Question

The tangent DB is produced to T.

Given ÐTBA = 62⁰. Find the values of x and y. The diagram shows a circle, centre O. Give reasons.

Answer 1

Answer:

RADII AND CHORDS

We begin by recapitulating the definition of a circle and the terminology used for circles. Throughout this module, all geometry is assumed to be within a fixed plane.

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.