Circle Theorems (Cambridge International IGCSE Mathematics): Exam Questions

i) Complete this statement.

$X,Y$and $Z$ are points on the .................................................. of the circle, centre $O$.

ii) Give a reason why angle $XYZ$ is 90°.

In the diagram,$A,B$ and $C$ are points on the circle, centre $O$.

i) On the diagram, draw a chord.

ii) Explain why angle $ABC$ is 90°.

In the diagram, $B$ and $D$ are points on the circumference of a circle, centre $O$. $AC$ is a straight line touching the circle at $B$ only and $BD$ is a straight line through $O$.

Complete the statement.

Angle $ABD$ = ................ because ...........................

.

$A,B$ and $C$ are points on the circumference of the circle, centre $O$. The straight line $DE$ touches the circle at $B$.

Write down the mathematical name for the line $DE$.

Extract:

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

On the circle, draw a radius.

Extract:

A, B and C are points on the circumference of the circle, centre O. The straight line DE touches the circle at B.

Complete the following statements.

i) Angle $ABD$ = ............... because

ii) Angle $ACB$ = ............... because

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

Write down the mathematical name for

i) the straight line $AC$,

ii) the straight line $AB$.

Extract:

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

Give a geometrical reason why angle $ABC=90^{\circ}$.

$A,B$ 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 $y$.

$x=..............................................\,y=..............................................$

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 $BAC=37^{\circ}$.

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]

Extract:

A, B and C are points on the circumference of a circle with diameter AB. A tangent is drawn at A.

Find angle $ABC$.

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

The diagram shows a circle, centre $O$, with diameter $PQ$. $R$ is a point on the circumference.

$PR=5.2\mathrm{cm}$, and $PR=\frac{1}{3}PQ$. Angle $PQR=x^{\circ}$.

Give a geometrical reason why angle $PRQ$ is 90°.

Extract:

The diagram shows a circle, centre O, with diameter PQ.

R is a point on the circumference.

PR = 5.2 cm, and PR = 1/3 PQ.

Angle PQR = x°.

Angle PRQ is 90°.

Find the value of $x$.

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

$P$ 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 $PQ$.

ii) Explain why angle $OPB$ is 90°.

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

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

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

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

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 $OAB=49^{\circ}$.

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

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

ii) Write down the reason why angle $ABO$ is $90^{\circ}$.

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

...................................................................................................................................................... [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]

The points $B$, $C$, and $D$ lie on the circumference of the circle, centre $O$.

The line $AB$ is a tangent to the circle at $B$.

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

$BC$ is $4$cm.

Hence, find the radius of the circle.

Round your answer to 3 significant figures.

Calculate the area of the circle.

Round your answer to 3 significant figures.