Revision mat · Circle Theorems

8 marks

Question mix weighted by 75 real exam questions from 55 GCSE papers (2017–24), so the most common question types get the most space. Show your working.

Angle at the centre $x = 2a$Angle in a semicircle $90°$Same segment $a = a$Cyclic quadrilateral $a + c = 180°$Tangent & radius $90°$

Alternate segment≈41% of real exam Qs

1In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle.
The angle between the tangent and the chord $TA$ is $42^\circ$.
Work out the size of angle $ABT$, marked $x$.

Answer: (2 marks)

Cyclic quadrilateral≈16% of real exam Qs

2$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$.
Angle CDA $= 112^\circ$.
Work out the size of angle ABC, marked $x$.

Answer: (2 marks)

Angle at centre≈11% of real exam Qs

3In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$.
The angle at the centre $AOB = 86^\circ$.
Work out the size of angle $x$.

Answer: (2 marks)

★ Exam capstoneeverything else ≈32% of real exam Qs

4$PA$ and $PB$ are tangents to the circle, centre $O$, at $A$ and $B$. $C$ is a point on the circle.
Angle $APB = 62^\circ$.
Work out the size of angle $AOB$.

Answer: (2 marks)

Mark scheme · Circle Theorems

Total: 8 marks

Award the marks shown for each correct step, then add up the total out of 8. A method mark counts even if the final answer is wrong.

Method

By the alternate segment theorem, the angle between the tangent and the chord equals the angle in the alternate segment.

$x = 42^\circ$.

Answer: $42$

Marks

✔1 markalternate segment theorem

✔1 mark42°

2$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$.
Angle CDA $= 112^\circ$.
Work out the size of angle ABC, marked $x$.[2 marks]

Method

Opposite angles of a cyclic quadrilateral add up to $180^\circ$.

$x = 180^\circ - 112^\circ = 68^\circ$.

Answer: $68$

Marks

✔1 markopposite angles of a cyclic quad sum to 180°

✔1 mark68°

3In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$.
The angle at the centre $AOB = 86^\circ$.
Work out the size of angle $x$.[2 marks]

Method

The angle at the centre is twice the angle at the circumference, so the angle at the circumference is half the angle at the centre.

$x = 86^\circ \div 2 = 43^\circ$.

Answer: $43$

Marks

✔1 markangle ACB = ½ × angle AOB

✔1 mark43°

4$PA$ and $PB$ are tangents to the circle, centre $O$, at $A$ and $B$. $C$ is a point on the circle.
Angle $APB = 62^\circ$.
Work out the size of angle $AOB$.[2 marks]

Method

$OA$ meets the tangent $PA$ at $90^\circ$, and $OB$ meets $PB$ at $90^\circ$ (a tangent meets a radius at $90^\circ$).

The angles of quadrilateral $OAPB$ add up to $360^\circ$: angle $AOB = 360^\circ - 90^\circ - 90^\circ - 62^\circ = 118^\circ$.

Answer: $118$

Marks

✔1 markangles OAP and OBP are 90° (tangent meets radius)

✔1 markangle AOB = 118°