An angle can be subtended by a line segment or curve. The legs of the subtended angle meet the endpoints of the line segment or curve.

An arc of a circle is part of the circle’s circumference.

A chord is any line segment joining two points on the circumference of a circle.

The diameter of a circle is any line segment that starts and ends on the circumference of the circle and passes through the centre of the circle.

The angle in a semicircle is a right angle

Maths

Checking and securing understanding of the parts of a circle

The angle at the centre of the circle is twice the angle at any point on the circumference

The perpendicular from the centre of a circle to a chord bisects the chord

The tangent at any point on a circle is perpendicular to the radius at that point

The angles in the same segment are equal

The alternate segment theorem

The opposite angles of a cyclic quadrilateral sum to 180°

The tangents from an external point are equal in length

Identifying which circle theorem to use

Problem solving with circle theorems

In the prior unit, pupils applied their understanding of angles to solve increasingly complex problems. In this unit, pupils will use their developed mathematical reasoning skills to derive and use various circle theorems.

In this unit, pupils will use their developed mathematical reasoning skills to derive and use various circle theorems.

Circle theorems

Find the size of angle $$a$$° on this 8-point geoboard (in degrees).

Find the size of angle $$b$$° on this 8-point geoboard (in degrees).

The size of the angle labelled $$c$$° on this 8-point geoboard is {{}}°.

The size of the angle $$(b+d)$$° on this 8-point geoboard is {{}}°.

The size of the angle labelled $$x$$° is {{}}°.

Chord AB has length of 4 cm.
The length of chord BC is {{}} cm (2 d.p.).

A chord has endpoints of A and B.
Where on the circle must points A and B lie?

The size of the angle labelled $$x$$° on this 20-point geoboard is {{}}°.

The size of the angle labelled $$y$$° on this 20-point geoboard is {{}}°.

The size of the angle labelled $$z$$° on this 20-point geoboard is{{}}°.

What is the size of the angle subtended by the chord AC (in degrees)?

The size of the angle labelled $$m$$° is {{}}°.

The angle in a semicircle is a right angle

Lesson outcome