# Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

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1 Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272

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3 hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion will introdue you to some properties tht will mke finding lines nd ngles inside irles esier. nswer these questions, efore working through the hpter. I used to think: Wht is "Theorem" in Geometry? Wht does the word 'sutend' men? Wht does it men to sy qudrilterl is 'yli'? nswer these questions fter you hve worked through the hpter. ut now I think: Wht is "Theorem" in Geometry? Wht does the word 'sutend' men? Wht does it men to sy qudrilterl is 'yli'? Wht do I know now tht I didn t know efore? 100% Geometry of the irle hords nd ngles K 15 1 TPI

4 hords nd ngles sis irle Terms Here is reminder of some terms whih relte to irle. dimeter entre sent minor segment rdius setor hord r Point of ontt tngent omplited Lnguge These re more terms relting to irles. The terms my sound omplited ut they hve simple menings. hord sutends + t the irumferene. This mens + is "stnding" on hord nd touhes the irumferene. "onentri irles" re irles with the sme entre nd different rdii. They're just irles inside eh other. r sutends + t the irumferene nd + t the entre. This mens + nd + re "stnding" on r. 2 K % Geometry of the irle hords nd ngles TPI

5 hords nd ngles Questions sis 1. Lel to h in the following digrm ording to where it lies in reltion to the irle. h d f e g e g d f h 2. omplete the sentenes ording to the following digrm y filling in the lnks. hord sutends + t the irumferene. hord sutends + t the irumferene. d hord sutends + t the entre. hord sutends + t the. e hord sutends + t the. 3. Wht is the differene etween sent nd hord? 100% Geometry of the irle hords nd ngles K 15 3 TPI

6 hords nd ngles Knowing More Using hords with irles hords n e used to solve prolems within irles, depending on their length nd position. Here re some theorems eplining how hords re used. Theorem 1: Equl hords sutend equl ngles t the entre Given: = To Prove: + = + Proof Proof In T nd T = = =... T / T... + = + (Equl rdii) (Equl rdii) (SSS) (orresponding ngles of ongruent tringles) This mens tht the ngles (t the entre) stnding on equl hords re equl to eh other. Theorem 1 (onverse of 1): Equl ngles t the entre sutend equl hords Given: + = + To Prove: = Proof Proof In T / + = + = =... T / T... = (Equl rdii) (Equl rdii) (SS) (orresponding sides of ongruent tringles) This mens tht if the ngles (t the entre) re equl to eh other, then the hords they re stnding on re equl. "onverse" mens "inverse". Find the length of hord PQ given is the entre of the irle P Q G m H is the entre +PQ = +GH = PQ = GH... PQ = 8 m (hords tht sutend equl ngles t the entre re equl) Theorem 1 4 K % Geometry of the irle hords nd ngles TPI

7 hords nd ngles Knowing More Theorem 2: perpendiulr line from the entre to hord isets the hord Given: + = + = 90, is the entre To Prove: = Proof Proof In T nd T + = + = 90 = is ommon... T / T... = (Equl rdii) (RHS) (ongruent tringles; T / T ) This mens tht perpendiulr line drwn from the entre to the hord, uts the hord in hlf. Theorem 2 (onverse of 2): line drwn from the entre to hord's midpoint is perpendiulr to the hord Given: = ; is the entre To Prove: = Proof Proof: In T nd T = = is ommon... T / T + = + ut = = + = = (Equl rdii) (SSS) (orresponding ngles of ongruent tringles) (Supplementry ngles) This mens tht line drwn from the entre, perpendiulr to the hord, uts the hord in hlf. lso, if line is perpendiulr to hord nd isets it it hs to pss through the entre. In the digrm elow is the entre, is 50 mm nd E is 30 mm. Find the length of E nd F E F E = E 2 = E = 40mm is the entre E = F... E = EF... E = EF = 40 mm F = E + EF = F = 80 mm (Pythgoren Theorem) (Perpendiulr from entre to hord isets hord) Theorem 2 100% Geometry of the irle hords nd ngles K 15 5 TPI

8 hords nd ngles Knowing More Theorem 3: Equl hords re equidistnt from the entre E F Given: E = F; = ; is the entre; F = To Prove: E = Proof: E 1 = F 2 1 = 2 ut = F ` E = In TEnd T E = + E= + = 90 = ` TE / T ` E = Proof (Perpendiulr from entre isets hord) (Perpendiulr from entre isets hord) Theorem 2 (Proved ove) ( = nd E = F ) (Rdii) (RHS) (orresponding sides in ongruent T 's ) This mens tht if two hords re equl, then they re the sme distne from the entre. Theorem 3 (onverse to 3): hords whih re equidistnt to the entre re equl E F Given: E = F; = ; is the entre; E = To Prove: F = Proof: In TEnd T E = + E = + = 90 = (Rdii) ` TE / T (RHS) ` E = ut E 1 = 2 F nd 1 = 2 ` 1 2 F 1 = 2 ` F = (orresponding sides in ongruent Proof T 's ) (Perpendiulr from entre isets hord) (Perpendiulr from entre isets hord) Theorem 2 This mens tht if two hords re the sme distne from the entre, then the hords re equl. is the entre of the irle. Find the length of PR in the digrm elow if Q = N = 15 m nd MN = 14 m P Q R 15 M 14 N S is the entre MN = NS = 14 m ` MS = MN + NS = 28 m ut PR = MS... PR = 28 m (Perpendiulr from entre isets hord) Theorem 2 (hords equidistnt from the entre re equl) Theorem 3 6 K % Geometry of the irle hords nd ngles TPI

9 hords nd ngles Knowing More Here re some emples using ll the ove theorems: If is the entre of the irle, find the length of if = 200 mm nd = 60 mm 60 mm is the entre... is dimeter... is rdius ` = 200mm ' 2= 100mm = ` = = 80mm (Pythgoren Theorem) = = = 80 mm (Perpendiulr from entre isets hord)... = + = = 160 mm Theorem 2 Find the sizes of ngles +, + nd + in the digrm elow given E = = F nd + F = + = + E = 90 E E = E = nd = = (hords equidistnt from the entre re equl) F Similrly: =... = =... T is equilterl...+ = + = + = 60 (hords equidistnt from the entre re equl) Theorem 2 (ll sides re equl) (Properties of equilterl tringle) is the entre in the irle elow. Find the length of E given nd + = + E = 50 nd + = E is the entre =... =... = + = = 20 m (Perpendiulr from entre isets hord) m + = +E = E =... E = 20 m (hords tht sutend equl ngles t the entre re equl) Theorem 1 100% Geometry of the irle hords nd ngles K 15 7 TPI

10 hords nd ngles Questions Knowing More 1. Find in eh of the following (ll lengths in m) d irle with dimeter 60 m hs hord MN 18 m from the entre. rw rough sketh of the irle nd hord in the o provided. How long is MN? 8 K % Geometry of the irle hords nd ngles TPI

11 hords nd ngles Questions Knowing More 3. The irle elow hs dimeter of 400 units. = 120 units nd P = 160 units. S Q Use the Pythgoren Theorem to find the length of Find the length of RS R P 4. Find the length of hord E 12 F 5. The irle elow hs dimeter 50 m. E = 30 m nd FG = 40 m. Find the distne etween the two hords. (Hint: Sketh in the distnes etween the hords nd the entre) E G F 100% Geometry of the irle hords nd ngles K 15 9 TPI

12 hords nd ngles Questions Knowing More E 6. nd re the entres of the ove irles (whih hve the sme rdius). Use the ove digrm to nswer the following questions. Show tht E = E. n the digrm onstrut lines,, nd. Prove tht =. 10 K % Geometry of the irle hords nd ngles TPI

13 hords nd ngles Questions Knowing More d Now show tht T E / TE nd TE / TE e Show tht E = E. f Show tht is rhomus. g omplete this sentene: When line is drwn etween the entres of different irles through ommon hord, then this line nd the hord eh other t n ngle of. 100% Geometry of the irle hords nd ngles K TPI

14 hords nd ngles Using ur Knowledge Using ngles with irles Here re some theorems showing how to find nd use ngles ppering inside irles. Theorem 4: n ngle sutended t the entre is twie the ngle sutended t the irumferene stnding on the sme r In these 3 digrms, the sme r () sutends n ngle t the entre ( + ) nd t the irumferene ( + ) igrm igrm igrm Proof M M M Given: is the entre of the irle To Prove: + = 2 # + Proof onstrut line M whih psses through the entre In ll 3 digrms = ` + = + ut + M = ` + M = 2+ similrly + M = 2+ (Equl rdii) (ngles opposite equl sides) (Eterior ngle of tringle) In igrm In igrm In igrm + = + M + + M refle + = + M + + M + = + M -+ M ` + = ` refle + = ` + = = 2( ) = 2( ) = 2( + -+ ) = 2+ = 2+ = 2+ Find the sizes of the ngles leled 35 = 2 # ngle tirumferene = 2# 35 = = # ngle tentre = # 40 2 = K % Geometry of the irle hords nd ngles TPI

15 hords nd ngles Using ur Knowledge Theorem 5: The ngle in semiirle is right ngle is the entre Proof + = 180 (Stright line) = = # 180= 90 2 (ngle t entre is twie ngle t irumferene on sme r) Theorem 4 This mens tht n ngle stnding on the dimeter of irle is 90. lso, if n ngle on the irumferene is 90 then it must e stnding on the dimeter. Theorem 6: ngles sutended on the irumferene y the sme r (in the sme segment) re equl Given: is the entre of the irle To Prove: + = + Proof Proof: onstrutrdii nd + 1 = = + 2 ` + = + (ngle t entre is twie ngle t irumferene on sme r) (ngle t entre is twie ngle t irumferene on sme r) Imgine there is hord joining nd. The ngles stnding on the sme r re equl if they re on the sme side of the imginry hord. The ngles stnding on t the irumferene elow, would NT e equl to the ngles meeting t the irumferene ove. If ngles re on the sme side of the imginry hord then they re "in the sme segment". Find the vlues of nd y in the following digrm 60 = 60 y = 120 (ngles in sme segment on sme r) (ngles in sme segment on sme r) Theorem y Notie nd y re not equl to eh other even though they re stnding on the sme r. This is euse is ove the imginry hord, nd y is elow the imginry hord. 100% Geometry of the irle hords nd ngles K TPI

16 hords nd ngles Using ur Knowledge Here re some emples using the ove theorems. is the entre of the irle elow. Find the sizes of the ngles leled, y nd z P y = 2+ PQR ` y = 2# 60 = 120 (ngle t entre is twie ngle t irumferene on sme r) S z y 60 Q P = R ` + PR = + RP = ` = 180 ` 2 = 60 ` = 30 (Equl rdii) (ngles opposite equl sides) (Sum of ngles in tringle) R refle + PR = 360- y = z = # refle + PR 2 1 ` z = # 240 = (ngles round point) (ngle t entre is twie ngle t irumferene on sme r) is the entre of the irle elow. Show tht +E = +EGF G + EGF 1 = + EF 2 nd + EF 1 = 2 + EF = E (ngle t entre is twie ngle t irumferene on sme r) (ngle t entre is twie ngle t irumferene on sme r) (Equl rdii) ` + E = + E (ngles opposite equl sides) E F ut + E = + EGF ` + E = + EGF (ngles in sme segment on sme r) (oth equl + E) In the following digrm is the entre of the irle nd +MPR = 90. Show tht +NPR = +NQP M N + MQP = 90 ` + NQP = MQN (ngle in semiirle) Q now + NPR = MNP ut + MQN = + MPN (ngles in sme segment on sme r) ` + NPR = MQN P R ` + NPR = + NQP 14 K % Geometry of the irle hords nd ngles TPI

17 hords nd ngles Questions Using ur Knowledge 1. Write only the sizes of nd y (no resons neessry) in eh of the following digrms: y = y = y 272 = y = y = y = d y = y = Find the size of % Geometry of the irle hords nd ngles K TPI

18 hords nd ngles Questions Using ur Knowledge 3. Find +QR in the digrm elow. R Q 30 P 4. is the entre of the irle elow nd LN is the dimeter. Find +LM. N Find +LNM. K Find +LMN. L 60 M d Let +KL =. Find +KL in terms of. e Prove +LK + +KN = K % Geometry of the irle hords nd ngles TPI

19 hords nd ngles Questions Using ur Knowledge E F 5. In the digrm ove is the entre of the irle. Let + = + =. Show tht =. Use + to find + in terms of. Find +. d Find +F in terms of. e Find +E in terms of nd prove tht E. 100% Geometry of the irle hords nd ngles K TPI

20 hords nd ngles Thinking More yli Qudrilterls These re lled "yli quds" for short. These re qudrilterls tht would llow irle to pss through ll its verties. This is yli qud. ll 4 verties touh the irumferene. This is NT yli qud. ne verte lies outside the irle. This is NT yli qud. ne verte does not touh the irumferene. yli quds hve their own properties nd theorems whih n e used to solve prolems. Theorem 7: pposite ngles of yli qud re supplementry y Given: is the entre of the irle To Prove: = 180 nd = 180 Proof Proof: 2 2y onstrutrdii nd Let+ = nd + = y ` refle + = 2 nd + = 2y ut 2+ 2y = 360 (ngle t entre is twie ngle t irumferene on sme r) (ngle t entre is twie ngle t irumferene on sme r) ` + y = 180 (ngles round point) ` = 180 ` Theoppositenglesre supplementry. Similrly, y onstruting rdii nd it n e shown = 180 This lso mens tht qudrilterl is yli if its opposite ngles dd up to 180 Find the sizes of ngle m nd n in the yli qudrilterl elow 80 n m + 80 = 180 ` m = = 100 (pposite ngles of yli qud re supplementry) Theorem m n = 180 ` n = = 65 (pposite ngles of yli qud re supplementry) 18 K % Geometry of the irle hords nd ngles TPI

21 hords nd ngles Thinking More Theorem 8: The eterior ngle of yli qud is equl to the interior opposite ngle Given: is the entre of the irle To Prove: + = +E Proof Proof: = E+ + = 180 (pposite ngles of yli qud re supplementry) (ngles on stright line) E ` + = + E (oth ngles re supplementry with +) This lso mens tht qudrilterl is yli if the eterior ngle is equl to its interior opposite ngle. Find the size of ngle nd y Q R S T = 110 y = 180 ` y = (Eterior ngle of yli qud equls interior opposite ngle) (pposite ngles of yli qud re supplementry) y = 53 P Find the following ngles if is the entre of the irle 55 E Find + is the dimeter... + = 90 Find = = = = 35 ( is the entre) (ngle in semiirle is right ngle) (Interior ngles of tringle) Find + +E + + = E + 35 = E = 145 Now + = +E... + = 145 (ngles on stright line) (Eterior ngle of yli qud equls interior opposite ngle) 100% Geometry of the irle hords nd ngles K TPI

22 hords nd ngles Questions Thinking More 1. Find the sizes of the ngle nd y in eh of the following y 77 y 61 d 102 y y e f y y K % Geometry of the irle hords nd ngles TPI

23 hords nd ngles Questions Thinking More 2. nswer the following questions out this digrm given = nd =. Show T / T. Find the sizes of + nd +. Is the dimeter of the irle? Prove it. 100% Geometry of the irle hords nd ngles K TPI

24 hords nd ngles Questions Thinking More 3. pposite ngles of yli qud re supplementry. Is trpezium yli qud? pposite ngles of yli qud re supplementry. Wht do you know out the opposite ngles of prllelogrm? When re the opposite ngles of prllelogrm supplementry? Is it possile for non-retngulr prllelogrm to e yli qudrilterl? 22 K % Geometry of the irle hords nd ngles TPI

25 hords nd ngles Questions Thinking More 5. Use the informtion given elow to nswer the questions tht follow. Given (fill this on your digrm) +HE = 50 +GHI = 20 is the entre of the lrger irle GFH is yli qud in the smller irle H Find +H. I G F Find +FGH. E Find +GFH. d Find +FH. 100% Geometry of the irle hords nd ngles K TPI

26 hords nd ngles Visul Theorems Visul Theorems Here is visul summry of ll the theorems in this hpter. Theorem 1: Equl hords sutend equl ngles t the entre Theorem 4: n ngle sutended t the entre is twie the ngle sutended t the irumferene y the sme r. 2 Theorem 1: Equl ngles t the entre sutend equl hords Theorem 5: The ngle in semiirle is right ngle Theorem 2: perpendiulr line from the entre to hord isets the hord imeter Theorem 2: line drwn from the entre to hord s midpoint is perpendiulr to the hord Theorem 6: ngles sutended y the sme r in the sme segment re equl Theorem 3: Equl hords re equidistnt to the entre Theorem 7: pposite ngles of yli qud re supplementry y y Theorem 3: hords whih re equidistnt to the entre re equl Theorem 8: The eterior ngle of yli qud is equl to the interior opposite ngle 24 K % Geometry of the irle hords nd ngles TPI

27 hords nd ngles nswers sis: Knowing More: 1. Setor hord 6. r d Tngent E e entre f imeter 2. g Sent h Point of ontt d entre g When line is drwn etween the entres of different irles through ommon hord, then this line nd the hord MEET eh other t n ngle of 90 e irumferene Using ur Knowledge: 3. sent is line tht uts through irle ut etends eyond the irumferene of the irle. 1. = 30 y = 52 = 88 y = 44 However, hord is line tht is formed etween two points on the irumferene. = 41 y = 41 d = 70 y = Knowing More: 20 m = 5 m = 2. + = 70 = 5 m d = QR = M MN = 48 m 4. + LM = LNM = 30 + LMN = 90 N d + KL = units = RS 320 units = 5. = = d + = 90 + F = M 20 m = e + E = 90-2 N 15 m = ` distne etween hords = 35 m 100% Geometry of the irle hords nd ngles K TPI

28 hords nd ngles nswers Thinking More: 1. = 77 = 44 y = 101 y = 56 = 61 d = 102 y = 29 y = 78 e = 114 f = 37 y = 66 y = = 90 + = is yli qudrilterl 4. They re equl. When prllelogrm is squre or retngle. No, euse opposite ngles re not supplementry, they re equl H = FGH = 80 + GFH = 80 d + FH = K % Geometry of the irle hords nd ngles TPI

29 hords nd ngles Notes 100% Geometry of the irle hords nd ngles K TPI

30 hords nd ngles Notes 28 K % Geometry of the irle hords nd ngles TPI

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32 Geometry of the irle - hords nd ngles

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