Understanding Engineering Mathematics

A

0 x 1 x 2

x 2 − x 1

y 2 − y 1

y

x

M

Q

P

B C

(x 2 , y 2 )

(x 1 , y 1 )

Figure 7.6Midpoint of a line segment.

Similarly for theycoordinate – it is the same as that ofP,whichis

y 1 +

y 2 −y 1

2

=

y 1 +y 2

2

More generally, again by using the intercept theorem or similar triangles (152
➤
), the
pointMthat divides the line joining (x 1 ,y 1 )to(x 2 ,y 2 ) in the ratio (14
➤
)μ:λis
(
λx 1 +μx 2
λ+μ

,

λy 1 +μy 2

λ+μ

)

It may help in remembering this to note that theλandμare adjacent to the ‘opposite’
points:
m M l
(x 1 , y 1 )(x 2 , y 2 )
Thegradient,orslopeof the line is given by

m=

AC

BC

=

y 2 −y 1

x 2 −x 1

Note that this is the same as tan( ABC). This holds for all values ofx 1 ,x 2 ,y 1 ,y 2 except
of course thatx 2 −x 1 must be non zero, i.e.x 1 =x 2 .Thecasewherex 1 =x 2 actually
corresponds to a vertical line, parallel to they-axis. Crudely, you might think of it as
having an infinite gradient.
If a line slopes upwards from right to left, i.e.yincreases asxincreases, then we have
a positive gradient. If it slopes downwards to the right, orydecreases asxincreases then
the gradient is negative.