Understanding Engineering Mathematics

the anglesubtended by the minor arcABat the centreOisAOB.Wesaytheangle
ACBstands on the minor arcAB, which is said tosubtendan angle ACBat the

circumference. Exactly the same statements can be made replacingCbyD. Implicit in
the definitions is the assumption that there is justoneangle subtended at the circumfer-
ence – i.e. that ACB= ADB. As we state below, this is in fact the case.
Now let’s add a chord to the circle. There are two important results to note regarding
chords.

• Chords equidistant from the centre are of equal lengths

To see this consider the triangles formed by connecting the ends of two such chords
to the centre. These two triangles have two corresponding sides (the radii) equal, and the
corresponding enclosed angles are equal. The triangles are therefore congruent, and so
their third sides (i.e. the length of the chords) are also equal.

• The perpendicular bisector of a chord of a circle passes through the
  centre of the circle

The proof is similar to that of the first result using congruent triangles, but the result is
sufficiently ‘self evident’ to gloss over here.
The next results concern the angles subtended by arcs at the centre and on the circum-
ference. Whilst it is important that you understand and can use the results, the proofs are
not important to us here.

• The angle subtended by an arc at the centre of a circle is twice the
  angle subtended at the circumference (Figure 5.22)

O O

a

a

2 a^2 a

Figure 5.22Angle subtended at the centre is twice that on the circumference.

• All angles subtended at the circumference by the same arc are equal


• The angle in a semicircle is 90°– i.e. the angle subtended by a diam-
  eter is a right angle

The last result follows from the fact that a semicircle subtends an angle of 180°at the
centre of a circle. Any angle subtended at the circumference must, by the above results,
be equal to half this, namely 90°.
Finally, we give some important results relating to the properties of tangents to circles.
Atangentto a circle is a straight line which touches it at exactly one point – thepoint
of contact. A straight line which cuts a circle at two distinct points is called asecant(a
chord is thus a segment of a secant). An important property of the tangent is: