Understanding Engineering Mathematics

AngleBAClooks like a right angle, but this doesn’t mean that it is.

In fact in this case itis, because, as you may have noticed, the sides

satisfy Pythagoras: 3^2 + 42 = 52 (➤ 154 ).

5.2.6 The intercept theorem

➤

145 162➤

The intercept theorem states thata straight line drawn parallel to one side of a triangle
divides the other two sides in the same ratio(14
➤
) (see Figure 5.16) i.e.AD/DC=
AE/EB.

D

CB

E

A

Figure 5.16The intercept theorem.

Thus, for example, if we knowAD,DC,AEsay then we can deduceEB.
The proof of this theorem may be found in most standard books on geometry and relies
on the fact thatABCandAEDare similar triangles.

Solution to review question 5.1.6

(i) By the intercept theorem, sinceEDis parallel toCBin Figure 5.6(i),

the lineEDdivides sidesACandABin the same ratio. Hence

AE

EC

=

4

6

=

AD

DB

=

5

DB

SoDB= 15 /2 units.

(ii) This is the case where the parallel intercept is actually outside the

triangle (Figure 5.6(ii)). This in fact makes no difference to the result

and again, by the intercept theorem,

AD

DC

=

3

1. 5

=

BD

DE

=

BD

2

SoDB=4 units.

5.2.7 The angle bisector theorem

➤

145 163➤

The intercept theorem is about dividing the sides of triangles. Here we consider a theorem
which divides an angle of a triangle:the line bisecting an angle of a triangle divides
the side opposite to that angle in the ratio(14
➤
)of the sides containing the angle
(see Figure 5.17).