Circle Theorems (Edexcel IGCSE Maths): Exam Questions

P, Q, R and S are points on a circle, centre O ROP is a diameter of the circle. Angle PRQ = 22°

(i) Find the size of angle RQP

(ii) Give a reason for your answer.

(i) Find the size of angle $PSQ$

(ii) Give a reason for your answer.

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

$PA$ and $PB$ are tangents.

Work out the size of angle $APB$.

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

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

$CD$ is a tangent to the circle.

Write down the size of angle $x$. Give a reason for your answer.

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

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

Circle the line that is a diameter of the circle.

In the diagram,

• E, F and D are points on the circumference of the circle
• G, D and H lie on a tangent to the circle
• angle EFD = 55°
• angle FDE = 43°.

Explain why angle HDF is 82°.

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

Angle BAD = 52°.

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

x = ................................... ° reason ..................................................................................

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

y = ................................... ° reason ..................................................................................

$A$ and $C$ are points on a circle, centre $O$. $AB$ and $CB$are tangents to the circle. Angle $ABC$ = 52°

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

$A$ and $C$ are points on a circle, centre $O$ $DA$ is the tangent to the circle at $A$ and $DC$ is the tangent to the circle at $C$

Angle $ADC=48^{\circ}$

Work out the size of reflex angle $AOC$

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

Angle $DAB=50^{\circ}$

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 $OPT=32^{\circ}$

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 $OCB=34^{\circ}$.

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 $ACD=62^{\circ}$

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 $ABD=40^{\circ}$

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 $EGD=42^{\circ}$

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 space 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 $ACD=32^{\circ}$.

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 $OAC=52^{\circ}$

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 $DAB=40^{\circ}$ Angle $CBE=75^{\circ}$

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 $CAB=25^{\circ}$ Angle $DEC=100^{\circ}$

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 $PSO=48^{\circ}$ $MP=MS$

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 $DAO=40^{\circ}$

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 $ABO=x^{\circ}$.

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 $ABD=39^{\circ}$ Angle $EFD=18^{\circ}$

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.

$\begin{gathered} B C & = ( 2 + \sqrt{5} ) \text{cm} \\ E D & = ( 4 + \sqrt{5} ) \text{cm} \\ D C & = 2 \sqrt{5} \text{cm} \end{gathered}$

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 $BCQ=x^{\circ}$

Prove that angle $CDA=2x^{\circ}$ 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 $ED$. $\mathrm{Angle}ABE=73^{\circ}$

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 $ABC=75^{\circ}$

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 $45^{\circ}--\frac{x}{4}$

$C,D$ 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