1. The Circle Label the diagrams using the key words Radius Diameter
Centre
Chord
Minor segment
Circumference
Arc
Major segment
Sector
Tangent
Tangent
Radius
Centre
Arc
Major segment
Sector
Diameter
Chord
Minor segment
Circumference

2. What’s so special about a circle?
Draw a triangle
Move the highest point horizontally…
and the angle decreases
Make the base be the diameter of a circle and the top any other point on the circumference…
and the angle is always 90o

3. and the opposite angles always add up to 180o
What’s so special about a circle?
Draw a quadrilateral
The opposite angles can add up to anything between zero and 360o
Make the corners at any points on the circumference of a circle…
and the opposite angles always add up to 180o
There are 8 special properties of circles that you need to learn in the GCSE.
You have seen two already, and we will go on to look at the proof of these rules

4. Circle theorems
Theorem 1: The angle subtended by a chord at the centre of the circle is twice the angle subtended by the same chord at any point on the circumference of the circle. P
AO, PO and OB are radii
ΔAPO and ΔBPO are isosceles O A B
‘The angle at the centre is twice the angle at the circumference’

5. Circle theorems
Theorem 2: The angle subtended by a diameter at any point on the circumference of the circle is 90o P
Or If the angle subtended by a chord at any point on the circumference of the circle is 90o then the chord must be a diameter
Using theorem 1, A O
But y is 180o, so x must be 90o B
‘The angle in a semi-circle is 90o’

6. Eg AB is a diameter of a circle, centre O
Find the size of angles x and y, giving reasons for your answers
The angle at the centre is twice the angle at the circumference, so x = 34o C
OB and OC are radii, so ΔOAC is isosceles
The angle in a semi-circle is 90o A B
So y = 90 – 34 = 56o O
NB the answer could be arrived at differently, but taking longer…
AOB makes a straight line, so COA = 112o
OB and OC are radii, so ΔOCB is isosceles
OA is also a radius, so ΔOAC is isosceles

7. Circle theorems 1 and 2 1. Find the size of angle x.
Give a reason for your answer
The angle in a semi-circle is 90o O
2. Find the size of angle x.
Give a reason for your answer O
The angle at the centre is twice the angle at the circumference

8. 3. Find the size of angle x.
Give a reason for your answer
The angle at the centre is twice the angle at the circumference O
4. Find the size of angle x.
Give a reason for your answer O
The angle at the centre is twice the angle at the circumference

9. 5. In triangle ABC, AC = 6cm and BC = 8cm.
Write down the length of AB, giving a reason for your answer A
The angle in a semi-circle is 90o O
Using Pythagoras’ theorem: B 6 8 C Q
6. PQ = QR. Prove that PQRS is a square P
As PR is a diameter, PQR = PSR = 90o O
As QS is a diameter, SPQ = SRQ = 90o R
Hence as all angles equal 90o, and PQ = QR, the shape is a square S

10. Can you prove theorem 2 in a different way?
Hint: make and P A O B

11. Circle theorems
Theorem 3: If ABCD are points on the circumference on a circle, the opposite angles in the cyclic quadrilateral ABCD add up to 180o B
Using theorem 1, A C O
‘Opposite angles in a cyclic quadrilateral sum to 180o’ D

12. Circle theorems
Theorem 4: The angle subtended by a chord at any point on the circumference within the same segment is equal P Q
APB = AQB
Using theorem 1,
If AOB = 2x O
Then APB = AQB = x A B
‘Angles in the same segment are equal’

13. Eg A,B, C D and E are points on the circumference of a circle
Find the size of angles x and y, giving reasons for your answers A
Opposite angles in a cyclic quadrilateral sum to 180o B
Angles in the same segment are equal
So EBC = x = 40o
Angles in a triangle sum to 180o E C
So y = 180 – ( ) = 45o D

14. Circle theorems 3 and 4 1. Find the size of angles x and y
Give reasons for your answers
Opposite angles in a cyclic quadrilateral sum to 180o
2. Find the size of angles a and b
Give reasons for your answers
The angle at the centre is twice the angle at the circumference O
Angles in the same segment are equal

15. 3. Find the size of angle x
Opposite angles in a cyclic quadrilateral sum to 180o
4. Find the size of angle x
Angles in the same segment are equal

16. 5. FX = EX
Prove that EF and GH are parallel F G
Triangle EFX is isosceles as FX = EX X
So XFE = XEF E
Angles in the same segment are equal
So EFH = EGH
and FEG = FHG
As alternate angles are equal, EF and GH must be parallel H
6. Find the size of angle x
Imagine point P on the circumference
The angle at the centre is twice the angle at the circumference, so angle at P is 54o O
Opposite angles in a cyclic quadrilateral sum to 180o P

17. Can you prove that ? Opposite angles are equal
Angles in the same segment are equal
Angles in the same segment are equal
So triangles are similar
Hence

18. Circle theorems
Theorem 5: A perpendicular line from the point where a tangent touches the circle passes through the centre of the circle
Consider another point P on the tangent
OP > OA as it is outside the circle.
The shortest distance from the tangent to O is found by going at right-angles from A O
But OA is a radius of the circle
Hence OA is at right-angles to the tangent. P A
‘A tangent is perpendicular to a radius’
tangent

19. Circle theorems
Theorem 6: A perpendicular bisector of a chord passes through the centre of the same circle
Or If a radius bisects a chord it makes an angle of 90o
OA = OB = radius of circle O
OM is a shared side
AMO = BMO = 90o B
ΔAOM and ΔBOM are congruent by RHS M
Hence AM = BM A
‘A perpendicular to a chord through the centre bisects the chord’

20. Eg PQ is a tangent to a circle, centre O
X is such that PX = XR
Find the size of angles v and w
A tangent is perpendicular to a radius, so OPQ = 90o
Angles in a triangle sum to 180o
So v = 180 – ( ) = 55o
If a radius bisects a chord it makes an angle of 90o O
As PX = XR and OX is a shared side
Then ΔPOX is congruent to ΔORX by RHS P X R
and w = v = 55o
b) Hence prove that PQ and OR are not parallel
PQ and OR are parallel if QPO and ROP make 180o (supplementary angles) Q
You may be asked to prove similarity, congruence or other properties of shapes & lines
QPO = 90o
ROP = v + w = 110o
So QPO + ROP = 200o
So angles are not supplementary, lines are not parallel

21. Circle theorems 5 and 6 1. AB is a tangent to the circle, centre O
Find the size of angle x, giving reasons for your answer
A tangent is perpendicular to a radius, so ABO = 90o A O
So BOA = 180 – 118 = 62o
The angle at the centre is twice the angle at the circumference B
2. ST is a tangent to the circle.
ST = 12cm and OT = 13cm
Find the radius of the circle S O
A tangent is perpendicular to a radius, so TSO = 90o 12
Using Pythagoras’ theorem: 13 T

22. 3. Construct a circle with J, K and L on its circumference
If JK, KL (and JL) are chords of the circle, their perpendicular bisectors will pass through the centre… J K L B
4. BD is a diameter to a circle, centre O.
BD intersects the chord AC at X. AX = CX.
Prove that triangles ABX and DCX are similar. O
Angles in the same segment are equal
So BAC = BDC
and ABD = ACD X A C
If a radius bisects a chord it makes an angle of 90o
Hence triangles similar by AAA D

23. Circle theorems
Theorem 7: The distance from the intersection of two tangents to a circle to the points where the tangents touch the circle are equal
If AP and BP are tangents, AP = BP
OA = OB = radius of circle
OP is a shared side B O
OAP = OBP = 90o using theorem 5
ΔAOP and ΔBOP are congruent by RHS
Hence AP = BP A
‘Tangents from a point are equal’ P

24. Circle theorems
Theorem 8: The angle between a chord and a tangent is equal to the angle subtended by the chord at any point in the other segment B
ΔPOB is isosceles A
Using theorem 1, O
PAB = ½ POB = 90 - y
But using theorem 5, P
tangent
‘Angles in alternate segments are equal ‘

25. Eg AD and BD are tangents to a circle
Use the Alternate Segment Theorem to prove that AD and BD are equal D
You may be asked to prove one circle theorem using another
Using AST, DAB = x
and BAD = x
Hence ΔADB is isosceles A B
Hence AD = BD C

26. Circle theorems 7 and 8 1. AB and AC are tangents to the circle. C
Find the size of angle x, giving a reason for your answer C
Tangents from a point are equal, so triangle is isosceles A B Q
2. QX and PX are tangents to the circle, centre O
Prove that quadrilateral POQX is a kite. O
OP = OQ = radius
Tangents from a point are equal, so PX = QX
A tangent is perpendicular to a radius, so OPX = OQP = 90o
2 pairs of adjacent sides equal and 1 pair of opposite angles equal, so POQX is a kite P X

27. C
3. AC and AB are tangents.
Find the size of angle x
Tangents from a point are equal A
Angles in alternate segments are equal, so x = 75o B
4. PX is a tangent to the circle, centre O
Find the size of angle x Q
Angles in alternate segments are equal, so QPX = 58o O
A tangent is perpendicular to a radius, so OPX = 90o P X

28. 5. AT is a tangent to the circle.
Prove that triangles BAT and ACT are similar A B C T
Angles in alternate segments are equal, so CAT = TBA
ATC = ATB22
Hence the 3rd angle must be the same and the triangles are similar by AAA A C A T B T

29. 6. In the diagram, O is the centre of the circle, AD is a diameter and AB is a tangent.
Angle ACE = xo
Find, in terms of x, the size of:
a) angle ADE
b) angle DAE
c) angle EAB
d) angle AOE D C E O A B
a) Angles in the same segment are equal, so ADE = x
b) The angle in a semi-circle is 90o, so DAE = 90 - x
c) A tangent is perpendicular to a radius, so EAB = DAE = x
Or angles in alternate segments are equal, so EAB = ACE = x
d) The angle at the centre is twice the angle at the circumference, so AOE = 2x

30. Circle theorems summary
‘The angle at the centre is twice the angle at the circumference’
‘The angle in a semi-circle is 90o’
‘Opposite angles in a cyclic quadrilateral sum to 180o’
‘Angles in the same segment are equal’

31. Circle theorems summary
‘A perpendicular to a chord through the centre bisects the chord’
‘A tangent is perpendicular to a radius’
‘Tangents from a point are equal’
‘Angles in alternate segments are equal’

32. The Circle Label the diagrams using the key words Radius Diameter
Centre
Chord
Minor segment
Circumference
Arc
Major segment
Sector
Tangent

33. Circle theorems summary
Write down the explanation you would need to give in the exam:

34. Circle theorems summary
Write down the explanation you would need to give in the exam:

35. Circle theorems 1 and 2 1. Find the size of angle x.
Give a reason for your answer O
2. Find the size of angle x.
Give a reason for your answer O

36. 3. Find the size of angle x.
Give a reason for your answer O
4. Find the size of angle x.
Give a reason for your answer O

37. 5. In triangle ABC, AC = 6cm and BC = 8cm.
Write down the length of AB, giving a reason for your answer A O B C Q
6. PQ = QR. Prove that PQRS is a square P O R S

38. Circle theorems 3 and 4 1. Find the size of angles x and y
Give reasons for your answers
2. Find the size of angles a and b
Give reasons for your answers O

39. 3. Find the size of angle x
4. Find the size of angle x

40. 5. FX = EX
Prove that EF and GH are parallel F G X E H
6. Find the size of angle x O

41. Circle theorems 5 and 6 1. AB is a tangent to the circle, centre O
Find the size of angle x, giving reasons for your answer A O B S
2. ST is a tangent to the circle.
ST = 12cm and OT = 13cm
Find the radius of the circle O T

42. 3. Construct a circle with J, K and L on its circumferenceB
4. BD is a diameter to a circle, centre O.
BD intersects the chord AC at X. AX = CX.
Prove that triangles ABX and DCX are similar. O X A C D

43. Circle theorems 7 and 8 1. AB and AC are tangents to the circle. C
Find the size of angle x, giving a reason for your answer C A B Q
2. QX and PX are tangents to the circle, centre O
Prove that quadrilateral POQX is a kite. O P X

44. C
3. AC and AB are tangents.
Find the size of angle x A B
4. PX is a tangent to the circle, centre O
Find the size of angle x Q O P X

45. 5. AT is a tangent to the circle.
Prove that triangles BAT and ACT are similar A B C T

46. 6. In the diagram, O is the centre of the circle, AD is a diameter and AB is a tangent.
Angle ACE = xo
Find, in terms of x, the size of:
a) angle ADE
b) angle DAE
c) angle EAB
d) angle AOE D C E O A B