Circle Theorems

Non-calculator

Work through in order — each question is a small step harder than the last. Show your working.

  1. 1In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$.
    The angle at the centre $AOB = 96^\circ$.
    Which circle theorem did you use to find $x$? Give a reason for your answer.    Circle, angle at the centre ()ABCOx96°
  2. 2$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter.
    Angle CBA $= 30^\circ$.
    Which circle theorem did you use to find $x$? Give a reason for your answer.    Circle with a diameter, angle in a semicircle ()ABCO30°x
  3. 3$A$, $B$, $C$ and $D$ are points on a circle.
    Angle $ACB = 29^\circ$.
    Work out the size of angle $ADB$, marked $x$.    Circle, angles in the same segment ()ABCD29°x
  4. 4$A$, $B$, $C$ and $D$ are points on a circle.
    Angle $ACB = 42^\circ$.
    Which circle theorem did you use to find $x$? Give a reason for your answer.    Circle, angles in the same segment ()ABCD42°x
  5. 5$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$.
    Angle CDA $= 100^\circ$.
    Which circle theorem did you use to find $x$? Give a reason for your answer.    Cyclic quadrilateral ()ABCD100°x
  6. 6$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle.
    Angle $TOP = 34^\circ$.
    Which circle theorem did you use to find $x$? Give a reason for your answer.    Tangent and radius meeting at 90° ()TPO34°x
  7. 7From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$.
    Angle $APB = 50^\circ$.
    Which circle theorem did you use to find $x$? Give a reason for your answer.    Two tangents from a point ()ABPO50°x
  8. 8In 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 $67^\circ$.
    Which circle theorem did you use to find $x$? Give a reason for your answer.    Tangent and chord, alternate segment ()TAB67°x
  9. 9In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$.
    The angle at the centre $AOB = 110^\circ$.
    Work out the size of angle $x$.    Circle, angle at the centre ()ABCOx110°
  10. 10$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter.
    Angle CBA $= 42^\circ$.
    Work out the size of angle CAB, marked $x$.    Circle with a diameter, angle in a semicircle ()ABCO42°x
  11. 11$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$.
    Angle ABC $= 109^\circ$.
    Work out the size of angle CDA, marked $x$.    Cyclic quadrilateral ()ABCD109°x
  12. 12$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle.
    Angle $TOP = 36^\circ$.
    Work out the size of angle $x$.    Tangent and radius meeting at 90° ()TPO36°x
  13. 13From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$.
    Angle $APB = 70^\circ$.
    Work out the size of angle $PAB$, marked $x$.    Two tangents from a point ()ABPO70°x
  14. 14In 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 $35^\circ$.
    Work out the size of angle $ABT$ in the alternate segment, marked $x$.    Tangent and chord, alternate segment ()TAB35°x
  15. 15In the diagram, the straight line through $T$ is a tangent to the circle and $TA$ is a chord. $P$ is a point on the major arc, and the angle $TPA$ is marked. $P$ then moves along the major arc to a new position $P'$ (shown dashed).Explain what happens to the size of angle $TPA$ as $P$ moves along the major arc.
    Give a reason for your answer.    Tangent at T, chord TA, point P on the arc ()TAPP'
  16. 16In the diagram, the straight line through $T$ is a tangent to the circle and $TA$ is a chord. $B$ is a point on the circle in the alternate segment.
    The angle between the tangent and the chord $TA$ is $46^\circ$.
    A student is asked to find angle $ABT$, marked $x$, and writes:
    “The tangent meets the chord at $46^\circ$.
    The alternate segment theorem says the angle at the circumference is twice the tangent–chord angle, so $x = 2 \times 46^\circ = 92^\circ$.”
    Explain the mistake in the student’s reasoning and give the correct value of $x$.    Tangent at T, chord TA, point B in the alternate segment ()TAB46°x
  17. 17$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii.
    Angle $OAB = 42^\circ$.
    Work out the size of the angle $AOB$ at the centre.    Circle, isosceles radii and angle at the centre ()ABCO42°x
  18. 18In the diagram, the straight line through $A$ is a tangent to the circle. $AB$ and $AC$ are chords.
    The angle between the tangent and the chord $AB$ is $t$, and the angle between the tangent and the chord $AC$ is $3t$.Show that angle $ABC$ is three times angle $ACB$ (that is, angle $ABC : $ angle $ACB = 3 : 1$).
    Give reasons.    Tangent at A with chords AB and AC ()ABCt3t
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