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�HKP 23. x = 30; AB = 50 25. x = 2; BC = 17 29. solution A 31. B 33. D 35. x = 5.5; yes; UV = WV= 41.5, and UT = WT = 33. TV = TV by the Reflex. Prop. of =. It is given that ∠VWT ≅ ∠VUT and ∠WTV ≅ ∠UTV. ∠WVT ≅ ∠ UVT by the Third Thm. Thus △TUV ≅ △TWV by the def. of ≅ . 39. 1 __ 9 41. rt. 43. 72° 45. 146° 4-4 ̶̶ AB ̶... |
K ≅ ̶̶ QP ≅ ̶̶̶ NQ ≅ ̶̶ CB c. ̶̶ LJ and ̶̶̶ MN ≅ ̶̶̶ MP by the Reflex. Prop. of ≅. ̶̶ DC d. Def. of ⊥ e. Rt. ∠ ≅ Thm. ̶̶ ̶̶ AB g. SAS Steps 2, 5, 6 AB ≅ Exercises 1. ∠T 3. It is given ̶̶ ̶̶̶ QP. MQ and that ̶̶̶ MP ≅ Thus △MNP ≅ △MQP by SSS. 5. When x = 4, HI = GH = 3, and ̶̶ ̶̶ IJ = GJ = 5. HJ by the Reflex. HJ ≅ Prop.... |
wing deflector match the lengths of the logs in the other wing deflector, the will be ≅ by SAS or SSS. 27. Yes; if each side is ≅ to the corr. side of the second △, they can be in any order. 29. G 31. J 35. x = 27; FK = FH = 171, so of ≅. ∠KFJ ≅ ∠HFJ by the def. of ∠ bisector. of ≅. So △FJK ≅ △FJH by SAS. 37. a < 4 ... |
̶ CB ̶̶ AC ≅ ̶̶ 4. Yes; it is given that DB. ̶̶ ≅ CB by the Reflex. Prop. of ≅. Since ∠ABC and ∠DCB are rt. , △ABC and △DCB are rt. , △ABC ≅ △DCB by HL. ̶̶ BC Exercises 1. The included side is enclosed between ∠ABC and ∠ACB. 3. Yes, the △ is determined by AAS. 5. No; you need to know that a pair of corr. sides are ≅.... |
4. △KJL ≅ △MJN (SAS Steps 2, 3) 5. ∠LKJ ≅ ∠NMJ (CPCTC) ̶̶̶ ̶̶ 6. MN (Conv. of Alt. KL ǁ Int. Thm.) 4. RJ = JL = √ 5, RS = JK = √ 10, and ST = KL = √ 17. So △JKL ≅ △RST by SSS. ∠JKL ≅ ∠RST by CPCTC. Exercises 1. corr. and corr. sides. 3a. Def. of ⊥ b. Rt. ∠ ≅ Thm. c. Reflex. Prop. of ≅ d. Def. of mdpt. e. △RX... |
̶ AC and ̶̶ BD. ̶̶ CE ; ̶̶ BE ≅ ̶̶ DE (Def. (Given) ̶̶ 2. AE ≅ of mdpt.) 3. ∠AEB ≅ ∠CED (Vert. Thm.) 4. △AEB ≅ △CED (SAS Steps 2, 3) 5. ∠A ≅ ∠C (CPCTC) ̶̶ ̶̶ 6. CD (Conv. of Alt. Int. AB ǁ Thm.) 17. 14 25. G 27. G 29. Any diag. on any face of the cube is the hyp. of a rt. △ whose legs are edges of the cube. Any 2 o... |
Prop. of ≅) Selected Answers S95 S95 ����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� ... |
Conv. of the Isosc. △ Thm., the created are isosc. and the dist. is the same. 2a. 66° 2b. 48° 3. 10 4. By the Mdpt. Formula, the coords. of X are (-a, b), the coords. of Y are (a, b), and the coords. of Z are (0, 0). By the Dist. Formula, XZ = YZ = ̶̶ √ XZ ≅ isosc. ̶̶ YZ and △XYZ is a 2 + b 2. So ̶̶ KJ and ̶̶ KL... |
̶̶ PC ≅ ̶̶ AC, ̶̶ AB ≅ ̶̶ 2. CB (CPCTC) 3. △ABC is isosceles (Def. of Isosc) 43. H 47. (2a, 0), (0, 2b), or any pt. ̶̶ AB 49. x = 3 on the ⊥ bisector of or x = 1 51. m = -3 53. m = 2 __ 3 By the Mdpt. Formula, the coords. of A are (0, a) and the coords. of B are (b, 0). By the Dist. Formula, PQ = √ (0 - 2b) 2 +... |
) 2 = 2c units. So EF = AD. 15a. b. 8.5 mi 17. 2s + 2t units; st square units 19. (p, 0) 21. AB ≈ 128 nautical miles; AP = BP ≈ 64 nautical miles; so P is the mdpt. ̶̶ AB. 23. By the Dist. Formula, AB ( and AM of = √ S96 S96 Selected Answers SGR ̶̶ DB ≅ ̶̶ XZ 9. ∠Q 10. 25 11. 7 ̶̶ ̶̶ AE (Given) DE, ̶̶ DA (Ref... |
pt. of ̶̶ AG. (Given) ̶̶̶ LM ≅ ̶̶ LN ≅ ̶̶ PQ ; ̶̶ GC ≅ ̶̶ HA ǁ ̶̶ AC (Def. of mdpt.) 2. ̶̶ 3. GB (Given) 4. ∠HAC ≅ ∠BGC (Alt. Int. Thm.) 5. ∠HCA ≅ ∠BCG (Vert. Thm.) 6. △HAC ≅ △BGC (ASA Steps 4, 2, 5) ̶̶ ̶̶̶ XZ, WX ⊥ ̶̶ ̶̶ ZX (Given) YZ ⊥ 2. ∠WXZ and ∠YZX are rt. . (Def. of ⊥) 3. △WZX and △YXZ are rt. . (Def. of r... |
) 6. ∠1 ≅ ∠2 (CPCTC) ���������������������������������������������������������������������� ̶̶ PQ ≅ ̶̶ PS ≅ ̶̶ QS ≅ ̶̶ RQ (Given) ̶̶ RS (Given) ̶̶ QS (Reflex. Prop. of ≅) 20. 1. 2. 3. 4. △PQS ≅ △RQS (SSS Steps 1, 2, 3) 5. ∠PQS ≅ ∠RQS (CPCTC) 6. bisect) ̶̶ QS bisects ∠PQR. (Def. of ̶̶ GJ, and L is ̶̶ KL (Def. of ≅) 21. ... |
(2a - a) 2 + (b - 0) 2 = √ a 2 + b 2, and GH = √ (0 - a) 2 + (b - 2b) 2 = √ a 2 + b 2. ̶̶̶ ̶̶ GH by the def. of ≅. EF ≅ So 26. Use coords. P (0, 2b), Q (0, 0), and R (2a, 0). Then, by the Mdpt. Formula, M (a, b). By the Dist. Formula, QM = √ PM = √ = √ a 2 + b 2, and RM = ... |
0) 2 = √ a 2 + b 2, (a - 0) 2 + (b - 2b) 2 (2 - 3) 2 + (5 - 5) 2 = 1. Since 10 2 ), or 9 + 1 = 10, the (2a - a) 2 + (0 - b) 2 = √ a 2 + b 2. Chapter 5 5-1 PQ, ̶̶ SQ ⊥ PS bisects ∠QPR. PR (Given) Check It Out! 1a. 14.6 1b. 10.4 2a. 3.05 2b. 126° 3. QS bisects ∠PQR. 4. y + 1 = - 2 __ 3 (x - 3) Ex... |
. Prop. of ≅) ̶̶ SQ ≅ ̶̶ PS ≅ 5-2 Check It Out! 1a. 14.5 1b. 18.6 1c. 19.9 2. (4, -4.5) 3a. 19.2 3b. 52° 4. By the Incenter Thm., the incenter of a △ is equidistant from the sides of the △. Draw the △ formed by the streets and draw the ∠ bisectors to find the incenter, point M. The city should place the monument at poi... |
136 5. 156 7. (4, 2) 9. (2, -3) 11. (-1, 2) 13. 7.2 15. 5.8 17. (0, -2) 19. (-2, 9) 21. 12 23. 5 25. 36 units 27. (10, -2) 29. 54 31. 48 33. Possible answer: ⊥ bisector of the base; bisector of the vertex ∠; median to the base; altitude to the base 35. A 37. A 41. D 43. D ̶̶ RS = c __ 45a. slope of ; slope of b ̶̶ ̶̶ ... |
Exercises 1. midpoints 3. 5.1 5. 5.6 7. 29° 9. less than 5 yd 11. 38 13. 19 15. 55° 17. yes 19. 17 21. n = 36 23. n = 8 25. n = 4 27. B 29. Possible answer: about 18 parking spaces 31. 11 33. 57° 35. 123° 37a. 2.25 mi b. 28.5 mi 39. D 41. D 43. equilateral and equiangular 45. 7 47a. 32; 16; 8; 4 b. 1 __ 4 c. 64 ( 1 __... |
mi < d < 72 mi ̶̶ EF, ̶̶ DE Exercises 3. Possible answer: Given: △PQR is an isosc. △ with base ̶̶ PR. Prove: △PQR cannot have a base ∠ that is a rt. ∠. Proof: Assume that △PQR has a base ∠ that is a rt. ∠. Let ∠P be the rt ∠. By the Isosc. △ Thm., ∠R ≅ ∠P, so ∠R is also a rt. ∠. By the def. of rt. ∠, m∠P = 90° S98 S98... |
∠K, ∠J 57. ∠J, ∠L, ∠K 59a. 0.4 h < t < 2 h b. no 61. 1 < n < 6 63. n > 0 65. n > 0.5 67a. def. of ≅ segs. b. Isosc. △ Thm. c. def. of ≅ d. m∠1 + m∠3 e. subst. f. m∠S g. Trans. Prop. of Inequal. 71. H 73. 3 __ 10, or 30% 77. -2x + y = 6 79. BC = 10, EF = 11, and m∠ABC = 102°, so △ABC ≅ △EFD by SAS. 81. (0, 0) ̶̶ BC. ... |
23. < 25. m∠RSV < m∠TSV 27. m∠YMX > m∠ZMX 31. D 33. Group A is closer to the camp. 37. 14; none 39. m∠2 = m∠6 = 36°; m ǁ n by the Conv. of the Corr. Post. 41. 2.5 43. 85° 5-7 Check It Out! 1a. x = 4 √ 5 1b. x = 16 2. 29 ft 1 in. 3a. 2 √ 41 ; no; 2 √ 41 is not a whole number. 3b. 10; yes; the 3 side lengths are... |
no b. yes. c. no d. no 57. x = -5 61. - 1 __ 3 < x < 2 5-8 Check It Out! 1a. x = 20 1b. x = 8 √ 2 2. 43 cm 3a. x = 9 √ 3 ; y = 27 3b. x = 5 √ 3 ; y = 10 3c. x = 12; y = 12 √ 3 3d. 34.6 cm Exercises 1. x = 14 √ 2 3. x = 9 5. x = 3; y = 3 √ 3 7. x = 21; y = 14 √ 3 9. x = 15 √ 2 _____ 11. x = 18 2 13. x =... |
̶ CP ⊥ AP ≅ CB, and bisector of ∠ABC by the Conv. of the ∠ Bisector Thm. 13. 42.2 14. 46 15. 57.6 16. 46 17. 18 18. 37° 19. (4, 3) 20. (-6, -3.5) 21. 16.4 22. 8.2 23. 5.8 24. 17.4 25. (-6, 0) 26. (1, 2) 27. (7, 4) 28. (3, 0) 29. (3, 4) 30. 35.1 31. 64.8 32. 32.4 33. 42° 34. 138° 35. 42° 36. V (-1, -1) ; W (6, 1) ; ̶̶ ̶... |
̶̶ AC, obtuse 52. not a triangle 53. triangle; right 54. triangle; acute 55. x = 26 √ 2 56. x = 6 √ 2 57. x = 32 58. x = 24; y = 24 √ 3 59. x = 6 √ 3 ; y = 12 60. x = 14 √ 3 _____ ; 3 y = 28 √ 3 _____ 3 61. 21 ft 3 in. 62. 15 ft 7 in. Chapter 6 6-1 Check It Out! 1a. not a polygon 1b. polygon, nonagon 1c. n... |
.) 3. ∠HJN ≅ ∠MJK (Vert. Thm.) 4. ∠N ≅ ∠K (≅ Supps. Thm.) Exercises 3. 36 5. 18 7. 70° 9. 24.5 11. 51° 13. (- 6, - 1) 15. 82.9 17. 82.9 19. 130° 21. 10 23. 28 25. (-1, 3) 27. PQ = QR = RS = SP = 21 29. PQ = RS = 17.5; QR = SP = 24.5 31a. ∠3 ≅ ∠1 (Corr. Post.); ∠6 ≅ ∠1 ( → opp. ≅); ∠8 ≅ ∠1 ( → opp. ≅) b. ∠2 is... |
̶ RS. So 1 which means that pair of opp. sides of PQRS are || and ≅. By Thm. 6-3-1, PQRS is a . 2a. Yes; possible answer: the diag. of the quad. forms 2 . 2 of 1 △ are ≅ to 2 of the other, so the third pair of are ≅ by the Third Thm. So both pairs of opp. of the quad. are ≅. By Thm. 6-3-3, the quad. is a .... |
6-3-1, RSTU is a . 9. BC = GH = 16.6, so CG = HB = 28, so both pairs of opp. sides of BCGH are ≅, BCGH is a by Thm. 6-3-2. 11. yes 13. no 15. Possible answer: slope of and ̶̶ RS = 5 __ 3 ; ̶̶ RS have the same slope, so ̶̶ ̶̶̶ GH. BC ≅ ̶̶ HB. Since ̶̶ PQ = slope of ̶̶ PQ ̶̶ PQ ̶̶ CG ≅ Selected Answers S99 S99 ̶̶ RS ... |
bisects∠QPS. (Rhombus → each diag. bisects opp. ) 3. ∠QPR ≅ ∠SPR (Def. of ∠ bisector) ̶̶ 4. PQ ≅ ̶̶ 5. PR ≅ 6. △QPR ≅ △SPR (SAS) 7. ̶̶ PS (Def. of rhombus) ̶̶ PR (Reflex. Prop. of ≅) ̶̶ RS (CPCTC) ̶̶ RQ ≅ ̶̶ TY ̶̶ RX ≅ ̶̶ XY (Reflex. Prop. of ≅) Exercises 1. rhombus; rectangle; square 3. 160 ft 5. 380 ft 7. 122° 9. P... |
� ∠RHX (Def. of ∠ bisector) ̶̶ 5. HX ≅ 6. △MHX ≅ △RHX (SAS) 7. ∠HMX ≅ ∠HRX (CPCTC) ̶̶ HX (Reflex. Prop. of ≅) 19. m∠1 = 54°; m∠2 = 36°; m∠3 = 54°; m∠4 = 108°; m∠5 = 72° 21. m∠1 = 126°; m∠2 = 27°; m∠3 = 27°; m∠4 = 126°; m∠5 = 27° 23. m∠1 = 64°; m∠2 = 64°; m∠3 = 26°; m∠4 = 90°; m∠5 = 64° 25. S 27. S 29. A 31. S 35a. Rect... |
, rect., rhombus, square 17. B 19. 21. (2, 6) 23. (-2, -2) 25. x = 3 ̶̶ 27. rhombus 29a. slope of AB = ̶̶ ̶̶ CD = - 1 __ 3 ; slope of AD = slope of ̶̶ ̶̶ CB = -3 b. Slope of AC = slope of ̶̶ BD = 1; the slopes are -1; slope of negative reciprocals of each other, so since it is a and its diags. are ⊥ (Thm. 6-5-4.) 3... |
G = 126° 35. 13 37. m∠PAQ = 108°; m∠OAQ = 130°; m∠OBP = 22° 41. kite 43. isosc. trap. 47. B 49. 18 51. AD = 7.08 in.; AB = CD = 5.08 in.; BC = 10.16 in. 53. 2x < x + 6; x < 6 55. rect., rhombus, square ̶̶̶ GH ≅ ̶̶ EF ≅ SGR 1. vertex of a polygon 2. convex 3. rhombus 4. base of a trapezoid 5. not a polygon 6. polygon; △... |
= 76°, m∠Q = 104°, and m∠R + m∠Q = 180°, so ∠R is supp. to ̶̶ RS ≅ ̶̶ RS ǁ ̶̶ BD ǁ ̶̶ BH ǁ ̶̶ FH and ̶̶ SU. ̶̶ SU ; slope of 5 ̶̶ DF ; by def., ∠Q. Since ∠R and ∠Q are a pair of same-side int. , and they are ̶̶ QT. So 1 pair of opp. supp., sides of QRST are || and ≅. By Thm. 6-3-1, QRST is a . 31. Yes; the diags. of... |
RSTU are ≅ ⊥ bisectors of each other. ̶̶ 49. EG = FH = 3 √ 2, so EG ≅ Slope of = 1, so ̶̶ EG and of the mdpt. of ̶̶ ( 7 __ 2, - 1 __ 2 ), so FH bisect each other. So the diags. of EFGH are ≅ ⊥ bisectors of each other. 50. Not valid; by Thm. 6-5-2, if the diags. of a are ≅, then the is a rect. By Thm. 6-5-4, if t... |
9 17. 3 __ 1 19. 3 __ 2 21. 72°; 108°; 72°; 108° 23. x = 20 25. m = 2 or m = -4 27. x = ± 8 29. 3 : 5 31. 5b 33. b __ 7 35. 1 __ 3 37. -3 ______ ______ = x in. 39a. 1.25 in. 9600 in. 15 in. or 66 ft 8 in. 41. 4 __ 9 45. H 47. First, cross multiply: 36x = 15 (72), or 36x = 1080. Then divide both sides by ____ 36: 36x _... |
the sides are not the same; 12 ___ 3.5 = 24 __ 7 ; 10 ___ 2.5 = 4; 6 ___ 1.5 = 4. 33a. rect. ABCD ∼ rect. BCFE b. ℓ __ 1 = 1 ____ ℓ - 1 c. ℓ = 1 + √ 5 ______ d. ℓ ≈ 1.6 35. 90° 2 37. 70° 39. 4 __ x = KL ___ TU 7-3 Check It Out! 1. By the △ Sum Thm., m∠C = 47°, so∠C ≅ ∠F. ∠B ≅ ∠E by the Rt. ∠ ≅ Thm. Therefore △ABC ∼ ... |
≅ ∠ACB. ∠A ≅ ∠A by the Reflex. Prop. of ≅. Therefore △AED ∼ △ACB by AA ∼. AB = 10 ̶̶̶ 7. 1. MN ǁ = 1 __ 2, so △DEF ∼ △JKL by ̶̶ KL (Given) 2. ∠JMN ≅ ∠JKL, ∠JNM ≅ ∠JLK (Corr. Post.) 3. △JMN ∼ △JKL (AA ∼ Step 2) 9. SAS or SSS ∼ Thm. 11. It is given that ∠GLH ≅ ∠K. ∠G ≅ ∠G by the Reflex. Prop. of ≅. Therefore △HLG ∼ △J... |
NL = 68°. So ∠JKL ≅ ∠MNL. Therefore △JKL ∼ △MLN by AA ∼. 7-5 Check It Out! 1. 15 ft 7 in. 2. 900 m, or 0.9 km 3. Check students’ work. The drawing should be 3.7 in. by 3 in. 4. P = 14 mm; A = 10 2 __ 3 mm 2 Exercises 1. indirect measurement 3. 12 ft 5. 60 ft 11. 27 cm 2 13. ≈ 61 km 19. 864 m 2 21. 175 ft 23. 375 ft 25.... |
work. The image of △MNP has vertices M′ (-6, 3), N′ (6, 6), and P′ (-3, -3). MP = √ 5, MN = √ 17, and PN = 3 √ 2. M′P′ = 3 √ 5, M′N′ = 3 √ 17, and P′N′ = ____ ____ = M′N′ 9 √ 2. M′P′ = 3. So MN MP △M′N′P′ ∼ △MNP by SSS ∼. = P′N′ ___ PN = JL __ JN = S′T ′ ___ ST = R′T ′ ___ RT = 3 __ 2. So △RST ∼ Exercises ... |
�. So △DEF ∼ △DGH by SAS ∼. 15. The image of △JKL has vertices J′ (-6, 0), K′ (-3, -3), and L′ (-9, -6). JK = √ 2, JL = √ 5, and LK = √ 5. J′K′ = 3 √ 2, J′L′ = 3 √ 5, and L′K′ = 3 √ 5. J′K′ ___ JK So △JKL ∼ △J′K′L′ by SSS ∼. 17. It is not a dilation; because it changes the shape of the figure. 21. A 23. A 2... |
N JM ̶̶̶ ̶̶ Since JK ___ = JL __ MN by the KL ǁ, JN JM Conv. of the △ Proportionality Thm. 24. EC ___ = 3 __ 7. Since EC ___ = ED ___ = EA EB ̶̶ ̶̶ ED ___ CD by the Conv. of the △ AB ǁ, EB Proportionality Thm. 25. SU = 4; SV = 6 26. 18 27. 4x + 8 28. 25 ft 4 in. 29. 3 ft 30. By the Dist. Formula, RS = 2 √ 2, RU = 4 √... |
. y = 3 or y = -5 16. yes; 5 __ 3 ; JKLM ∼ PQRS 17. yes; 2; △TUV ∼ △WXY 18. 1. JL = 1 __ 3 JN, JK = 1 __ 3 JM (Given) Chapter 8 8-1 Check It Out! 1. △LJK ∼ △JMK ∼ △LMJ 2a. 4 2b. 10 √ 3 2c. 6 √ 2 3. 27; 3 √ 10 ; 9 √ 10 4. 148 ft Exercises 1. 8 is the geometric mean of 2 and 32. 3. △BED ∼ △ECD ∼ △BCE 5. 10 7. 2 9... |
1a. 24 __ 25 = 0.96 1b. 24 __ 7 ≈ 3.43 1c. 24 __ 25 = 0.96 2. s _ s = 1 3a. 0.19 3b. 0.88 3c. 0.87 4a. 21.87 m 4b. 7.06 in. 4c. 36.93 ft 4d. 6.17 cm 5. 14.34 ft 2 3. 4 __ 5 = 0.8 5. 4 __ 5 = Exercises 1. LK ___ JL 0.8 7. 4 __ 3 ≈ 1.33 9. 1 __ 2 11. √ 2 ___ 2 13. 0.39 15. 0.03 17. 0.16 19. 9.65 m 21. 7 ft 6 in. 23. 1... |
8.75; m∠D = 32° 4. RS = ST = 7; RT ≈ 9.90; m∠S = 90°; m∠R = m∠T = 45° 5. 21° Exercises 1. ∠1 3. ∠1 5. ∠2 7. 65° 9. 34° 11. 38° 13. RP ≈ 9.42; m∠P ≈ 19°; m∠R ≈ 71° 15. YZ ≈ 13.96; m∠Y ≈ 38°; m∠Z ≈ 52° 17. RS = 5; ST = 6; RT ≈ 7.81; m∠S = 90°; m∠R ≈ 50°; m∠T ≈ 40° 19. AB = 2; BC = 4; AC ≈ 4.47; m∠B = 90°; m∠A ≈ 27°; m∠C... |
. F 81. -1 83. 0.89 85. 2.05 8-4 Check It Out! 1a. angle of depression 1b. angle of elevation 2. 6314 ft 3. 1717 ft 4. 32,300 ft Exercises 1. elevation 3. angle of elevation 5. angle of elevation 7. 18 ft 9. 64.6 m 11. angle of elevation 13. angle of depression 15. 1962 ft 17. T 19. F 21. ∠1 and ∠3 25a. 424 ft b. 276 f... |
1 8-6 Check It Out! 1a. 〈-3, -4〉 1b. 〈7, 1〉 2. 3.2 3. 23° PQ = 4a. XY ǁ MN 5. 4.4 mi/h; 58°, or N 32° E RS 4b. PQ ǁ RS ; XY UV RS = RS = CD = LM 29. Exercises 1. equal 3. magnitude 5. 〈8, -8〉 7. 4.1 9. 5.8 11. 11° 13. EF 15. 17. 4.6 mi; 20°, or N 70° ... |
� 2, and the direction of the resultant is tan -1 ( 10 __ 10 ) = 45°. 53. 〈3.5, 1〉; 3.6; 16° 55. 〈4, 4〉; 5.7; 45° 57a. 〈1, 3〉; 〈2, 6〉 b. √ 10 ; 2 √ 10 ; the magnitude v is twice the magnitude of of 2 c. 72°; 72°; the direction of 2 same as the direction of d. Multiply each component by k. e. - v = -1〈x, y〉 = 〈-... |
. 20.1 27. 56° 28. 〈-7, 2〉 29. 〈1, -6〉 30. 〈-2, -5〉 31. 32. 5.8 33. 2 34. 5.7 35. 51° 16° 36. 641.6 mi/h; 32°, or N 58° E Chapter 9 formula for y and substitute the expression into the perimeter formula. Graph, and find the minimum value. 63. -2 ≤ y ≤ 2 65. P = 2x + 8; A = 7x __ 2 67. 〈6, 8〉 9-2 Check It Out! 1. A = (4... |
= 0.5 yd 2. A = 96 m 2 3. d 2 = 8y m 4. P = (4 + 4 √ 2 ) cm; A = 4 cm 2 Check It Out! 1. A = 1781.3 m 2 2. A ≈ 10.3 in 2 3. 23,296.5 gal 4. A ≈ 12 ft 2 Exercises 1. A ≈ 40.5 units 2 3. isosceles triangle; P = (6 + 6 √ 2 ) units; A = 9 units 2 5. rectangle; P = 28 units; A = 40 units 2 7. A = 20 units 2 9. A = 6 un... |
5) ft 2 15. h = 20 cm 17. A = 196 √ 3 in 2 19. A = (12 x 2 + 34x + 20) ft 21. A = 4.5 in 2 23. A = 30 √ 3 cm 2 25. A = 300 in 2 27. A = x 2 √ 3 ____ 2 b. A = 561.6 in 2 c. 734.4 in 2 31. 8; 50 33. 9; 24 35. h = 5 cm 37. 9 39. 100 41. A = 108 ft 2 43a. A = 1 __ 2 (a + b) 2 b. 1 __ 2 ab; 1 __ 2 ab; 1 __ 2 c 2 c. 1... |
= 25 units 2 3. A = 48 units 2 Exercises 1. The area is doubled. 3. The perimeter is tripled. The area is multiplied by 9. 5. The side length is multiplied by √ 2. 7. $147.00 9. The area is multiplied by 2 __ 3. 11. The circumference is multiplied by 3 __ 5. The area is multiplied by 9 __ 25. 13. The side length is ... |
. 4 m 10 55. By the Distance Formula, AB = 2 √ 5, AC = 2 √ 5, BC = 4, AD = 4 √ 5, AE = 4 √ 5, and DE = 8. AB ___ = 1 __ 2, so △ABC ∼ = BC ___ DE AD △ADE by SSS. 57. A ≈ 10.6 in 2 = AC ___ AE SGR 1. apothem 2. center of a circle 3. geometric probability 4. A = 81 in 2 5. P = 22 cm 6. h = 3 x 2 in. 7. h = 8 ft 8.... |
. 0.66 33. 35. Chapter 10 10-1 37 a. pentagonal prism b. 2 pentagons and 5 rectangles c. ̶̶ VY, ̶̶ TV, ̶̶ UV, Check It Out! 1a. cone; vertex: N; edges: none; base: ⊙M 1b. triangular prism; vertices: T, U, ̶̶ ̶̶̶ TU, TW, V, W, X, Y; edges: ̶̶ ̶̶̶ ̶̶̶ ̶̶ UX, XY ; bases: △TUV, WY, WX, △WXY 2a. triangular pyramid 2b. cylin... |
one 53. y = x 2 + 6 55. largest: ∠B; smallest: ∠C 57. largest: ∠I; smallest: ∠H 59. yes; 10:17 Selected Answers S105 S105 ������������������ 10-2 Check It Out! 1. 11. no 13. no 15. c. 17. 19. 21. 23. yes 25. no 27. 29c. 9 31. 35. B 39. 41a. b. 2. 3a. 3b. 4. no Exercises 1. perspective 3. 5. 7. 9. S106 S106 Selected An... |
2.8 units; M (2, 2, 2) 47. 25. d ≈ 10.3 units; M (5.5, 6.5, 8.5) 27. 6557 ft 29. 6 31. 12 33. V = n + 1; E = 2n; F = n + 1; (n + 1) - 2n + (n + 1) = 2 35. 6 √ 3 37. 6 √ 3 39. 41. 43. d ≈ 6.8 units; M (3.5, 4, 6.5) 49. d ≈ 4.6 units; M (4, 1.5, 7) 51. Possible answer: z = 9 53. Possible answer: 1.8 in. 55. AB = 11,... |
5 m 27. S ≈ 121.5 units 2 29. 836.58 31. 1057.86 33. Multiply the radius and height by 1 __ 2. 35. < 4.86 cm 2 37a. AB = 7 in.; BC = 4 √ 2 in. ≈ 5.7 in. b. 4.1 in. c. 97.6 in 2 39. F 41. h = 18 cm 43. 198 cm 2 45. 70 ≤ s ≤ 110 47. 77° 49. 10-5 Check It Out! 1. L = 90 ft 2 ; S ≈ 105.6 ft 2 2. L = 80π cm 2 ; S ≈ 144π c... |
2 = r _ d. The area of the ℓ = 2πr ___ 2πℓ ℓ 10-6 Check It Out! 1. V = 157.5 yd 3 2. 859,702 gal; 7,161,318 lb 3. V = 1088π in 3 ≈ 3418.1 in 3 4. The volume is multiplied by 8. 5. V ≈ 51.4 cm 3 Selected Answers S107 S107 ��������������������������������������������������������������������������������������������������... |
. The volume is multiplied by 27. 11. V = 2592 cm 3 13. V = 160 ft 3 15. V = 384 ft 3 17. V = 1107π m 3 ≈ 3477.7 m 3 19. V = 144π ft 3 ≈ 452.4 ft 3 21. The volume is multiplied by 216. 23. V = 150 ft 3 ____ 6 m 3 27. V = 240π cm 3 25. V = 25π 29. 1350 m 3 31. 166.3 cm 3 33. C = 10π √ 3 cm 35. V = 1280 in 3 37. V = 17... |
85 in.; 19.63 in 2 ; 8.18 in 3 33. Possible answer: 14,293 in 3 35. about 1408 times as great 37. The surface area of Saturn is greater. 39. The cross section of the hemisphere is a circle with radius √ π ( r 2 - x 2 ). The cross section of the cylinder with the cone removed has an outer radius of r and an inner ra... |
; S = 224π m 2 25. L = 150π in 2 ; S = 250π in 2 26. S = 800 ft 2 27. S = 448π m 2 28. V = 1080 ft 3 29. V ≈ 1651.7 cm 3 30. V = 900π in 3 31. V = 45π m 3 32. V = 112 m 3 33. V ≈ 10.4 cm 3 34. V = 120π cm 3 35. V = 48π ft 3 36. V = 512π ft 3 ____ 3 37. V ≈ 1533.3 cm 3 38. V = 500π 39. S = 144π in 2 40. d = 16 ft 41. S... |
C: 2; radius of ⊙D: 4; pt. of tangency: (-4, 0) ; eqn. of tangent line: x = -4 15. 413 km 17. 7 ̶̶ AC 27. 45° 19. N 21. A 23. 31. 8 33. 22 35a. rect.; ∠BCD and ∠EDC are rt. because a line tangent to a ⊙ is ⊥ to a radius. It is given that ∠DEB is a rt. ∠. ∠CBE must also be a rt. ∠ because the sum of the of a quad. i... |
. of diam.) ̶̶ JK is a diam. of ⊙A. ̶̶̶ GH. 45. Solution A 47a. 13.5 in.; 6.5 in. b. 11.8 in. c. 23.7 in. 49. F 51. 48.2° 53a. 90°; 60°; 45° b. 3 __ 4 π; 3 __ 2 π 55. b 3 __ 16 57. 31 59. 28 61. 9 11-3 Check It Out! 1a. π __ 4 m 2 ; 0.79 m 2 1b. 25.6π in 2 ; 80.42 in 2 2. 203,575 ft 2 3. 4.57 m 2 4a. 4 __ 3 π m; 4.19 m... |
. ∠; therefore △FBC is a rt. △. 33. 72°; 99°; 108°; 81° 35a. AB 2 + AC 2 = BC 2, so by the Conv. of the Pyth. Thm., △ABC is a rt. △ with rt. ∠A. Since ∠A is an inscribed rt. ∠, it intercepts a semicircle. This means ̶̶ BC is a diam. b. 120° 39. D that 41. C 45. 133° 49. 13 __ 7 51. 5 __ 2 53. 3 m 2 11-5 Check It Out! 1... |
m∠ABC = 90 - (90 - 1 __ 2 m∠AXB). Simplifying gives m∠ABC = 1 __ 2 m∠AXB. m∠AXB = m ⁀ AB because ∠AXB is a central ∠. Thus m∠ABC = 1 __ 2 ⁀ AB. 47. 95° 49. yes 51. no 53. 96π cm 3 ≈ 301.6 cm 3 55. 37° 57. 53° 11-6 Check It Out! 1. 3.75; AB = 11; CD = 11.75 2. 3 2 __ 3 in. 3. z = 14; JG = 27; LG = 39 4. 7 2 __ 7 Exerci... |
2.5) 2 = 3 13. (x - 1) 2 + (y + 2) 2 = 45 15. 17. 19. (x - 1) 2 + (y + 2) 2 = 4 21a. 80 ft b. x 2 + y 2 = 1600 23. T 25. T 29a. E (-3, -1) ; G (-6, 2) b. 6 c. (x + 3) 2 + (y - 2) 2 = 9 31. (0, -15) ; 5 33. A = 9π; C = 6π 35. A = 25π; C = 10π 37. (-200, -100) 39. (x - 1) 2 + (y + 2) 2 = 16 43. H 45a. (x - 2) 2 + (y + 4... |
̶̶ UV ; tangent: ̶̶ QS, ̶̶ PS ; secant: UV ; 1. segment of a circle 2. central angle 3. major arc 4. concentric circles 5. chords: ̶̶ PQ, ℓ; radii: ̶̶ ̶̶̶ ̶̶ QS 6. chords: MN ; KH, diam.: ̶̶ ̶̶ ̶̶ ̶̶ tangent: KL ; radii: JM, JK, JH, JN ; ̶̶ ̶̶̶ KH 7. 25 secant: MN ; diams.: MN, 8. 12 9. 7 10. 1.8 11. 8... |
. 11. (8.7, 5) 13. yes 15. no around the center of rotation by the same ∠, pts. that are farther from the center of rotation move a greater distance than pts. that are closer to the center of rotation. 39. A′ (-2, 3), B′ (-3, 0), C′ (0, -3), D′ (3, 0), E′ (2, 3), 43. H 45. 160° 47. Use the fact that the rotation of a s... |
�ABC ≅ △A′B′C′ by SSS. 51. (4, 5) 53. x = 15, y = 5 55. M′ (-2, 0), N′ (-3, -2), P′ (0, -4) 57. M′ (0, -2), N′ (2, -3), P′ (4, 0) PQ 12-3 Check It Out! 1a. no 1b. yes 2. 17. 19. 23. 25. 2. a translation in direction ⊥ to n and p, by distance of 6 in. 3. Exercises 1. Draw a figure and translate it along a vector. T... |
. 5 cm 67. P′ (6, -5) 69. P′ (0,-4) 3b. line symmetry; 51.4°; order: 7 4a. both 4b. neither 12-6 11c. 13. Exercises 1. The line of symmetry is the ⊥ bisector of the base. 3. yes; 2 lines of symmetry 5. no 7. no 9. 72°; order: 5 11. both 13. yes; 1 line of symmetry 15. no 17. yes; 72°; order: 5 19. 90; order: 4 21. neit... |
MNPQR 27. 13. no 14. yes 15. no 16. no 17. 29. 31. B 35. -4.5 × 10 -12 37. k = -2; A′ (4, -4), B′ (-2, -6) 39. k = 1 and k = -1 47. H 49. no 51. y = -x + 6 53. P = 24 units; A = -28 uni ts 2 55. yes SGR 1. reg. tessellation 2. frieze pattern 3. isometry 4. composition of transformations 5. yes 6. no 7. no 8. yes 9. 18... |
o no comparten puntos internos. adjacent arcs (p. 757) Two arcs of the same circle that intersect at exactly one point. arcos adyacentes Dos arcos del mismo círculo que se cruzan en un punto exacto. alternate exterior angles (p. 147) For two lines intersected by a transversal, a pair of angles that lie on opposite side... |
) A segment from the vertex to the plane of the base that is perpendicular to the plane of the base. SPANISH altura de un prisma Segmento con sus extremos en los planos de las bases que es perpendicular a los planos de las bases. altura de una pirámide Segmento que se extiende desde el vértice hasta el plano de la base... |
superior. angle of rotation (p. 840) An angle formed by a rotating ray, called the terminal side, and a stationary reference ray, called the initial side. ángulo de rotación Ángulo formado por un rayo rotativo, denominado lado terminal, y un rayo de referencia estático, denominado lado inicial. angle of rotational sym... |
squares of a given size that will exactly cover the interior of a plane figure. área Cantidad de cuadrados unitarios de un determinado tamaño no superpuestos que cubren exactamente el interior de una figura plana. arrow notation (p. 50) A symbol used to describe a transformation. notación de flecha Símbolo utilizado p... |
�ngulos que tienen como lado la base del triángulo. base of a cone (p. 654) The circular face of the cone. base de un cono Cara circular del cono. base of a cylinder (p. 654) One of the two circular faces of the cylinder. base de un cilindro Una de las dos caras circulares del cilindro. base of a geometric figure (p. 4... |
a en dirección norte y cuyo lado terminal se determina por una rotación en el sentido de las agujas del reloj. between (p. 14) Given three points A, B, and C, B is between A and C if and only if all three of the points lie on the same line, and AB + BC = AC. entre Dados tres puntos A, B y C, B está entre A y C si y sól... |
the lines that connect each point of the image with the corresponding point of the preimage. centro de dilatación Intersección de las líneas que conectan cada punto de la imagen con el punto correspondiente de la imagen original. Glossary/Glosario S119 S119 ������������������������������������ ENGLISH center of rotati... |
rculo dividido en sectores no superpuestos. circumcenter of a triangle (p. 307) The point of concurrency of the three perpendicular bisectors of a triangle. circuncentro de un triángulo Punto donde se cortan las tres mediatrices de un triángulo. circumference (p. 37) The distance around the circle. circunferencia Dista... |
E y se expresan In the experiment of rolling a number cube, the complement of rolling a 3 is rolling a 1, 2, 4, 5, or 6. complementary angles (p. 29) Two angles whose measures have a sum of 90°. ángulos complementarios Dos ángulos cuyas medidas suman 90°. component form (p. 559) The form of a vector that lists the ver... |
era del polígono. concentric circles (p. 747) Coplanar circles with the same center. círculos concéntricos Círculos coplanares que comparten el mismo centro. conclusion (p. 81) The part of a conditional statement following the word then. conclusión Parte de un enunciado condicional que sigue a la palabra entonces. If x... |
ent angles (p. 22) Angles that have the same measure. SPANISH ángulos congruentes Ángulos que tienen la misma medida. EXAMPLES ∠ABC ≅ ∠DEF congruent arcs (p. 757) Two arcs that are in the same or congruent circles and have the same measure. arcos congruentes Dos arcos que se encuentran en el mismo círculo o en círculos... |
computación de geometría o plegando papeles. contraction (p. 873) See reduction. contracción Ver reducción. contrapositive (p. 83) The statement formed by both exchanging and negating the hypothesis and conclusion of a conditional statement. contrapuesto Enunciado que se forma al intercambiar y negar la hipótesis y la... |
-axis. plano cartesiano Plano dividido en cuatro regiones por una línea horizontal denominada eje x y una línea vertical denominada eje y. The coordinate of point A is 3. The coordinates of point B are (1, 4). coordinate proof (p. 267) A style of proof that uses coordinate geometry and algebra. prueba de coordenadas Ti... |
corresponding sides. S124 S124 Glossary/Glosario ����������������������������������������������������������������� ENGLISH SPANISH EXAMPLES cosecant (p. 532) In a right triangle, the cosecant of angle A is the ratio of the length of the hypotenuse to the length of the side opposite A. It is the reciprocal of the sine ... |
dadero. degree (p. 20) A unit of angle measure; one degree is 1 ___ 360 of a circle. grado Unidad de medida de los ángulos; un grado es 1 ___ círculo. 360 de un denominator (p. 451) The bottom number of a fraction, which tells how many equal parts are in the whole. denominador El número inferior de una fracción, que in... |
directo Proceso de razonamiento que comienza con una hipótesis verdadera y elabora un argumento lógico para demostrar que una conclusión es verdadera. S126 S126 Glossary/Glosario ������������������������ ENGLISH SPANISH EXAMPLES direct variation (p. 501) A linear relationship between two variables, x and y, that can b... |
p. 95) A curve or segment that joins two vertices of the graph. arista de una gráfica Curva o segmento que une dos vértices de la gráfica. edge of a three-dimensional figure (p. 654) A segment that is the intersection of two faces of the figure. arista de una figura tridimensional Segmento que constituye la intersecció... |
entes. triángulo equilátero Triángulo con tres lados congruentes. Euclidean geometry (p. 726) The system of geometry described by Euclid. In particular, the system of Euclidean geometry satisfies the Parallel Postulate, which states that there is exactly one line through a given point parallel to a given line. geometrí... |
an angle (p. 20) The set of all points outside an angle. exterior de un ángulo Conjunto de todos los puntos que se encuentran fuera de un ángulo. exterior of a polygon (p. 225) The set of all points outside a polygon. exterior de un polígono Conjunto de todos los puntos que se encuentran fuera de un polígono. exterior... |
that each term is the sum of the two previous terms. sucesión de Fibonacci Sucesión infinita de números que comienza con 1, 1, … de forma tal que cada término es la suma de los dos términos anteriores. 1, 1, 2, 3, 5, 8, 13, 21, … Glossary/Glosario S129 S129 ������������������������������� ENGLISH flip (p. 50) See refl... |
comes between two given nonconsecutive terms of the sequence. media geométrica Dados los números positivos a y b, el número positivo x tal que a __ x = x __ sucesión geométrica, un término que está entre dos términos no consecutivos dados de la sucesión.. En una b geometric probability (p. 630) A form of theoretical p... |
segmento en dos partes de forma tal que la razón entre la longitud de todo el segmento y la de la parte más larga sea igual a la razón entre la longitud de la parte más larga y la de la parte más corta, entonces dicha razón se denomina razón áurea. La razón áurea es igual a 1 + √ 5 ______ 2 ≈ 1.618. golden rectangle... |
AB = AB _ BC Create segment such that AC _ AB ≈ 1.62 and AB _ BC ≈ 1.62 A � D � E � F ����� B C Glossary/Glosario S131 S131 ���������������������������������������ABC ENGLISH heptagon (p. 382) A seven-sided polygon. SPANISH heptágono Polígono de siete lados. EXAMPLES hexagon (p. 382) A six-sided polygon. hexágono Polí... |
donde se encuentran las tres bisectrices de los ángulos de un triángulo. P is the incenter. included angle (p. 242) The angle formed by two adjacent sides of a polygon. ángulo incluido Ángulo formado por dos lados adyacentes de un polígono. ∠B is the included ̶̶ angle between AB and ̶̶ BC. S132 S132 Glossary/Glosario ... |
a vector (p. 559) The starting point of a vector. punto inicial de un vector Punto donde comienza un vector. initial side (p. 570) The ray that lies on the positive x-axis when an angle is drawn in standard position. lado inicial Rayo que se encuentra sobre el eje x positivo cuando se traza un ángulo en posición están... |
o Ángulo formado por dos lados de un polígono con un vértice común. interior of a circle (p. 746) The set of all points inside a circle. interior de un círculo Conjunto de todos los puntos que se encuentran dentro de un círculo. ⁀ DF is the intercepted arc. ∠1 is an interior angle. �������� interior of an angle (p. 20)... |
ratio is known. SPANISH seno inverso Medida de un ángulo cuya razón seno es conocida. EXAMPLES If sin A = x, then sin -1 x = m∠A. inverse tangent (p. 534) The measure of an angle whose tangent ratio is known. tangente inversa Medida de un ángulo cuya razón tangente es conocida. If tan A = x, then tan -1 x = m∠A. irrat... |
exactly two pairs of congruent consecutive sides. cometa o papalote Cuadrilátero con exactamente dos pares de lados congruentes consecutivos. Koch snowflake (p. 882) A fractal formed from a triangle by replacing the middle third of each segment with two segments that form a 60° angle. copo de nieve de Koch Fractal for... |
isosceles triangle. cateto de un triángulo isósceles Uno de los dos lados congruentes del triángulo isósceles. length (p. 13) The distance between the two endpoints of a segment. longitud Distancia entre los dos extremos de un segmento. line (p. 6) An undefined term in geometry, a line is a straight path that has no t... |
cumple con una condición determinada. logically equivalent statements (p. 83) Statements that have the same truth value. enunciados lógicamente equivalentes Enunciados que tienen el mismo valor de verdad. M magnitude (p. 560) The length of a vector, written ⎜ AB ⎟ or ⎜ v ⎟. magnitud Longitud de un vector, que s... |
measured in degrees. A degree is 1 ___ 360 of a complete circle. medida de un ángulo Los ángulos se miden en grados. Un grado es 1 ___ 360 de un círculo completo. measure of a major arc (p. 756) The difference of 360° and the measure of the associated minor arc. medida de un arco mayor Diferencia entre 360° y la medid... |
points are on or in the interior of a central angle. arco menor Arco de un círculo cuyos puntos están sobre un ángulo central o en su interior. N natural number (p. 80) A counting number. ⁀ AC is a minor arc of the circle. número natural Número de conteo. 1, 2, 3, 4, 5, 6, … negation (p. 82) The negation of statement ... |
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