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Geometry of the Circle - Chords and Angles Geometry of the Circle Chords and Angles Curriculum Ready www.mathletics.com Chords and Angles CHORDS AND ANGLES The Circle is a basic shape and so it can be found almost anywhere. This section will introduce you to some properties that will make finding lines and angles inside circles easier. Answer these questions, before working through the chapter. I used to think: What is a "Theorem" in Geometry? What does the word 'subtend' mean? What does it mean to say a quadrilateral is 'cyclic'? Answer these questions after you have worked through the chapter. But now I think: What is a "Theorem" in Geometry? What does the word 'subtend' mean? What does it mean to say a quadrilateral is 'cyclic'? What do I know now that I didn’t know before? 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 1 Chords and Angles Basics Circle Terms Here is a reminder of some terms which relate to a circle. diam eter minor segment secant centre radius chord sector arc Point of contact tangent Complicated Language These are more terms relating to circles. The terms may sound complicated but they have simple meanings. C C O A A B Chord AB subtends +ACB at the circumference. This means +ACB is "standing" on chord AB and touches the circumference. 2 K 15 SERIES TOPIC B "Concentric circles" are circles with the same centre and different radii. Arc AB subtends +ACB at the circumference and +AOB at the centre. They're just circles inside each other. This means +ACB and +AOB are "standing" on arc AB. 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Questions Basics 1. Label a to h in the following diagram according to where it lies in relation to the circle. h a a f e d b g c a b c d e f g h 2. Complete the sentences according to the following diagram by filling in the blanks. C a Chord AB subtends + _________ at the circumference. b Chord BC subtends + _________ at the circumference. c Chord _____ subtends + AOC at the centre. d Chord BC subtends + BOC at the _____________________. e Chord _____ subtends + ABC at the ___________________. O A B 3. What is the difference between a secant and a chord? 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 3 Chords and Angles Knowing More Using Chords with Circles Chords can be used to solve problems within circles, depending on their length and position. Here are some theorems explaining how chords are used. Theorem 1a: Equal chords subtend equal angles at the centre A C Given: AB = CD To Prove: +AOB = +COD O B D Proof In TAOB and TCOD AB = CD AO = CO BO = DO ... TAOB / TCOD ... + AOB = + COD Proof (Given) (Equal radii) (Equal radii) (SSS) (Corresponding angles of congruent triangles) This means that the angles (at the centre) standing on equal chords are equal to each other. Theorem 1b (Converse of 1a): Equal angles at the centre subtend equal chords A Given: + AOB = + COD To Prove: AB = CD C O B D Proof In TAOB / COD + AOB = + COD AO = CO BO = DO ... TAOB / TCOD ... AB = CD Proof (Given) (Equal radii) (Equal radii) (SAS) (Corresponding sides of congruent triangles) This means that if the angles (at the centre) are equal to each other, then the chords they are standing on are equal. "Converse" means "inverse". Find the length of chord PQ given O is the centre of the circle P O is the centre + POQ = + GOH = 40c ... PQ = GH Q 40c O 40c 8 cm H G 4 K 15 SERIES TOPIC (Given) (Given) (Chords that subtend equal angles at the centre are equal) ... PQ = 8 cm 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Theorem 1b Chords and Angles Knowing More Theorem 2a: A perpendicular line from the centre to a chord bisects the chord Given: + ABO = + CBO = 90c, O is the centre To Prove: AB = BC O A C B Proof In TABO and TCBO + ABO = + CBO = 90c AO = CO BO is common ... TAOB / TCBO ... AB = BC Proof (Given) (Equal radii) (RHS) (Congruent triangles; TAOB / TCOD ) This means that a perpendicular line drawn from the centre to the chord, cuts the chord in half. Theorem 2b (Converse of 2a): A line drawn from the centre to a chord's midpoint is perpendicular to the chord Given: AB = BC; O is the centre To Prove: OB = AC O A C B Proof: In TABO and TCBO AB = BC AO = CO BO is common ... TAOB / TCOB + ABO = + CBO Proof (Given) (Equal radii) But + ABO + + CBO = 180c ... + ABO = + CBO = 90c ... OB = AC (SSS) (Corresponding angles of congruent triangles) (Supplementary angles) This means that a line drawn from the centre, perpendicular to the chord, cuts the chord in half. Also, if a line is perpendicular to a chord and bisects it — it has to pass through the centre. In the diagram below O is the centre, OD is 50 mm and OE is 30 mm. Find the length of DE and DF DE2 = OD2 OE2 = 502 302 DE = 40 mm D O E F O is the centre OE = DF ... DE = EF ... DE = EF = 40 mm (Pythagorean Theorem) (Given) (Given) (Perpendicular from centre to chord bisects chord) Theorem 2a DF = DE + EF = 40 + 40 ... DF = 80 mm 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 5 Chords and Angles Knowing More Theorem 3a: Equal chords are equidistant from the centre Given: OE = DF; OB = AC; O is the centre; DF = AC To Prove: OE = OB F E D Proof: DE = 12 DF AB = 12 AC O A C B But AC = DF ` DE = AB Proof (Perpendicular from centre bisects chord) (Perpendicular from centre bisects chord) (Given) In TODE and TOAB DE = AB +OED = +OBA = 90c OD = OA ` TODE / TOAB ` OE = OB Theorem 2a (Proved above) (OB = AC and OE = DF ) (Radii) (RHS) (Corresponding sides in congruent T's ) This means that if two chords are equal, then they are the same distance from the centre. Theorem 3b (converse to 3a): Chords which are equidistant to the centre are equal Given: OE = DF; OB = AC; O is the centre; OE = OB To Prove: DF = AC F E Proof: In TODE and TOAB OE = OB +OED = +OBA = 90c OD = OA ` TODE / TOAB D O A C B Proof (Given) (Given) (Radii) (RHS) ` DE = AB but DE = 12 DF (Corresponding sides in congruent T's ) and AB = 12 AC (Perpendicular from centre bisects chord) ` 1 2 DF = 1 2 (Perpendicular from centre bisects chord) AC Theorem 2a ` DF = AC This means that if two chords are the same distance from the centre, then the chords are equal. O is the centre of the circle. Find the length of PR in the diagram below if OQ = ON = 15 cm and MN = 14 cm R M 14 O Q N O is the centre MN = NS = 14 cm ` MS = MN + NS = 28 cm (Given) (Perpendicular from centre bisects chord) Theorem 2a 15 S P 6 K 15 SERIES TOPIC But PR = MS ... PR = 28 cm (Chords equidistant from the centre are equal) Theorem 3b 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Knowing More Here are some examples using all the above theorems: If O is the centre of the circle, find the length of BD if AB = 200 mm and OC = 60 mm O is the centre ... AB is a diameter ... OB is a radius ` OB = 200 mm ' 2 = 100 mm D C B 60 mm BC2 = OB2 - OC2 ` BC = 1002 - 602 = 80 mm O A OC = BD BC = CD = 80 mm (Given) (Pythagorean Theorem) (Given) (Perpendicular from centre bisects chord) ... BD = BC + CD = 80 + 80 = 160 mm Theorem 2a Find the sizes of angles +A , +B and +C in the diagram below given EO = OD = OF and +OFB = +ODA = +OEA = 90c OE = OD OE = AC and OD = AB AB = AC (Given) D Similarly: BC = AC B ... AB = AC = BC ... TABC is equilateral ... + A = + B = + C = 60c (Chords equidistant from the centre are equal) Theorem 2a A E O C F (Given) (Chords equidistant from the centre are equal) (All sides are equal) (Properties of equilateral triangle) O is the centre in the circle below. Find the length of DE given and +AOC = +DOE = 50c and +OBA = 90c D O 50c E O is the centre OB = AC ... AB = BC ... AC = AB + BC = 10 + 10 = 20 cm (Given) (Given) (Perpendicular from centre bisects chord) 50c A 10 cm B C + AOC = + DOE = 50c (Given) ... DE = AC (Chords that subtend equal angles at the centre are equal) ... DE = 20 cm Theorem 1b 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 7 Chords and Angles Questions Knowing More 1. Find x in each of the following (all lengths in cm). a b 10 x x 10 c d x 42c 42c x 5 25c 2. A circle with diameter 60 cm has a chord MN 18 cm from the centre. a Draw a rough sketch of the circle and chord in the box provided. b How long is MN? 8 K 15 SERIES TOPIC 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Questions Knowing More 3. The circle below has a diameter of 400 units. OB = 120 units and AP = 160 units. Q S a Use the Pythagorean Theorem to find the length of OA b Find the length of RS O B A 120 160 R P 4. Find the length of chord AB. C A 47c B 11 86c O 88c E D 12 F 5. The circle below has diameter 50 cm. ED = 30 cm and FG = 40 cm. Find the distance between the two chords. (Hint: Sketch in the distances between the chords and the centre) G E O D F 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 9 Chords and Angles Questions Knowing More B A E C D 6. A and C are the centres of the above circles (which have the same radius). Use the above diagram to answer the following questions. a Show that BE = ED. b On the diagram construct lines AB, BC, CD and DA. c Prove that BD = AC . 10 K 15 SERIES TOPIC 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Questions Knowing More d Now show that TBEC / TDEC and TAEB / TAED e Show that AE = EC. f Show that ABCD is a rhombus. g Complete this sentence: When a line is drawn between the centres of different circles through a common chord, then this line and the chord _________ each other at an angle of _____. 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 11 Chords and Angles Using Our Knowledge Using Angles with Circles Here are some theorems showing how to find and use angles appearing inside circles. Theorem 4: An angle subtended at the centre is twice the angle subtended at the circumference standing on the same arc In these 3 diagrams, the same arc (AB) subtends an angle at the centre ( +AOB ) and at the circumference ( +ACB ) Diagram b Diagram a Diagram c Proof C M O O M A B A B O M C B A C Given: O is the centre of the circle To Prove: +AOB = 2 # +ACB Proof Construct line CM which passes through the centre O In all 3 diagrams OC = OA ` +OCA = +OAC but +AOM = +OCA + +OAC ` +AOM = 2+OCA similarly +BOM = 2+BCO In Diagram a +AOB = +AOM + +BOM ` +AOB = 2+OCA + 2+BCO = 2 (+OCA + +BCO) = 2+ACB (Equal radii) (Angles opposite equal sides) (Exterior angle of a triangle) In Diagram b reflex +AOB = +AOM + +BOM ` reflex +AOB = 2+OCA + 2+BCO = 2 (+OCA + +BCO) = 2+ACB In Diagram c +AOB = +BOM - +AOM ` +AOB = 2+BCO - 2+OCA = 2 (+BCO - +OCA) = 2+ACB Find the sizes of the angles labeled x a b x 12 35c K 15 SERIES TOPIC x = 1 # angle at centre x = 2 # angle at circumference = 2 # 35c = 70c = 40c 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning x 2 1 2 # 40c = 20c Chords and Angles Using Our Knowledge Theorem 5: The angle in semicircle is a right angle C O is the centre B A O Proof +AOB = 180c (Straight line) +ACB = 1 +AOB (Angle at centre is twice angle at circumference on same arc) 2 180c +ACB = 1 # 180c = 90c 2 Theorem 4 This means that an angle standing on the diameter of a circle is 90c. Also, if an angle on the circumference is 90c then it must be standing on the diameter. Theorem 6: Angles subtended on the circumference by the same arc (in the same segment) are equal D C Given: O is the centre of the circle To Prove: +ACB = +ADB Proof: Construct radii AO and BO +ACB = 12 +AOB O B A +ADB = 12 +AOB ` +ACB = +ADB Proof (Angle at centre is twice angle at circumference on same arc) (Angle at centre is twice angle at circumference on same arc) Imagine there is a chord joining A and B. The angles standing on the same arc are equal if they are on the same side of the imaginary chord. The angles standing on AB at the circumference below AB, would NOT be equal to the angles meeting at the circumference above. If angles are on the same side of the imaginary chord then they are "in the same segment". Find the values of x and y in the following diagram 60c x (Angles in same segment on same arc) x = 60c y = 120c (Angles in same segment on same arc) Theorem 6 120c y Notice x and y are not equal to each other even though they are standing on the same arc. This is because x is above the imaginary chord, and y is below the imaginary chord. 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 13 Chords and Angles Using Our Knowledge Here are some examples using the above theorems. O is the centre of the circle below. Find the sizes of the angles labeled x, y and z P y = 2+PQR ` y = 2 # 60c = 120c S z y O 60c x Q OP = OR ` +PRO = +RPO = x ` x + x + 120c = 180c ` 2x = 60c ` x = 30c (Angle at centre is twice angle at circumference on same arc) (Equal radii) (Angles opposite equal sides) (Sum of angles in a triangle) reflex +POR = 360c - y = 240c (Angles around a point) R z = 1 # reflex +POR (Angle at centre is twice angle at `z= 2 1 2 circumference on same arc) # 240c = 120c O is the centre of the circle below. Show that + DEO = + EGF D G +EGF = 12 +EOF (Angle at centre is twice angle at circumference on same arc) and +EDF = 12 +EOF (Angle at centre is twice angle at circumference on same arc) O OD = OE ` +ODE = +DEO but +ODE = +EGF E F ` +DEO = +EGF (Equal radii) (Angles opposite equal sides) (Angles in same segment on same arc) (Both equal + ODE) In the following diagram O is the centre of the circle and + MPR = 90c. Show that + NPR = + NQP M +MQP = 90c N (Angle in a semicircle) ` +NQP = 90c - +MQN now +NPR = 90c - +MNP Q O but +MQN = +MPN (Angles in same segment on same arc) ` +NPR = 90c - +MQN ` +NPR = +NQP P 14 K 15 SERIES TOPIC R 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Questions Using Our Knowledge 1. Write only the sizes of x and y (no reasons necessary) in each of the following diagrams: a b x= x y x= y 272c y= 30c x y= 52c c d y x= x= x y x y= y= 35c 41c 2. Find the size of + ACB. C A 20c O B 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 15 Chords and Angles Questions Using Our Knowledge 3. Find + QOR in the diagram below. R Q P 30c O 4. O is the centre of the circle below and LN is the diameter. a Find + LOM. b Find + LNM. N O K x L c Find + LMN. d Let + KLO = x. Find + KOL in terms of x. e Prove + LKO + + KNO = 90c. 16 K 15 SERIES TOPIC 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning 60c M Chords and Angles Questions Using Our Knowledge A E B F O x C x D 5. In the diagram above O is the centre of the circle. Let + ODC = + ACB = x. a Show that AB = BC. b Use + CAB to find + BOC in terms of x. c Find + BCD. d Find + OCF in terms of x. e Find + DEC in terms of x and prove that DE || CA. 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 17 Chords and Angles Thinking More Cyclic Quadrilaterals These are called "cyclic quads" for short. These are quadrilaterals that would allow a circle to pass through all its vertices. This is a cyclic quad. This is NOT a cyclic quad. This is NOT a cyclic quad. All 4 vertices touch the circumference. One vertex lies outside the circle. One vertex does not touch the circumference. Cyclic quads have their own properties and theorems which can be used to solve problems. Theorem 7: Opposite angles of a cyclic quad are supplementary A Given: O is the centre of the circle To Prove: + C + + A = 180c and + B + + D = 180c y Proof Proof: Construct radii OB and OD 2x O D 2y Let +C = x and +A = y ` reflex +BOD = 2x and +BOD = 2y B but 2x + 2y = 360c x (Angle at centre is twice angle at circumference on same arc) (Angle at centre is twice angle at circumference on same arc) (Angles around a point) ` x + y = 180c ` +C + +A = 180c ` The opposite angles are supplementary. Similarly, by constructing radii OA and CD it can be shown + B + + D = 180c C This also means that a quadrilateral is cyclic if its opposite angles add up to 180c Find the sizes of angle m and n in the cyclic quadrilateral below A m + 80c = 180c n ` m = 180c - 80c = 100c Theorem 7 B 80c 115c m n + 115c = 180c D ` n = 180c - 115c = 65c C 18 K 15 SERIES (Opposite angles of cyclic quad are supplementary) TOPIC 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning (Opposite angles of cyclic quad are supplementary) Chords and Angles Thinking More Theorem 8: The exterior angle of a cyclic quad is equal to the interior opposite angle B Given: O is the centre of the circle To Prove: + ABC = + CDE C Proof Proof: +ABC + +ADC = 180c +CDE + +ADC = 180c E D A ` +ABC = +CDE (Opposite angles of cyclic quad are supplementary) (Angles on a straight line) (Both angles are supplementary with + ADC) This also means that a quadrilateral is cyclic if the exterior angle is equal to its interior opposite angle. Find the size of angle x and y R 127c Q x T 110c S x = 110c (Exterior angle of cyclic quad equals interior opposite angle) y + 127c = 180c (Opposite angles of cyclic quad are supplementary) ` y = 180c - 127c = 53c y P Find the following angles if O is the centre of the circle a O A 55c D Find + ACD AD is the diameter ... + ACD = 90c (O is the centre) (Angle in semicircle is right angle) E b Find + ADC + ADC + + ACD + + CAD = 180c (Interior angles of a triangle) ... + ADC = 90c + 55c = 180c ... + ADC = 35c c Find + ABC + CDE + + ADC = 180c ... + CDE + 35c = 180c ... + CDE = 145c B C Now + ABC = + CDE (Angles on a straight line) (Exterior angle of cyclic quad equals interior opposite angle) ... + ABC = 145c 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 19 Chords and Angles Questions Thinking More 1. Find the sizes of the angle x and y in each of the following. a b 101c 136c 124c y 77c x x y c d x 102c 61c x y e y f x 88c 111c y 60c 132c y x 20 K 15 SERIES TOPIC 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Questions Thinking More 2. Answer the following questions about this diagram given AB = BC and AD = DC. a A Show TABD / TCBD. B C D b Find the sizes of + DAB and + DCB. c Is BD the diameter of the circle? Prove it. 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 21 Chords and Angles Questions Thinking More 3. Opposite angles of a cyclic quad are supplementary. Is trapezium ABCD a cyclic quad? A 30c B D 40c 70c C 4. Opposite angles of a cyclic quad are supplementary. a What do you know about the opposite angles of a parallelogram? b When are the opposite angles of a parallelogram supplementary? c Is it possible for a non-rectangular parallelogram to be a cyclic quadrilateral? 22 K 15 SERIES TOPIC 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Questions Thinking More 5. Use the information given below to answer the questions that follow. Given (fill this on your diagram) • + HED = 50c • + GHI = 20c • O is the centre of the larger circle • GFOH is a cyclic quad in the smaller circle a H D Find + HOD. O I G F E b Find + FGH. c Find + GFH. d Find + FHO. 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 23 Chords and Angles Visual Theorems Visual Theorems Here is a visual summary of all the theorems in this chapter. Theorem 1a: Equal chords subtend equal angles at the centre Theorem 4: An angle subtended at the centre is twice the angle subtended at the circumference by the same arc. x 2x Theorem 1b: Equal angles at the centre subtend equal chords Theorem 5: The angle in a semicircle is a right angle Theorem 2a: A perpendicular line from the centre to a chord bisects the chord Dia me ter Theorem 6: Angles subtended by the same arc in the same segment are equal Theorem 2b: A line drawn from the centre to a chord’s midpoint is perpendicular to the chord Theorem 3a: Equal chords are equidistant to the centre Theorem 7: Opposite angles of a cyclic quad are supplementary y 180c - x x 180c -y Theorem 3b: Chords which are equidistant to the centre are equal 24 K 15 SERIES TOPIC Theorem 8: The exterior angle of a cyclic quad is equal to the interior opposite angle 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Answers Basics: 1. a c Knowing More: Sector b Chord Arc d Tangent 6. b A e Centre f Diameter g Secant h Point of contact g 2. a +ACB b +BAC c +AOC d Centre e Circumference 1. a c When a line is drawn between the centres of different circles through a common chord, then this line and the chord MEET each other at an angle of 90c x = 30c x = 41c y = 41c Knowing More: 2. a b x = 88c y = 44c d x = 70c y = 145c 2. +ACB = 70c x = 20 cm b x = 5 cm x = 5 cm d x = 130c 3. +QOR = 60c b MN = 48 cm 4. a +LOM = 60c 30 b +LNM = 30c 30 c +LMN = 90c d +KOL = 180c - 2x M C y = 52c However, a chord is a line that is formed between two points on the circumference. c E Using Our Knowledge: 3. A secant is a line that cuts through a circle but extends beyond the circumference of the circle. 1. a B 18 N 3. a OA = 120 units 4. a AB = 11 b RS = 320 units 5. OM = 20 cm 5. b BOC = 2x c +BCD = 90c d +OCF = 90c - 2x e +DEC = 90c - 2x ON = 15 cm ` distance between chords = 35 cm 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 25 Chords and Angles Answers Thinking More: 1. a x = 77c b y = 56c y = 101c c x = 61c d x = 114c f b x = 37c y = 51c y = 66c 2. x = 102c y = 78c y = 29c e x = 44c +DAB = 90c +DCB = 90c 3. ABCD is a cyclic quadrilateral 4. a They are equal. b When a parallelogram is a square or rectangle. c No, because opposite angles are not supplementary, they are equal. 5. a +HOD = 100c c 26 +GFH = 80c K 15 SERIES TOPIC b +FGH = 80c d +FHO = 30c 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Chords and Angles Notes 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning K 15 SERIES TOPIC 27 Chords and Angles 28 K 15 SERIES TOPIC Notes 100% Geometry of the Circle – Chords and Angles Mathletics 100% © 3P Learning Geometry of the Circle - Chords and Angles www.mathletics.com