Circle Theorems (Edexcel IGCSE Maths A: Higher): Exam Questions

$B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$AB$ and $AD$ are tangents to the circle.

Angle $DAB=50°$

Work out the size of angle $BCD$.
Give a reason for each stage in your working.

A and B are points on a circle, centre O.

AC and BC are tangents to the circle.

Angle ACB = 48° .

Work out the size of angle $x$.

Give a reason for each stage of your working

$x=$............°

$S$ and $T$ are points on the circumference of a circle, centre $O$.
$PT$ is a tangent to the circle.
$SOP$ is a straight line.

Angle $OPT=32°$

Work out the size of the angle marked $x$.
Give reasons for your answer.

$A,B$ and $D$ are points on the circumference of a circle, centre $O$.
$BOD$ is a diameter of the circle. $BC$ and $AC$ are tangents to the circle.
Angle $OCB=34°$.

Work out the size of angle $DOA$.

$A,B,C$ and $D$ are points on the circumference of a circle, centre $O$.

Angle $AOC=y$.

Find the size of angle $ABC$ in terms of $y$.
Give a reason for each stage of your working.

$A,B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$BOD$ is a straight line.
Angle $ACD=62°$

Find the size of angle $ADB$.
Give a reason for each stage in your working.

The points $A,B,C$ and $D$ lie on a circle. $CDE$ is a straight line.

$BA=BD$
$CB=CD$
Angle $ABD=40°$

Work out the size of angle $ADE$.
You must give a reason for each stage of your working.

APB and CPD are chords of a circle.

AP = 14 cm            PB = 6cm             CP = 10 cm

Calculate the length of PD.

A, B, D and E are points on a circle. EDC and ABC are straight lines.

ED = 24 cm                       DC = 15 cm                       BC = 18 cm

Work out the length of AB.

$D,E,F$ and $G$ are points on a circle, centre $O$

$EOG$ is a diameter of the circle.

Angle $EGD=42°$

Calculate the size of angle $DFG$
Give a reason for each stage of your working.

Angle DFG = ....................................................... 

$PTQ$is a diameter of a circle.
$RTS$ is a chord of the circle.

Calculate the radius of the circle.

........................................... cm

$APB$and $CPD$are chords of a circle.
$AP$ = 9 cm $PB$ = 6 cm $CP$ = 8 cm

Calculate the length of $PD$.

......................cm

$ABCD$ is a quadrilateral.

Prove that $ABCD$  is not a cyclic quadrilateral.

$C,D$ and $E$ are points on a circle.

$XC=XD$.

Is $X$the centre of the circle? Tick a box.

$\square$ Yes            $\square$ No

Show working to support your answer.

$A$, $B$ and $C$ are points on a circle.

$DCB$ is a straight line.

$PAQ$ is a tangent to the circle.

Sam is trying to work out the size of angle $m$. Here is his working.

Make a criticism of his working.

The diagram shows a circle, centre O.

Points A, B, C and D lie on the circumference of the circle.
EDF is a tangent to the circle.
Angle ABC = 82° and angle ODC = 57°.

Work out the value of x.

x = ......................

Work out the value of y.

y = ..................

A, B, C and D are points on the circumference of a circle.

PQ is a tangent to the circle at D.
Angle BDQ = 72° and angle ABD = 63°.

Work out angle x.
Give a reason for your answer.

^{Angle x = ...................° because ......................}

Work out angle y.
Give a reason for your answer.

Angle y = ...................° because .....................

$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. $DOB$ is a diameter of the circle. Angle $ACD=32°$.

Work out the size of angle $ADB$.

Give a reason for each stage in your working.

A, B, C and D are points on a circle with centre O.
DOB is a diameter of the circle.
Angle ACD = 43°

Work out the size of angle ADB.
Give a reason for each stage in your working.

P, Q, S and T are points on a circle.
TSR and PQR are straight lines.

Work out the length of PQ.

$A,B$ and $C$are points on a circle, centre $O$

Angle $OAC=52°$

Find the size of angle $ABC$
Give reasons for your working.

$B,C$ and $D$ are points on the circumference of a circle, centre $O$. $ABE$ and $ADF$ are tangents to the circle.

Angle $DAB=40°$
Angle $CBE=75°$

Work out the size of angle $ODC$.

$A,B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$AC$ is a diameter of the circle.
$AC$ and $BD$ intersect at E.

Angle $CAB=25°$
Angle $DEC=100°$

Work out the size of angle $DAC$.
You must show all your working.

$P,M$ and $S$ are points on a circle, centre $O$. $RST$ is a tangent to the circle.

Angle $PSO=48°$
$MP=MS$

Work out the size of angle $MST$.
Give reasons for each stage of your working.

$B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$ABO$ is a straight line.
$AD$ is the tangent at $D$ to the circle.
Angle $DAO=40°$

Work out the size of angle $BCD$.
Give a reason for each stage of your working.

$A$, $B$, $C$ and $D$ are points on a circle. $ABC$ and $EDC$ are straight lines.

Work out the length of $AB$.

$A$ and $B$ are points on a circle, centre $O$.

$BC$ is a tangent to the circle.
$AOC$ is a straight line.
Angle $ABO=x°$.

Find the size of angle $ACB$, in terms of $x$.
Give your answer in its simplest form.
Give reasons for each stage of your working.

$A,B,C$ and $D$ are four points on the circumference of a circle.

$AEC$ and $BED$ are straight lines.
Prove that triangle $ABE$ and triangle $DCE$ are similar.
You must give reasons for each stage of your working.

A, B and C are points on a circle.
ABP is a straight line.
QCP is a tangent to the circle.

AB = 10 cm                        BP = 8 cm

Work out the length of CP.

A, B and C are points on a circle, centre O.
OPC is a straight line.
APB is a chord of the circle, and bisects OC at the point P.

AP = 7.2 cm                     PB = 5.4 cm                     OP = PC.

Work out the area of the circle.

Give your answer correct to 3 significant figures.

The points A, B, C and D lie on a circle.

ABY and DCY are straight lines.

AB = 13 cm               BY = x cm             DC = x cm              CY = 9 cm

Show that x² + 4x – 81 = 0

Find the length of DY.
Show your working clearly.
Give your answer correct to 3 significant figures.

$B,D,E$ and $F$ are points on a circle.

$ABC$ is the tangent at $B$ to the circle.
Angle $ABD=39°$
Angle $EFD=18°$

Work out the size of angle $BDE$.
Give reasons for your working.

$P,Q,R$ and $S$ are points on a circle.

$PXR$and $QXS$are straight lines.
$PX=SX$

Prove that $QS$ is not a diameter of the circle.

$A,B$and $C$ are points on a circle.

• $BC$ bisects angle $ABQ$.

• $PBQ$ is a tangent to the circle.

Angle $CBQ=x$

Prove that $AC=BC$

In the diagram,

• A, B and C are points on the circumference of a circle

• O is the centre of the circle

• angle OAB = 15°

• angle BCO = 10°.

Calculate the acute angle AOC.

A, B, C and D are points on the circumference of a circle, centre O.
AC is a diameter of the circle.
Angle ABD = 58°.
Angle CDB = 22°.

Work out the sizes of angle ACD and ACB, giving reasons for your answers.

Angle ACD = ...........................°

Angle ACB = ...........................°

$A,B,D$ and $E$ are points on a circle.
$ABC$ and $EDC$ are straight lines.

Prove that triangle $BCD$ is similar to triangle $ECA$.
You must give reasons for your working.

$A,B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$FDE$ is a tangent to the circle.

Show that $y-x=90$
You must give a reason for each stage of your working.

Dylan was asked to give some possible values for $x$ and $y$.

He said, "$y$ could be 200 and $x$ could be 110, because 200 − 110 = 90"

Is Dylan correct?
You must give a reason for your answer.

$PQRST$ is a regular pentagon.
$R,U$ and $T$ are points on a circle, centre $O$.
$QR$ and $PT$ are tangents to the circle.
$RSU$ is a straight line.

Prove that $ST=UT$.

A, B, C and D are points on a circle.
APB, CPD, and BD are chords of the circle.
APB and CPD intersect perpendicularly at P.

Work out the values of x and y.
Show your working clearly.
Give your answers correct to 3 significant figures.

A, B, C and D are points on a circle, centre O.
APBE, COP and ODE are straight lines.

AP = 4 cm, PB = 5 cm, CP = 10 cm, DE = 10 cm and BE = x cm

Work out the value of x.
Show your working clearly.

$A,B,D$ and $E$ are points on a circle.

$ABC$and $EDC$ are straight lines.

$\begin{array}{rcl}BC& =& (2+\sqrt{5})\mathrm{cm}\\ ED& =& (4+\sqrt{5})\mathrm{cm}\\ DC& =& 2\sqrt{5}\mathrm{cm}\end{array}$

Show that the length of $AB$ is $(p\sqrt{5}+q)\mathrm{cm},$ where $p$ and $q$ are integers whose values are to be found.
Show your working clearly.

$A,B,C$ and $D$ are points on a circle.
$PCQ$ is a tangent to the circle.
$AB=CB.$

Angle $BCQ=x°$

Prove that angle $CDA=2x°$
Give reasons for each stage in your working.

$B,C,D$ and $F$ are points on a circle.
$ABC$, $AFD$, $BFE$and $CDE$are straight lines.

Work out the size of angle $x$.
Show your working clearly.

$x$ = ......................

$A,D,B$ and $E$ are points on a circle, centre $O$.
$AFBC$, $OEC$ and $OFD$ are straight lines.

$AF=7\mathrm{cm},FB=4\mathrm{cm},BC=5\mathrm{cm},FD=2\mathrm{cm}\mathrm{and}CE=x\mathrm{cm}.$

Work out the value of $x$.
Show your working clearly.

$x=$....................................................... 

$B,C,D$ and $E$ are points on a circle.

$AB$ is the tangent at $B$ to the circle.
$AB$ is parallel to $ED$.
$\mathrm{Angle}ABE=73°$

Work out the size of angle $DCE$.
Give a reason for each stage of your working.

$A,B$ and $C$ are points on a circle with centre $O$.

Angle $ABC=75°$

The area of the shaded segment is 200cm²

Calculate the radius of the circle.
Give your answer correct to 3 significant figures.

...................................................... cm

$A,B$ and $C$ are three points on the circumference of a circle, centre $O$.

$BD$ and $CD$ are tangents to the circle.
$ABDC$ is a kite.
Angle $BDC$ is $x$

Prove that angle $ABO$is  $45°–\frac{x}{4}$

$C,D$ and $E$ are points on a circle.

$CE=DE$
The tangent at $D$ is shown.
$ACD$ and $BCE$ are straight lines.

Prove that   $y=3x$

$A,B$ and $C$ are points on a circle.

$CD$ is a tangent.

Assume that triangle $ABC$ is isosceles with $AC=BC$

Prove that $AB$ is parallel to $DC$.

In fact, triangle $ABC$ is equilateral.

Tick the two boxes for the statements that must be correct. 

Here is a cyclic quadrilateral.

$x:y=5:7$

Work out the size of angle $w$.

.....................degrees

$A,B,C,D$ and $E$ are points on a circle.

$BFD$ and $AFC$ are straight lines.

$DC=DF$

Work out the size of angle$x$.

You must show your working which may be on the diagram.

..........................degrees 

A, B, C and D are points on the circumference of a circle, centre O.

Angle BAD = 112° and angle DCO = 33°.

Show that angle y = 35°.
Give reasons for each stage of your working.

Work out angle z.

Give reasons for your answer.

Angle z = ......................... ° because ........................................................................................

A, B, C and D are points on the circumference of a circle, centre O.

Angle CAD = 28° and CD = 6.4cm.
BD is a diameter of the circle.

Calculate the area of the circle.

...................... cm²