Preparing for the Regents Examination Geometry, AK

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3. mCDB 1 _ m ( 2 CB ) 4. mE mCDB 5. CBD FCE (AA) 6. EF _ EC CD _ DB (Corresponding sides of similar triangles are in proportion.) 14 1. AZT MZH 2. HMT TAH 3. MZH AZT (AA) 4. MZ _ AZ ZH _ ZT (Corresponding sides of similar triangles are in proportion.) 15 a 1. BAC BDC 2. ABD ACD (Congruent inscribed angles) 3. ACD ABE b BE 6 (AA) 13-8 Circles in the Coordinate Plane (pages 386–387) 1 (2) (x 4) 2 (y 3) 2 36 2 (4) (13, 1) 3 a (x 1) 2 (y 4) 2 25 b (x 3) 2 (y 2) 2 49 c (x 4) 2 (y 1) 2 121 d (x 2) 2 (y 5) 2 64 e (x 2) 2 y 2 121 f x 2 y 2 144 4 a x 2 y 2 49 b (x 4) 2 y 2 25 c x 2 y 2 41 d (x 2) 2 y 2 36 e (x 3) 2 (y 2) 2 9 f (x 1) 2 (y 3) 2 25 5 For parts a–d, check students’ graphs. The center and radius of each circle are listed below. a C(2, 4), r 3 b C(3, 4), r 5 c C(0, 2), r 4 d C(6, 0), r 2 6 a C(0, 0), r 9 b C(0, 0), r 11 c C(0, 0), r 20 d C(0, 0), r 1 e C(0, 0), r √ 2 7 Note: Students’ answers may vary for the two other points on the circle. a C(5, 3), r 8; (5, 11), (5, 5) b C(4, 0), r 10; (6, 0), (14, 0) c C(2, 5), r 4; (2, 5), (6, 5) d C(3, 7), r 2 √ 3 ; (3, 7 2 √ 3 ), (3, 7 2 √ 3 ) 8 a x 2 y 2 25 b x 2 y 2 4 c x 2 y 2 34 d x 2 y 2 37 e x 2 y 2 68 9 a (x 2) 2 (y 3) 2 2 b (x 3) 2 (y 7) 2 80 10 a x 2 (y 0.5) 2 12.25 b (x 3) 2 (y 3) 2 5 11 a Circumference 8, area 16 b Circumference 12, area 36 c Circumference 2 √ 13 , area 13 12 a (3, 0) b (x 3) 2 (y 1) 2 1 c 13 a C(1, 3), r 2 √ 2 b Area 8, circumference 4 √ 2 13-9 Tangents, Secants, and the Circle in the Coordinate Plane (pages 392–393) 1 (1) 0 2 a (3, 11), (3, 1) b (3) x 1 3 y 0 4 y 0.5x 2.5 5 y x 4 6 y 3 7 y 4 8 y 2x 10 9 a (4, 3), (3, 4) b secant 10 a (0, 4), (4, 0) b secant 11 a (2, 3), (2, 5) b secant 12 a (0, 3) b tangent 13 a (0, 2), (2, 0) b secant 14 a (5, 5) b tangent 15 a (1, 4), (1, 4) b secant 13-9 Tangents, Secants, and the Circle in the Coordinate Plane 85
16 a (2 √ 2 , 2 √ 2 ), (2 √ 2 , 2 √ 2 ) b secant 17 a (10, 0) and (10, 0) b secant 18 a (7, 7) and (7, 7) b secant 19 a (6, 0) b tangent 20 a (8, 6) and (6, 4) b secant 21 a (0, 5) and (4, 3) b secant 22 a Sub-in the given points in (x 3) 2 (y 2) 2 25. b Midpoint of −−− ME is (3.5, 2.5); distance √ 0.5 . 23 a y x 8 b y x 8 c (0, 8) 24 a y 3x 10 b x 10 c P(10, 20) d PA √ 160 4 √ 10 , PB √ 1000 10 √ 10 , PM 20 PA PB PM 2 4 √ 10 10 √ 10 20 2 400 400 Chapter Review (page 393–397) 1 (1) All chords in a circle are congruent. 2 (2) (x 4) 2 (y 2) 2 9 3 (2) 1 4 (3) 8 5 (3) 6 and 15 6 (2) 12 7 (1) 12 8 (2) 6 9 (2) 68 10 (3) 122 11 (1) 43 12 (3) 86 13 (2) 2 : 1 14 (1) 41 15 (3) 10 inches 16 (3) 125 17 (1) 40 18 (4) 100 19 (2) 230 20 16 21 90 22 104 23 20 24 48 25 12 26 a 40 b 40 c 40 d 40 e 110 f 110 g 35 86 Chapter 13: Geometry of the Circle 27 a 40 b 70 c 20 d 45 e 25 f 65 28 a 30 b 45 c 15 d 135 e 90 29 a C(0, 0), r 20 b Answers may vary. (0, 20), (0, 20) 30 a C(7, 11), r 9 b Answers may vary. (7, 20), (7, 2) 31 a C(3, 13), r 6 b Answers may vary. (3, 19), (3, 7) 32 a C(3, 5), r 2 √ 5 b Answers may vary. (3, 5 2 √ 5 ), (3, 5 2 √ 5 ) 33 (x 3) 2 (y 2) 2 16, C(3, 2), r 4 34 a (x 4) 2 (y 2) 2 49 b (x 5) 2 y 2 2 c x 2 y 2 169 d C(3, 0), r 5, (x 3) 2 y 2 25 e C(4, 4) r √ 10 , (x 4) 2 (y 4) 2 10 2 ) 2 35 6 2 6 2 (6 √ 36 y 1 _ x 10 3 37 a (4, 3) and (4, 3) b secant 38 a (0, 3) and (3, 0) b secant 39 a (8, 2) and (2, 8) b secant 40 a (0, 5) and (3, 4) b secant Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 41 1. AOB COB 2. −− AB −− CB 3. AB CB 42 1. −−− DA is the diameter of circle O. 2. Radii −−− OA , −−− OC , and −−− OD 3. −−− OA −−− OC −−− OD (Radii are equal.) 4. −−− AD −−− CD 5. AOD COD (SSS SSS) 6. AD CD 7. ABD CBD (Congruent arcs are intercepted by congruent inscribed angles.)
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ANSWER KEY Preparing for the REGENT
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Contents Chapter 1: Essentials of G
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1-4 Angles (pages 9-10) 1 (4) It is
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6 True 7 True 8 True 9 True 10 Answ
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3 1. e ∨ ~f 2. ~f → g 3. ~e 4.
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CHAPTER 3 3-1 Inductive Reasoning (
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6. BAC DAE 6. Transitive property.
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CHAPTER 4-1 Setting Up a Valid Proo
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6 Given: BA bisects CBE 1 2 2 4
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1 1. −−−− ABCD 2. 1 2 3. _
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18 1. I is the midpoint of −− E
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5-3 Isosceles and Equilateral Trian
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12. −−− HG −−− DC 13.
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5 1. −− FG is the perpendicular
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Chapter Review (pages 84-85) Note:
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20 Use constructing congruent angle
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6-3 Line Reflections and Symmetry (
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11 A(0, 8), B(2, 2), C(6, 4) (8) 10
Page 37 and 38: 5 (3) (x, y) → (x, 2y) 6 (1) tran
Page 39 and 40: 11. mABC mADC 12. 2mABD 2mADB 13.
Page 41 and 42: 6. mDAB mCAD mDCB mACD 7. mCAB
Page 43 and 44: g e 17 a _ f d b Undefined c a _
Page 45 and 46: 5 BIG is isosceles because it has t
Page 47 and 48: ___ 27 a M KA (5, 1), M ___ AT (4
Page 49 and 50: 8 106 9 mA 75, mC 67 10 57 11 60
Page 51 and 52: 9-5 The Sum of the Measures of the
Page 53 and 54: 9-7 The Converse of the Isosceles T
Page 55 and 56: 12 1. ___ CE ___ BA , ___ BD ___
Page 57 and 58: 12 a mx 45, my 45 b mx 98, my 8
Page 59 and 60: Quadrilaterals 10-2 The Parallelogr
Page 61 and 62: 7. MAD RCB 8. MAD RCB (SAS SAS)
Page 63 and 64: 5. RSQ TSV (Vertical angles) 6. QR
Page 65 and 66: 3 4x 2 3x 3 x 5 RS 18 4 Perime
Page 67 and 68: Note: Since there are many variatio
Page 69 and 70: 28 Enclose PAT in a large rectangle
Page 71 and 72: Geometry of Three Dimensions 11-1 P
Page 73 and 74: 11-6 Volume of a Prism (pages 269-2
Page 75 and 76: 14 47.5 in. 2 15 h 4 in. 16 25 cm
Page 77 and 78: 14 a 3 : 2 b QR 10, RS 20, ST 12
Page 79 and 80: 4. A A 5. ADE ABC b (AA) AC _ AB
Page 81 and 82: 12 7 √ 2 13 4 14 5 √ 3 15 x
Page 83 and 84: 13-2 Arcs and Chords (pages 350-351
Page 85 and 86: 27 1. Common external tangents, −
Page 87: 8. mHCT 1 _ m 2 TH mCHT 1 _ CT 2
Page 91 and 92: 15 a Use constructing a congruent a
Page 93 and 94: 5 Two horizontal lines y 11 and y
Page 95 and 96: Each review has a total of 58 possi
Page 97 and 98: Part III For each question, use the
Page 99 and 100: Chapters 1-3 (pages 438-441) Part I
Page 101 and 102: Part IV For each question, use the
Page 103 and 104: 16 Score Explanation 4 a 91, b 35
Page 109 and 110: 20 Score Explanation 6 The followin
Page 113 and 114: Part II For each question, use the
Page 117 and 118: Part III For each question, use the
Page 123 and 124: 12 Score Explanation 2 31, and an a
Page 127 and 128: 16 Score Explanation 4 DN √ 106
Page 129 and 130: 20 Score Explanation 2 a 3, and an
Page 131 and 132: 17 Score Explanation 4 6 √ 5 , a
Page 135 and 136: 16 Score Explanation 4 mADE 93, mB
Page 137 and 138: 20 Score Explanation Chapters 1-14
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Part IV For each question, use the
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Practice Regents Examination One (p
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36 Score Explanation 4 36, and an a
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32 Score Explanation 2 108, and an
Page 147 and 148:
Part II For each question, use the
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Part IV For each question, use the
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Preparing for the Regents Examination Geometry, AK

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