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.

The tangents from an external point are equal in length

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°

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

A triangle is formed on the circle from a segment of a tangent, a radius and another line segment. The size of angle $$a$$° is {{}}°.

Points T and U are points where two separate tangents intersect the circle.
Find the size of angle $$b$$° (in degrees).

Points G and H are points where two separate tangents intersect the circle.
The size of angle $$c$$° is {{}}°.

Points Q and P are points where two separate tangents intersect the circle.
The size of the angle labelled $$d$$° is {{}}°.

Points J and K are points where two separate tangents intersect the circle.
The size of angle $$e$$° is {{}}°.

Points L and N are points where two separate tangents intersect the circle. ∠LMN = 118°.
$$g$$° = {{}}°.

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

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

∠ATC = 51°
∠ABT is subtended by the chord AT.
The size of the angle labelled $$e$$° is {{}}°.

∠PEF is subtended by the chord PF.
∠PFE is subtended by the chord PE.
Find ∠EPD (in degrees).

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

The tangents from an external point are equal in length

Lesson outcome