A circle is the set of all points in the plane that are a fixed distance (the radius) from a fixed point (the centre).
Any interval joining a point on the circle to the centre is called a radius. By the definition of a circle, any two radii have the same length. Notice that the word ‘radius’ is being used to refer both to these intervals and to the common length of these intervals.
An interval joining two points on the circle is called a chord.
A chord that passes through the centre is called a diameter. Since a diameter consists of two radii joined at their endpoints, every diameter has length equal to twice the radius. The word ‘diameter’ is use to refer both to these intervals and to their common length.
A line that cuts a circle at two distinct points is called a secant. Thus a chord is the interval that the circle cuts off a secant, and a diameter is the interval cut off by a secant passing through the centre of a circle centre.
Symmetries of a circle
Circles have an abundance of symmetries:
A circle has every possible rotation symmetry about its centre, in that every rotation of the circle about its
centre rotates the circle onto itself.
If AOB is a diameter of a circle with centre O, then the
reflection in the line AOB reflects the circle onto itself.
Thus every diameter of the circle is an axis of symmetry.
As a result of these symmetries, any point P on a circle
can be moved to any other point Q on the circle. This can
be done by a rotation through the angle θ = anglePOQ about
the centre. It can also be done by a reflection in the diameter
AOB bisecting anglePOQ. Thus every point on a circle is essentially
the same as every other point on the circle − no other figure in
the plane has this property except for lines.
EXERCISE 1
a
Identify all translations, rotations and reflections of the plane that map a line script_l onto itself.
b
Which of the transformations in part a map a particular point P on script_l to another particular point Q on script_l.
Congruence and similarity of circles
Any two circles with the same radius are congruent− if one circle is moved so that its centre coincides with the centre of the other circle, then it follows from the definition that the two circles will coincide.
More generally, any two circles are similar − move one circle so that its centre coincides with the centre of the other circle, then apply an appropriate enlargement so that it coincides exactly with the second circle.
A circle forms a curve with a definite length, called the circumference, and it encloses a definite area. The similarity of any two circles is the basis of the definition of π, the ratio of the circumference and the diameter of any circle. We saw in the module, The Circles that if a circle has radius r, then
circumference of the circle = 2πr and area of the circle = πr2
Radii and chords
Let AB be a chord of a circle not passing through its
centre O. The chord and the two equal radii OA and
BO form an isosceles triangle whose base is the chord.
The angle angleAOB is called the angle at the centre
