Understanding Engineering Mathematics

A

C PDM B

AP = Altitude

AM = Median

AD = Angle bisector

Figure 5.17Altitude, median and angle bisector.

So the angle bisector theorem states that

AC

AB

=

CD

DB

The angle bisector should not be confused with two other lines dropped from a vertex of
a triangle – the altitude and the median, also shown in Figure 5.17. Thealtitudeis the
line drawn from a vertex of a triangle, perpendicular to the opposite side. Amedianof a
triangle is the line joining a vertex to the midpoint of the opposite side. You might like
to explore the circumstances under which two or more of these lines are in fact the same
thing.

Solution to review question 5.1.7

By the angle bisector theorem we have, in Figure 5.7.

AC

AB

=

CD

DB

So DB=

CD×AB

AC

=

2 × 9 / 10

1

=

9

5

5.2.8 Pythagoras’ theorem

➤

146 163➤

For any right-angled triangleABCPythagoras’ theorem tells us that

(BC)^2 +(AC)^2 =(AB)^2

AC

B

Figure 5.18Pythagoras’ theorem.