Understanding Engineering Mathematics

5.2 Revision

5.2.1 Division of a line in a given ratio

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143 160➤

Points and lines are regarded asundefined primitive concepts(that is, we assume we
all know what is meant by them) in geometry, in terms of which theaxiomsor rules of
geometry may then be expressed. Apointis a geometrical element which has a position,
but no size or extent. Alineis a straight one-dimensional geometrical figure of infinite
length and no thickness. There is a unique straight line passing through two specified
pointsAandB.Aline segmentis a finite portion of a line between two fixed points.
Its length is the shortest distance between the points in a plane. Note that we are talking
simply about geometry on a plane here, rather than, for example spherical geometry, which
deals with geometrical properties on the surface of a sphere.
Much elementary geometry depends on the division of a line by a point. We say a point
P on a lineABdivides the lineinternally in the ratiop:qifP is betweenAandB

andAP:PB=p:q,or

AP

PB

=

p

q

( 14

➤

).

If a pointP is onABproduced (that is, extended in the directionAB), thenP is said

to divideABexternally in the ratiop:qifAP:PB=p:qor

AP

PB

=

p

q

.

Solution to review question 5.1.1

For division internally we have, lettingAP=xcm, say:

ABP

x 30 − x

ThenPB=( 30 −x)cm, and since

AP:PB=3:2

we have

x

30 −x

=

3

2

or 2x= 3 ( 30 −x)= 90 − 3 x

or

5 x= 90

Sox=18 and hence

AP=18 cm andPB=12cm.

Alternatively, if we let the length of AP be 3y units and that of

PB 2 yunits, then 3y+ 2 y= 5 y=30, from whichy=6. Thus,APis

18 cm andPB12 cm as before.

For divisionexternallyletAP=xthen, from the figure below,

AP

PB

=

x

x− 30

=

3

2

or

x= 90