, 5)

Q(3, 2)

0

1

2

3

4

5

6

7

1 2 3 4

Figure 2.10