Circle Theorems (Cambridge (CIE) IGCSE International Maths: Core): Exam Questions

$A,B$ 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 $DOA$.

$A$ and $B$ are points on the circumference of a circle, centre $O.$
$CA$ and $CB$ are tangents to the circle.

By showing that at least one angle and all three sides are equal, show that triangle $OAC$ is congruent to triangle$OBC.$

$A,B$ and $C$ are points on a circle.
$AC$ is a diameter of the circle.

Find the value of $a$.

$a=$.............................

The points $E,F$ and $G$ lie on the circumference of a circle, centre$O$.
$JGH$ is a tangent to the circle.

Find the value of $x$ and the value of $y$.

$\begin{array}{l}x=..............................................\\ y=..............................................\end{array}$

The diagram shows the vertices of a triangle lying on the circumference of a circle with centre $O$.

Find the value of $b$.
Give a reason for your answer.

$b$ = .................... because.............................. 

The points $A,B$ and $C$ lie on the circumference of a circle, centre $O$.
$PBR$ is a tangent to the circle and angle $BAC=37°$.

Find the value of $j$ and the value of $k$.

$j=$ .................................    

$k=$................................ [3]

$A$, $B$  and $C$  are points on the circumference of a circle with diameter $AB$.

A tangent is drawn at $A$.

Find angle $BAC$.

$\mathrm{Angle}BAC=$ ................................ [1] 

Find angle $ABC$.

$\mathrm{Angle}ABC=$................................ [2] 

$S$ 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 $x$.

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

Angle $DAB=40°$
Angle $CBE=75°$
Angle $BCD=70°$

i) Explain why reflex angle $BOD=220°$.

[3]

ii) Work out the size of angle $ODC$.

[2]

$P$ and $Q$ are points on the circle, centre $O$. $APB$ is a tangent to the circle at $P$.

i) Write down the mathematical name for the line $PQ$.

[1]

ii) Explain why angle $OPB$ is 90°.

[1]

iii) Find the value of $x$.

$x$ = ................................................. [3]

The diagram shows a circle, centre $O$, with diameter $CE$.
$A,B,C,D$ and $E$ lie on the circumference of the circle.

i) Find the value of $x$.
Give a reason for your answer.

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

[3]

ii) Find the value of $y$.
Give a reason for your answer.

 $y$ = ..................... because ....................................

[2]

iii)Draw a tangent to the circle at $A$.

[1]

The diagram shows a circle, centre $O,$ with diameter $EF$.
Angle $DFE$ = 63°.

i) Find angle $DEF$.

Angle $DEF$ = ................................................... [2]

ii) $EF$ = 12 cm

Calculate $DF$.

$DF$ = ............................................. cm [2]

K, L and M are points on the circle.
KS is a tangent to the circle at K.
KM is a diameter and triangle KLM is isosceles.  

Find the value of $z$.  

$z=$................................................ [2]

The diagram shows a circle, centre $O$, with points $B$ and $D$ on the circumference.
The line $AC$ touches the circle at $B$. $OB$ is parallel to $DC$ and angle $OAB=49°$.

i) Write down the mathematical name of the line $OB$.

................................................. [1]

ii) Write down the reason why angle $ABO$ is $90°$.

......................................................................................................................................................     

...................................................................................................................................................... [1]

iii) Find angle$AOB$.

$\mathrm{Angle}AOB=$ ................................................ [1]

iv) Write down the reason why angle $ADC$ = angle $AOB$.

...................................................................................................................................................... [1]

v) Complete the statement using a mathematical word.

Triangle $AOB$ is ................................................................. to triangle $ADC$. [1]

$AB=5.4\mathrm{cm}$.

Calculate

via) $OB$,

$OB=$ ......................................... cm [2]

vib) $OA$,

$OA=$ ......................................... cm [2]

via) the area of triangle $OAB$.

.........................................cm² [2]