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.

Checking and securing understanding of the parts of a circle

Maths

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

The angle in a semicircle is a right angle

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

What is the dashed line segment on this circle called?

What is the shaded region on this circle called?

A circular sector is made from exactly one radius on the circle and an arc.

A circular sector is made from exactly two radii on the circle and an arc.

A circular sector is made from exactly one chord on the circle and an arc.

A circular sector is made from exactly two chords on the circle.

A circular sector is made from either radii or chords on the circle and an arc.

Which of these statements are always correct?

The size of the angle subtended by chord AB is {{}} degrees.

The size of the angle subtended by chord AC is {{}} degrees.

∠XOY is a 60° angle at the centre of a circle subtended by the arc XY.
The circle has a radius of 42 cm.
The area of the sector OXY is {{}} cm² (2 s.f.)

The length of the diameter of this circle is {{}} cm.

The diameter of a circular fence is greater than 8 metres long. Which of these are possible lengths of the radius of this fence?

A semicircular sector has been cut from a full circle of equal radius.
What fraction of the full circle is this sector?

This sector has been cut from a circle.
What fraction of the full circle is this sector?

This circular sector is broken down into two straight line segments and an arc.
The perimeter of this arc is {{}} cm.

Checking and securing understanding of the parts of a circle

Lesson outcome