Circle theorems I: past paper questions

Where the questions come from

• Question 15May/June 2026, paper 22Structured5 marksHas a diagram

  $A$, $B$, $C$ and $D$ are points on a circle. $EF$ is a tangent to the circle at $A$. Angle $ADB={20}^{\circ }$ and angle $BCD={75}^{\circ }$.

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• Question 14Oct/Nov 2025, paper 21Structured9 marksHas a diagram

  $A,B,C,D$ and $E$ lie on the circle. $AC$ and $BD$ intersect at $X$. Angle $ACD={55}^{\circ }$ and angle $CXD={88}^{\circ }$.

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• Question 17Oct/Nov 2024, paper 22Structured2 marksHas a diagram

  $A$, $B$ and $C$ are points on the circumference of a circle with centre $O$. $DA$ and $DC$ are tangents to the circle. Angle $ABC={64}^{\circ }$.

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• Question 17Oct/Nov 2023, paper 21Structured3 marksHas a diagram

  $A$, $B$ and $C$ are points on the circumference of a circle, centre $O$. Tangent $DE$ touches the circle at $C$. Angle $BCE={53}^{\circ }$ and angle $ACO={20}^{\circ }$.

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• Question 13May/June 2026, paper 23Structured3 marksHas a diagram

  $A$, $B$ and $C$ are points on a circle with centre $O$. $AOC$ is a straight line and $M$ is the midpoint of $AB$. $AB=6\text{ cm}$ and $OM=4\text{ cm}$.

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• Question 13Oct/Nov 2025, paper 23Structured4 marksHas a diagram

  $A$, $B$, $C$ and $D$ are points on the circumference of a circle with centre $O$. $ED$ and $EB$ are tangents to the circle. $AC$ is parallel to $EB$.

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• Question 14Oct/Nov 2024, paper 23Structured4 marksHas a diagram

  $A,B,C$ and $D$ lie on the circle. $TAS$ is a tangent to the circle at $A$. Find the value of $x$.

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• Question 17Oct/Nov 2023, paper 22Structured4 marksHas a diagram

  $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $OBA={28}^{\circ }$. Find angle $ACB$.

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• Question 15May/June 2026, paper 23Structured3 marksHas a diagram

  The diagram shows a cyclic quadrilateral $ABCD$. Find angle $ACB$

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• Question 21Oct/Nov 2025, paper 23Structured4 marksHas a diagram

  The diagram shows a cyclic quadrilateral. The ratio $p:m=2:3$. Find the value of $y$.

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• Question 4Oct/Nov 2024, paper 42Structured15 marksHas a diagram

  The angles of a quadrilateral are $w^{\circ }$, $x^{\circ }$, $y^{\circ }$ and $z^{\circ }$. The ratio $w:(x+y+z)=3:5$. Find the value of $w$.

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• Question 5Oct/Nov 2023, paper 41Structured14 marksHas a diagram

  $ACD$ is a right-angled triangle. $B$ is on $AC$ and $BC=54.5\text{ m}$. $AD=83.2\text{ m}$ and angle $ABD={38}^{\circ }$.

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• Question 15Oct/Nov 2025, paper 41Structured3 marksHas a diagram

  $A$, $B$ and $C$ lie on a circle, centre $O$. $TA$ and $TB$ are tangents to the circle at $A$ and $B$. Calculate angle $ATB$.

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• Question 13May/June 2024, paper 21Structured3 marksHas a diagram

  The diagram shows a cyclic quadrilateral. Find the value of $p$.

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• Question 10Oct/Nov 2023, paper 42Structured7 marksHas a diagram

  $ABCDEF$ is a regular hexagon. $DF$, $DA$ and $DB$ are diagonals. Complete the following statements using three different triangles.

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• Question 12Oct/Nov 2025, paper 42Structured4 marksHas a diagram

  $A$, $B$ and $C$ lie on a circle centre $O$. $DE$ is a tangent to the circle at $C$. Angle $ABC={52}^{\circ }$ and angle $BCE={65}^{\circ }$.

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• Question 14May/June 2024, paper 23Structured3 marksHas a diagram

  $P$, $Q$, $R$ and $T$ are points on the circle. $AB$ is a tangent to the circle at $T$. Angle $ATP={50}^{\circ }$, angle $PTR={48}^{\circ }$ and $PQ=QR$.

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• Question 4Oct/Nov 2023, paper 43Structured7 marksHas a diagram

  Find the size of one interior angle of a regular 10-sided polygon.

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• Question 10May/June 2025, paper 21Structured4 marksHas a diagram

  $A$, $B$ and $C$ are three points on a circle, centre $O$. $DE$ is a tangent to the circle at $A$. Angle $ACO={35}^{\circ }$ and angle $BCO={40}^{\circ }$.

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• Question 2May/June 2024, paper 42Structured9 marksHas a diagram

  The diagram shows a straight line intersecting two parallel lines. Find the value of $a$ and the value of $b$.

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• Question 12May/June 2023, paper 21Structured4 marksHas a diagram

  $AO$, $OB$ and $OC$ are all radii of the circle. $AB=BC$. Therefore triangle $AOB$ is congruent to triangle $COB$.

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• Question 12May/June 2025, paper 22Structured2 marksHas a diagram

  $P$, $Q$ and $R$ lie on a circle. $QR$ is a diameter. Find angle $PRQ$.

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• Question 13May/June 2023, paper 22Structured2 marksHas a diagram

  $A$, $B$, $C$ and $D$ are points on a circle. $AB$ is parallel to $DC$ and angle $ACD={32}^{\circ }$. Chords $AC$ and $DB$ intersect at $E$.

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• Question 13May/June 2025, paper 23Structured4 marksHas a diagram

  $A$, $B$, $C$ and $D$ are points on a circle. $EF$ is a tangent to the circle at $A$. $AB$ is parallel to $DC$.

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• Question 1May/June 2023, paper 42Structured18 marksHas a diagram

  The diagram shows an isosceles triangle with the base extended. Find the value of $x$.

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• Question 18May/June 2025, paper 42Structured4 marksHas a diagram

  $A$, $B$, $C$, $D$ and $E$ lie on a circle. Find the values of $x$, $y$ and $z$.

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• Question 4May/June 2023, paper 43Structured13 marksHas a diagram

  $A$, $B$ and $C$ are three towns and the bearing of $C$ from $A$ is ${114}^{\circ }$. $B$ is due south of $A$ and $AC=BC$. Calculate the bearing of $B$ from $C$.

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