Basic Engineering Mathematics, Fifth Edition

172 Basic Engineering Mathematics

A right-angled triangleABCis shown in Figure 20.19.

The point of intersection of two lines is called a vertex

(pluralvertices); the three vertices of the triangle are

labelled asA,BandC, respectively. The right angle

is angleC. The side opposite the right angle is given

the special name of thehypotenuse. The hypotenuse,

lengthABin Figure 20.19, is always the longest side of

a right-angled triangle. With reference to angleB, AC

is theoppositeside andBCis called theadjacentside.

With reference to angleA, BCis theoppositeside and

ACis theadjacentside.

Often sides of a triangle are labelled with lower case

letters,abeing the side opposite angleA,bbeing the

side opposite angleBandcbeing the side opposite

angleC. So, in the triangleABC, lengthAB=c, length

BC=aand lengthAC=b. Thus,cis the hypotenuse

in the triangleABC.

∠is the symbol used for ‘angle’. For example, in the

triangle shown,∠C= 90 ◦. Another way of indicating

an angle is to use all three letters. For example,∠ABC

actually means∠B; i.e., we take the middle letter as the

angle. Similarly,∠BACmeans∠Aand∠ACBmeans

∠C.

Here are some worked examples to help us understand

more about triangles.

Problem 18. Name the types of triangle shown in

Figure 20.20

2

2.6

2.8

2.5

2.5

2.1

2.1

398

1078

518

2

(b)

(d) (e)

(a) (c)

2

Figure 20.20

(a) Equilateral triangle(since all three sides are

equal).

(b) Acute-angled scalene triangle (since all the

angles are less than 90◦).

(c) Right-angled triangle(39◦+ 51 ◦= 90 ◦; hence,

the third angle must be 90◦, since there are 180◦

in a triangle).

(d) Obtuse-angled scalene triangle(since one of the

angles lies between 90◦and 180◦).

(e) Isosceles triangle(since two sides are equal).

Problem 19. In the triangleABCshown in

Figure 20.21, with reference to angleθ, which side

is the adjacent?

A B



C

Figure 20.21

The triangle is right-angled; thus, side AC is the

hypotenuse. With reference to angleθ, the opposite side

isBC. The remaining side,AB,is the adjacent side.

Problem 20. In the triangle shown in

Figure 20.22, determine angleθ

568



Figure 20.22

The sum of the three angles of a triangle is equal to

180 ◦.

The triangle is right-angled. Hence,

90 ◦+ 56 ◦+∠θ= 180 ◦

from which, ∠θ=^180 ◦−^90 ◦−^56 ◦=^34 ◦.

Problem 21. Determine the value ofθandαin

Figure 20.23