Circle Theorems

and  are points on the circumference of a circle, centre .
 is a tangent to the circle.
 is a straight line.

Angle 

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

and  are points on the circumference of a circle, centre .
 is a diameter of the circle.
and  are tangents to the circle.
Angle .

Work out the size of angle .

and  are points on the circumference of a circle, centre .

Angle .

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

and  are points on the circumference of a circle, centre .
 is a straight line.
Angle 

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

The points  and  lie on a circle.
 is a straight line.

Angle 

Work out the size of angle .
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.

and are points on a circle, centre

is a diameter of the circle.

Angle

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

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

is a diameter of a circle.
is a chord of the circle.

Calculate the radius of the circle.

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

and are chords of a circle.

= 9 cm  = 6 cm = 8 cm

Calculate the length of .

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

is a quadrilateral.

Prove that   is not a cyclic quadrilateral.

and are points on a circle.

and are tangents.

Work out the size of angle .

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

and are points on a circle.

.

Is the centre of the circle?
Tick a box.

Yes             No

Show working to support your answer.

,  and  are points on a circle.

 is a straight line.

 is a tangent to the circle.

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

angles in the same segment are equal angles at a point on a straight line add up to 180°

Make a criticism of his working.

and  are points on a circle.

 is a tangent to the circle.

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

............................degrees 

and 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°.

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

and  are points on the circumference of a circle, centre .
is a tangent to the circle.

Show that
You must give a reason for each stage of your working.

Dylan was asked to give some possible values for  and .

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

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

 is a regular pentagon.
and  are points on a circle, centre .
and  are tangents to the circle.
is a straight line.

Prove that .

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.

BD = 7 cm,

AP = 5 cm,

CP = 3 cm,

PB = x cm,

and

PD = y cm.

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.

and  are points on a circle.

and  are straight lines.

Show that the length of  is where  and  are integers whose values are to be found.
Show your working clearly.

and  are points on a circle.
 is a tangent to the circle.

Angle 

Prove that angle 
Give reasons for each stage in your working.

and  are points on a circle.
, , and are straight lines.

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

= ......................

 and are points on a circle, centre .
, and are straight lines.



Work out the value of .
Show your working clearly.

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

and  are points on a circle.

is the tangent at  to the circle.
is parallel to .

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

and are points on a circle with centre .

Angle

The area of the shaded segment is 200cm²

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

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

and are three points on the circumference of a circle, centre .

and are tangents to the circle.
is a kite.
Angle is

Prove that angle is 

and are points on a circle.

The tangent at is shown.
and are straight lines.

Prove that   

and are points on a circle.

is a tangent.

Assume that triangle is isosceles with

Prove that is parallel to .

In fact, triangle is equilateral.

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

	is parallel to
	bisects angle
	bisects angle

Here is a cyclic quadrilateral.

Work out the size of angle .

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

and are points on a circle.

and are straight lines.

Work out the size of angle.

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