Circle Terminology

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Year 10 Interactive Maths - Second Edition

Circle Terminology

In this section we will consider arcs, subtended angles, the angle at the circumference and the Angle at Centre Theorem.

Arcs

An arc is a curved line along the circumference of a circle.

For example, AXB is a minor arc, and AYB is a major arc.

In the circle, AXB is known as the minor arc and AYB is known as the major arc.

Subtended Angles

The angle RAT is said to be the angle subtended by the tree TR at A. So, the tree subtends an angle of 25º at A. This can be described as follows.
'An angle of 25º is subtended at A by the tree'.

Angle RAT of size 25 degrees is subtended by TR.

Angle at the Circumference

If the end points of an arc are joined to a third point on the circumference of a circle, then an angle is formed.
For example, the minor arc AB subtends an angle of 45º at C. The angle ACB is said to be the angle subtended by the minor arc AB (or simply arc AB) at C.

In the circle, the angle subtended by the minor arc AB at C is 45 degrees.

The angle ACB is an angle at the circumference standing on the arc AB.

Angle at the Centre

If the end points of an arc are joined to the centre of a circle, then an angle is formed. For example, the minor arc AB subtends an angle of 105º at O. The angle AOB is said to be the angle subtended by the minor arc AB (or simply arc AB) at the centre O.

The minor arc AB subtends the angle at the centre, AOB, of size 105 degrees.

The angle AOB is an angle at the centre O standing on the arc AB.

Angle at Centre Theorem

Theorem

Use the information given in the diagram to prove that the angle at the centre of a circle is twice the angle at the circumference if both angles stand on the same arc.

In the circle, angle AOB at the centre is subtended by the arc AB and angle ACB at the circumference is also subtended by the arc AB. The diameter of the circle passing through O and touching C forms angles of size a degrees and b degrees near the point C and x degrees and y degrees near the centre O.

Given:

To prove:

Proof:

From triangle OAC, x = a + a {Exterior angle of a triangle}. So, x = 2a ...(1). From triangle OBC, y = b + b {Exterior angle of a triangle}. So, y = 2b ...(2). Adding (1) and (2) gives x + y = 2a + 2b = 2(a + b). So, angle AOB = 2(angle ACB) as required.

In general:

The angle at the centre of a circle is twice the angle at the circumference if both angles stand on the same arc. This is called the Angle at Centre Theorem.

We also call this the basic property, as the other angle properties of a circle can be derived from it.

Example 22

Find the value of the pronumeral in the following circle centred at O.

In the circle, the angle AOB at the centre subtended by arc AB is 4x degrees and the angle ACB at the circumference subtended by arc AB is 60 degrees.

Solution:

Example 23

Find the value of the pronumeral in the following circle centred at O.

Solution:

Angle in a Semi-Circle

In general:

The angle in a semi-circle is a right angle.

Example 24

Find the value of the pronumeral in the following circle centred at O.

Solution:

Key Terms

arc, minor arc, major arc, subtended angle, angle at the circumference, angle at the centre, Angle at Centre Theorem, basic property of a circle, angle in a semi-circle

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