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.

The radius is any line segment that joins the centre of a circle to any point on its circumference.

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

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

Maths

Checking and securing understanding of the parts of a circle

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

The size of angle $$x$$° on this 12-point geoboard is {{}}°.

The size of angle $$y$$° on this 12-point geoboard is {{}}°.

Angle $$a$$° is subtended by chord TU on this 12-point geoboard.
$$a$$° = {{}}°

Angle $$b$$° is subtended by chord TU at the circumference of this 12-point geoboard.
$$b$$° = {{}}°

The size of the angle marked $$d$$° is {{}}°.

What is the dashed line segment on this circle called?

The line segment on this circle passes through its centre.
What is the shaded region on this circle called?

The circumference forms part of a segment.

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

“circle with diameter 8 more than its radius”

“circle with radius 12 less than its diameter”

A 36° angle at the centre (O) of a circle has been subtended by the arc XY.
The circle has a radius of 50 cm.
The area of the shape bound the arc XY and the legs of the 36° angle is {{}} cm² (2 s.f.).

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

Lesson outcome