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

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

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

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’

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’

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

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

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

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

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

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

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’

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 – (95 + 40) = 45o D

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

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

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

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

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

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’

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 – (90 + 35) = 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

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

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

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

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 ‘

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

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

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

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

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 = 90 - 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

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’

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’

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

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

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

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

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

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

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

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

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

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

3. Construct a circle with J, K and L on its circumference 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 X A C D

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

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

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

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