A tangent of a circle is a line that intersects the circle exactly once.

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

Two lines are perpendicular if they meet at right angles.

Points A and B are equidistant from a third point C if the distance AC is equal to the distance BC.

Identifying which circle theorem to use

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 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

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

Points X and Y are points where two separate tangents intersect the circle.
$$a$$° = {{}}°.

Points X and Y are points where two separate tangents intersect the circle.
Calculate the size of the angle labelled $$b$$°.

Points A and B are points where two separate tangents intersect the circle.
Calculate the size of the angle labelled $$c$$°.

Points A and B are points where two separate tangents intersect the circle.
$$d$$° = {{}}°.

The tangent shown intersects the circle at point T.
$$x$$° = {{}}°.

Point T is a point on the circumference where a tangent intersects the circle.
$$y$$° = {{}}°.

∠AKC = 21°
∠AWK is subtended by the chord AK.
$$a$$° = {{}}°.

The image shows a circle, a tangent to the circle and three line segments forming a triangle inside the circle. The size of the angle labelled $$b$$° is {{}}°.

Find the size of angle $$c$$° (in degrees).

Calculate the size of angle $$g$$° (in degrees).

Starter Quiz

∠AKC = 21° ∠AWK is subtended by the chord AK. $$a$$° = °.