Circle Properties A. a semicircle. 2) Copy each of the following diagiams and find the marked angles

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Gregory Burns
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1 Circle Properties A a semicircle 1) Draw a citcle of any size, centie O. (You will be taking measutements fiom the diagiam so make the radius of your c cle greater than 5 cm) Diaw the diametei AB At any point on the circumference mark the point C Diaw the lines AC and BC Measuie and record all the angles Repeat these instiuctions foi anothei two circles W te yom findings 2) Copy each of the following diagiams and find the marked angles The lettei O lepiesents the centie of the circle Diagrams not to scale w\vw 1 Clicks com u Page 27 of 138 Workshee s li en ed o Kooweeruo Seeoradan/ Colleae. Exnire 11 /I 0/2014

2 The angle at the centre lylu I) Diaw a c cle of any size, centie O Diaw two radii OA and OB At any point on the ciicumfeience maik the point C Diaw the lines AC and BC Measuie and tecoid all the angles Repeat these instructions foi another two circles Wi ite yout findings ' ACB and Z AOB are said to be angles subtended by the arc AB. 2) Copy each of the following diagiams and find the maiked angles The lettei O lepresents the centre of the ciicle Diagiams not to scale Page 28 of 138 Worksh ts: licensed o ICooweerun Secondan/ C SIeae. Exnires 11/10/2014

3 h). - k). Page 29 of 138 or sheets licensed o Kooweenuin Secondar CoSleoe. Exnires 11/10/2014,

4 Angies in Cyclic Quadrilateral. 1) Page 30 of 138 www 1 Oticks com au WorkishootB lie nsed ICooweeruo Setsondan/ Colleo. Eifoi ob 11/10/2014,

5 Page 31 of 138 Workssh ts lic nsed to Kooweerun S condar Colloa. Ex iros 11/10/2014.

6 Cir ft Properties 4 Mixed Questions Copy each of the following diagiams and find the marked angles and lengths The lettei 0 tepiesents the centie of the citcle A Diagiams ot \ww loticks com au Page 33 of 138 Wor sho tss licensed to ICoowe ruo Secondarv Colle e. Exnires 11/10/2014

7 23) 24) 27). Page 34 of 138 orksheets lice ed to Kooweerun Seconderu Colieae. Evnire 11/10/2014.

8 Revision Exercise Copy each of the following c agiams and find the maiked angles and lengths The lettei O repi esents the cenhe of the ciicle Diagiams not to scale Page 37 of 138 Worksheet?; licensed o Conweerurs Se orsdaru Colleae. Exoires 11/10/2014

9 SBT is a tangent to the circle at B The segment containing ZABS is known as the alte ate segment of ZACB, because ZABS and ZACB aie on alte ate sides of the chotd AB The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. 1) In Fig 1 UV is a tangent to the c cle N In Fig 2 construct MN pe pendicular to the tangent UV Prove that Za = Zb. a + c = 90 MN passes through the centre of the circle and is a diameter. e = 90 (angle m a semicircle) Now finish the proof showing Za = Zb. 2) In the diagram find a). ZZXY, b) ZSZX, c). ZYZX. a). ZZXY = 73 (Angle in alternate segment) Now finish parts b) and c) Copy each of the following diagr ms and find the marked angles. The letter O represents the centre of the circle Diagrams not to scale. 3). 4). 5). Page 38 of 138 www 1 Oticks com a Workshee s licensed o ICoowee uo Secondary Colleae. Exnire 11/10/2014

10 Page 39 of 138 wwvv 1 Oticks com a Work hee s licen ed o Cooweerun Secon er / Coll e. Exnires 11/1 0/2014

11 CIRCLE THEOREMS ANSWERS Circle Properties. Angle in a semicircle. (Page 27) 1) Angle in a semicircle is 90 2) a) x = 90, y = 22 b) x = 90, y = 54 c) x 90, y = 63 d) x = 90, y = 41 e) x = 90, y = 52 f) x = 90, y = z = 45 g) u = 42, v = 96, w = 84, x = y = 48 h) u = 31, v = 31, w = 62, x = 59, y = 59 i) u = 53, v = 74, w 37 x 106, y = 37 J) u v = w 60, x = 30, y = 120, z = 30 k) u = v = 45, w = 45, x = 90, y = 90, z = 45 l) u = v = 55, w = 35, x = 35, y = 35, z = 35 Circle Properties. Angle at centre. (Page 28) 1) Angle at centre is twice that at circumference outside subtended arc 2) a) x = 262, y = 49 b) x = 108, y = 252 C) x = 49, y = 82 z = 98 d) x = 118, y = 59 e) x = 120, y = 60 f) x = 53, y = 106 g) u = 56, v = 22, w = 56, x = 22, y = 156 h) u = 33, v = 33, w = 41, x = 148, y = 41, z= 139 i) u = 34, v = 48, w = 96, x = 14 j) u = 32, v = 32, w = 116, x = 58 k) u = 58 5, v = 58 5, w = 63, x = y = 58 5, z = 63 l) w= 114, x = 57, y = 57 Circle Properties. Angles in same Segment. (Page 29) 1) Angle in the same segment are equal 2) a) x = 64 b) x = 41 c) u = 47, v = 35, w = 98, x = 98 d) u = 45, v = 94, w = 41, x = 41, y = 86 e) x = 62, y = 62 f) u = 45, v = 38, w = 45, x = 38, y = 97, z = 83 g) u = 53, v = 37, w = 37, x = 53, y = 106, z = 74 h) u = 43, v = 43, w = 94, x = 43, y = 47, z = 47 i) u = 74, v = 37, w = 37, x = 106, y = 53, z = 37 j) v = 60, w = 120, x = 32, y = 28, z = 28 k) u = 37, v = 37, w = 106, x = 37, y = 106 l) r = 32, s = 109, t = 39, u = 39, v = 44, w = 71, x = 65, y = 65, z = 71 Circle Properties. Cyclic Quadrilateral. (Page 30) 1) Opposite angles in a cyclic quadrilateral add up to 180 Exterior angle in cyclic quadrilateral is equal to the interior opposite angle 2) a) x = 65, y = 78 b) x = 97, y = 72, z = 108 c) u = 98, v = 121, w = 82, x = 59 d) u =108, v = 108, w = 72, x = 102 e) x = 82, y = 98, z= 196 f) u = 128, v = 64, w = 116, x = 64 g) u = 72, v = 72, w = 108, x = 72 h) u = 80, v = 109, w = 80, x = 71, y = 29 i) u = 47, v = 133, w = 47, x = 97, y = 83, z = 97 j) u = 32, v = 48, w = 48, x = 40, y = 40, z = 60 k) p = 42, q = 43, r = 42, s = 57, t = 57, u = 80, v = 80, w = 100, x = 38, y = 38, z = 100 I) x= 116, y = 128, z = 52

12 Circle Properties. Tangents. (Page 31) 1) Tangent to radii is 90, two tangents from exterior point equal in length 2) a) x = 90, y = 53 b) x = 90, y = 13 cm c) x = 90, y = 13 4 cm d) x = 90, y = 10 4 m e) x = 90, y = 62, z = 59 f) x = 56, y = 68, z = 34 g) u = 56, v = 68, w = 34, x = 22 h) u = 50, v = 50, w = 80, x = 40, y = 10 i) u 90, v = 21, w = 42, x = 48, y = 138, z = 21 j) u = 27, v = 27, w = 126, x = 63, y = 54, z = 43 mm k) r = 90, s = 52, t = 90, u = 52, v = 38, w 52, x = 19 cm l) p = 60, q = 60, r = 60, s = 60, t = 60, u = 60, v = 30 Circle Properties. Circle constructions. (Page 32) Section A 1) Angles in a semi circle are 90 2/3) Correct construction Section B) 1) Correct construction Section C 1/2). Correct construction Circle Properties. Mixed Questions. (Page 33) 1) x = 90, y = 17 2). x = 94, y = 266 3) x = 90, y = 9 8 cm 4) x = 94, y = 68, z= 112 5) x = 90, y= 56 6) x = 46 7) u = 52, v= 101, w = 27, x = 27, y = 79 8) x = 124, y = 62 9) x = 90, y = 45, z = 45 10) x = 63, y = 54, z = 27 11) p = q = r = s = t = u = 60, v = 30 12) u = 93, v= 117, w = 87, x = 63 Circle Properties. Mixed Questions. (Page 34) 13) u = 39, v = 102, w = 78, x = 51, y = 51 14) u = 49, v = 35, w = 49, x = 35, y = 96, z = 84 15) u = 52, v = 52, w = 76, x = 38, y = 14 16). x = 61, y = ) v = 36, w = 108, x = 27, y = 45, z = 45 18). u = 38, v = 38, w = 76, x = 52, y = 52 19) u = 48, v = 48, w = 42, x = 42, y = 42, z = 42 20) r = 90, s = 51, t = 90, u = 51, v = 39, w = 51, x = 9 4 cm 21). r = 33, s = 118, t = 29, u = 29, v = 47 w =62, x = 71, y = 71, z = 62 22) u = 124, v = 62, w = 118, x = 62 23) u = 36, v = 43, w = 43, x = 39, y = 39, z = 62 24) u = 61, v = 61, w = 58, x = 61, y = 61, z = 58 25) u = 41, v = 41, w = 37, x= 156, y = 37, z = ). u = 44, v = 136, w = 44, x = 87, y = 93, z = 87 27) u = 31, v = 31, w = 118, x = 59, y = 62, z = 47 mm Symmetry in Circles (Page 35) 1) a) 17 3 cm b) 18 8 cm c) 21.2 cm d) cm 2) a) 2 cm b) 8 cm c) 6 cm d) 4 4 cm 3). a) 38 cm b) 88 cm c) 69 6 cm d) 55 7 cm

13 Circle Theory (a) (Page 36) Thm 1 OB = OC radii Therefore OBA = BAO base s isos triangle BOD = OBA + BAO ext of AOB = 2 BAO Similarly COD - OAC + OCA from AOC = 2CAO BOC = BOD + COD BOC = 2BAO + 2CAO = 2(BAO + CAO) = 2 BAG Thm 2 AOB = 2ACB at centre AOB = 180 st line ACB = 90 Thm 3 AOB = 2ABD at centre AOB = 2ACD at centre Therefore 2ABD = 2ACD ABD = ACD Thm 4 Join B and C to centre O BCD = y, BAD = x BOD = 2y at centre, reflex BOD = 2x at centre 2x + 2y = 360 at point x + y = 180 BAD + BCD = 180 Thm 5 s are congruent RHS Therefore AP = BP, OAP = OBP APO = OPB therefore OP bisects APB Circle Theory (b) (Page 37) I) x = 264, y = 48 2) x = 90, y = 22 3) x = 36, y = 84, z = 60 4) a = 109, b = 86, c = 94 5) a = 52, b = 46, c = 46, d = 98, e = 31 6) a = 49, b = 41, c = 82, d = 41, e = 98 7) a = 35, b = 35, c = 26, d = 134 8) a = 28 9) a = 127, b = 106, c = 21 10) a = 24, b = 90, c = 66, d = 228, II) a= 49, b = ) a = 53, b = 127, c = 53, d = 91, e = 89, = 66 = 91 Alternate Segment Theorem (a) (Page 38) 1) d + c = 90 angle sum triangle a = d = b (in same seg) Therefore UML = MWL 2) b) SZX = 24 ( alt seg) c) YZX = 83 st line 3) a = 24, b = 63, c = 93 4) d = 31, e = 71, f = 71 5) x = 42, y = 68, z = 70 Alternate Segment Theorem (b) (Page 39) 6) a = 65, b = 38, c = 77 7) u = 76, v = 40, w = 40 8) a = 51, b = 78, c = 78, d = 51 9) g = h = i=j = 49 10) s = 78, t = 78, u = 24, v = 78 11) a = 50, b = 50, c = 50 12) u = 63, v = 54, w = 63 13) a = 64, b = 58, c = 58, d = 58 14) z = 64 15) c = 90, d = 69, e = 69 16) u = 90, v = 26, w = 64, v = 64 17) a = 42, b = 73, c = 17 18) a = 37, b = 53, c = 53, d = 90 19) c = 29, d = 90, e = 61, 20) a = 46, b = 46, c = 46, d = 92, e = 44, f = 44 f = 61

Properties of the Circle

9 Properties of the Circle TERMINOLOGY Arc: Part of a curve, most commonly a portion of the distance around the circumference of a circle Chord: A straight line joining two points on the circumference