Split (1)

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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 ̶̶ BC ≅ ̶̶ AC ≅ Check It Out! 1. It is given that ̶̶ ̶̶ DA. By the Reflex. ≅ CD and ̶̶ Prop. of ≅, AC. So △ABC ≅ △CDA by SSS. 2. It is given that ̶̶ BD and ∠ABC ≅ ∠DBC. By the ≅ ̶̶ BA ̶̶ BC ≅ ̶̶ BC. So ̶̶ DB ≅ ̶̶ DB by ̶̶ DC by def. of ≅. Reflex. Prop. of ≅, △ABC ≅ △DBC by SAS. 3. DA = ̶̶ DA = DC = 13, so m∠ADB = m∠CDB = 32°, so ∠ADB ≅ ∠CDB by def. of ≅. the Reflex. Prop. of ≅. Therefore △ADB ≅ △CDB by SAS. ̶̶ 4. 1. QS (Given) 2. 3. ∠RQP ≅ ∠SQP (Def. of bisector) ̶̶ ̶̶ 4. QP (Reflex. Prop. of ≅) QP ≅ 5. △RQP ≅ △SQP (SAS Steps 1, 3, 4) ̶̶ QR ≅  QP bisects ∠RQS. (Given) ̶̶ GL. ̶̶ KJ ≅ ̶̶ NP ≅ ̶̶ G
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. of ≅. Therefore △GHJ ≅ △IHJ by SSS. 7a. Given b. ∠JKL ≅ ∠MLK c. Reflex. Prop. of ≅ d. SAS Steps 1, 2, 3 9. It is given ̶̶ ̶̶ GJ GJ ≅ that by the Reflex. Prop. of ≅. So △GJK ≅ △GJL by SSS. 11. When y = 3, NQ = NM = 3, and QP = MP = ̶̶̶ 4. So by the def. of ≅, NM ̶̶̶ MP. m∠M = m∠Q = 90°, and so ∠M ≅ ∠Q by the def. of ≅. Thus △MNP ≅ △QNP by SAS. ̶̶ ̶̶ 13a. Given b. AB DB ≅ ⊥ f. 15. SAS 17. neither 19. QS = TV = √  5. SR = VU = 4. QR = TU = √  13. The  are ≅ by SSS. 21a. Given b. Def. of ≅ c. m∠WVY = m∠ZYV d. Def. of ≅ e. Given ̶̶ f. VY g. SAS Steps 6, 5, 7 25. Measure the lengths of the logs. If the lengths of the logs in 1
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 ̶̶ FH by the def ̶̶ FJ by the Reflex. Prop. ̶̶ VY ≅ ̶̶ FK ≅ ̶̶ FJ = 43. 34° �� 4-5 ̶̶ LN ≅ Check It Out! 1. Yes; the △ is uniquely determined by AAS. 2. By the Alt. Int.  Thm., ∠KLN ̶̶ ≅ ∠MNL. LN by the Reflex. Prop. of ≅. No other congruence relationships can be determined, so ASA cannot be applied. 3. Given: ≅ ∠M. Prove: △JKL ≅ △JML ̶̶ JL bisects ∠KLM, and ∠K ̶̶ DF, ̶̶ BC ≅ ̶̶ ≅ EF, and ∠C and ∠F are rt. . ∠C ≅ ∠F by the Rt. ∠ ≅ Thm. Thus △ABC ≅ △DEF by SAS. 27. J 29. G 31. Yes; the sum of the ∠ measures in each △ must be 180°, which makes it possible to solve for x and y. The value of x is 15, and the value of y is 12. Each △ has  measuring 82°, 68°, and ̶̶ VU by the Reflex. Prop. 30°. of ≅. So △VSU ≅ △VTU by ASA or AAS. 35. 2; -6 37. 1; 5 39. 36.9° ̶̶ VU ≅ 4-6 Check It Out! 1. 41 ft 2. ̶
̶ 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 ≅. 7. Yes; it is given that ∠D and ∠B ̶̶ ̶̶ BC. △ABC and are rt.  and AD ≅ ̶̶ ̶̶ CA by AC ≅ △CDA are rt.  by def. the Reflex. Prop. of ≅. So △ABC ≅ △CDA by HL. 9. ̶̶ BE ̶̶ AE ≅ ̶̶ CE, and 11. No; you need to know that ∠MKJ ≅ ∠MKL. 13a. ∠A ≅ ∠D b. Given c. ∠C ≅ ∠F d. AAS 15. Yes; E is a mdpt. So by def., ̶̶ DE. ∠A and ∠D ≅ are ≅ by the Rt. ∠ ≅ Thm. By def. △ABE and △DCE are rt. . So △ABE ≅ △DCE by HL. 17. △FEG ≅ △QSR; rotation 19a. No; there is not enough information given to use any of the congruence theorems. b. HL 21. It is given that △ABC and △DEF are rt.. ̶̶ AC 3. 1. J is the mdpt. of ̶̶̶ KM and ̶̶ NJ ≅ ̶̶ LJ (Def. of ̶̶ NL. (Given) ̶̶ ̶̶ 2. MJ, KJ ≅ mdpt.) 3. ∠KJL ≅ ∠MJN (Vert.  Thm.)
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. △RXS ≅ △RXT f. CPCTC 5. EF = JK = 2 and EG = FG = JL = KL = √  10. So △EFG ≅ △JKL by SSS. ∠EFG ≅ ∠JKL by CPCTC. 7. 420 ft ̶̶̶ 9. 1. WX ≅ ̶̶ 2. ZX ≅ 3. △WXZ ≅ △YZX (SSS) 4. ∠W ≅ ∠Y (CPCTC) ̶̶ ̶̶̶ XY ≅ ZW (Given) ̶̶ ZX (Reflex. Prop. of ≅) ̶̶ YZ ≅ 11. 1. ̶̶̶ LM bisects ∠JLK. (Given) 3, 2, 4) ̶̶̶ ̶̶ 6. KM (CPCTC) JM ≅ 7. M is the mdpt. of of mdpt.) ̶̶ JK. (Def. 13. AB = DE = √  13, BC ≠ EF ± 5, and AC = DF = √  18 ≠ 3 √  2. So △ABC ≅ △DEF by SSS. ∠BAC ≅ ∠EDF by CPCTC. 15. 1. E is the mdpt. of ̶
̶ 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 of these  are ≅ by SAS. Therefore any 2 diags. are ≅ by CPCTC. 33. 94 35. reflection across the x-axis 37. Yes; it is given that ∠B ≅ ∠D and the Vert. ∠ Thm., ∠BCA ≅ ∠DCE. Therefore △ABC ≅ △EDC by ASA. ̶̶ DC. By ̶̶ BC ≅ 4-7 Check It Out! 1. Possible answer:, 6 + 0 ____ 2 2. △ABC is a rt. △ with height AB and base BC. The area of △ABC is 1 __ 2 (4) (6) = 12 square units. By the Mdpt. Formula, the coords. of D ̶̶ are ( 0 + 4 ) = (2, 3). With ____ 2 AB as the base of △ADB, the x-coord. of D gives the height of △ADB. The area of △ADB = 1 __ 2 bh = 1 __ 2 (6) (2) = 6 square units. Since 6 = 1 __ 2 (12), the area of △ADB is the area of △ABC. 3. Possible answer: 2. ∠JLM ≅ ∠KLM (Def. of ∠ bisector) ̶̶ ̶̶ 3. JL ≅ KL (Given) ̶̶̶ 4. LM ≅ 5. △JLM ≅ △KLM (SAS Steps ̶̶̶ LM (Reflex.
Prop. of ≅) Selected Answers S95 S95 �� 4. △ABC is a rt. △ with height 2j and base 2n. The area of △ABC = 1 __ 2 bh = 1 __ 2 (2n) (2j) = 2nj square units. By the Mdpt. Formula, the coords. of D are (n, j). The base of △ABD is 2j units and the height is n units. So the area of △ADB = 1 __ 2 bh = 1 __ 2 (2j) (n) = nj square units. Since nj = 1 __ 2 (2nj), the area of △ADB is 1 __ 2 the area of △ABC. 2 2 - y 1 + y 2 _____ 2 y 1 + y 2 _____ 2  _____ 2  2 = √ + ( ___ 2 ) ( ___ _____ - 2 = √  1 __ __ __ 2 √  (. So AM = 1 __ 2 AB. 27. B 29. D 31. (a + c, b) 35. x = -2.5 or x = 0.25 37. x = 2 or x = -1.67 39. 22 Exercises 7. 4-8 Check It Out! 1. 4.2 × 10 13 ; since it is 6 months between September and March, the ∠ measures will be the same between Earth and the star. By the
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 ; ̶̶ BC, and ̶̶ AB ≅ ̶̶ AC. By the Mdpt. Exercises 1. legs: ̶̶ JL ; base : ∠J and ∠L 3. 118° base: 5. 27° 7. y = 5 9. 20 11. It is given that △ABC is rt. isosc., X is the mdpt. of Formula, the coords. of X are(a, a). By the Dist. Formula, AX = BX = a √  2. So △AXB is isosc. by def. of an isosc. △. 13. 69° 15. 130° or 172° 17. z = 92 19. 26 21. It is given that △ABC is isosc., P is the mdpt. of mdpt. of the coords. of P are (a, b) and the coords. of Q are (3a, b). By the Dist. Formula, PC = QB = √  9 a 2 + b 2, ̶̶ QB by the def of ≅ segs. so 23. S 25. N 27a. 38° b. m∠PQR = m∠ PRQ = 53° 29. m∠1 = 127°; m∠2 = 26.5°; m∠3 = 53° 33. 20 39. 1. △ABC ≅ △CBA (Given) ̶̶ AB ≅ ̶̶ AB, and Q is the ̶̶ AC. By the Mdpt. Formula,
̶̶ 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 + (2a) 2 = √  (-2b) 2 + (2a) 2 = √  4b 2 + 4a 2 = 2 √  b 2 + a 2 units. AB = √  (0 - b) 2 + (a - 0) 2 = √  (-b) 2 + a 2 = √  So AB = 1 __ 2 PQ. 13. b 2 + a 2 units. By the Mdpt. Formula, the coords. of E are (0, a) and the coords. of F are (2c, a). By the Dist. Formula, AD = √  (2c - 0) 2 + (2a - 2a) 2 = √  (2c) 2 = 2c units. Similarly, EF = √  (2c - 0) 2 + (a - a) 2 = √  (2c
) 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 (Reflex. Prop. of ≅) 1. isosceles triangle 2. corresponding angles 3. included side 4. equiangular; equil. 5. obtuse; scalene 6. 60° 7. 66.5° 8. ̶̶ 12. 1. AB ≅ ̶̶ 2. DA ≅ 3. △ADB ≅ △DAE (SSS Steps 1, 2) ̶̶ GJ bisects ̶̶ GJ. (Given) bisects ̶̶ ̶̶ ̶̶ 2. FK ≅ GK ≅ JK, seg. Bisect) 3. ∠GKF ≅ ∠JKH (Vert.  Thm.) 4. △FGK ≅ △HJK (SAS Steps 2, 3) ̶̶ HK (Def. of ̶̶ FH, and 13. 1. ̶̶ FH ̶̶ BC ≅ ̶̶ YZ ; ∠C ≅ ̶̶ XZ ; so △ABC ≅ △XYZ 14. BC = (-6) 2 + 36 = 72; YZ = 2 (-6) 2 = 72; ̶̶ AC ≅ ∠Z; by SAS. 15. PQ = 25 - 1 = 24; QR = 25; PR = 25 2 - (25 - 1) 2 - 42 = ̶̶ ̶̶ ̶̶̶ PR ; so QR ; MN ≅ 7; △LMN ≅ △PQR by SSS. 16. 1. C is the md
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 rt. △) 4. 5. 6. △WZX ≅ △YXZ (HL Steps 5, 4) ̶̶ XZ (Reflex. Prop. of ≅) ̶̶ YX (Given) ̶̶ XZ ≅ ̶̶̶ WZ ≅ 17. 1. ̶̶ RT ≅ 18. 1. ∠S and ∠V are rt. . (Given) 2. ∠S ≅ ∠V (Rt. ∠ ≅ Thm.) 3. RT = UW (Given) ̶̶̶ 4. UW (Def. of ≅) 5. m∠T = m∠W (Given) 6. ∠T ≅ ∠W (Def. of ≅) 7. △RST ≅ △UVW (AAS Steps 2, 6, 4) 19. 1. M is the mdpt. of ̶̶ BD. (Given) ̶̶̶ MB ≅ ̶̶ BC ≅ ̶̶̶ CM ≅ ̶̶̶ DM (Def. of mdpt.) ̶̶ DC (Given) ̶̶̶ CM (Reflex. Prop. of ≅) 2. 3. 4. 5. △CBM ≅ △CDM (SSS Steps 2, 3, 4
) 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. 1. H is mdpt. of line ̶̶̶ MK. (Given) mdpt. of 2. GH = JH, ML = KL (Def. of mdpt.) ̶̶̶ ̶̶ ̶̶̶ 3. ML ≅ GH ≅ JH, ̶̶ ̶̶̶ KM (Given) 4. GJ ≅ ̶̶̶ ̶̶ KL (Div. Prop. of ≅) 5. GH ≅ ̶̶̶ ̶̶ 6. KJ, ∠G ≅ ∠K (Given) GM ≅ 7. △GMH ≅ △KJL (SAS Steps 5, 6) 8. ∠GMH ≅ ∠KJL (CPCTC) 22. (0, 0), (r, 0), (0, s) 23. (0, 0), (2p, 0), (2p, p), (0, p) 24. (0, 0), (8m, 0), (8m, 8m), (0, 8m) 25. Use coords. A (0, 0), B (2a, 0), C (2a, 2b), and D (0, 2b). Then, by the Mdpt. Formula, E (a, 0), F (2a, b), G (a, 2b), and H (0, b). By the Dist. Formula, EF = √ 
 (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 = √  So QM = PM = RM. By def. M is equidistant from the vertices of △PQR 27. To be a rt. △, the side lengths must have lengths such that a 2 + b 2 = c 2. √  (3 - 3) 2 + (5 - 2) 2 = 3, √  (3 - 2) 2 + (2 - 5) 2 = √  10, and √  3 2 + 1 2 = ( √  triangle is a rt. △. 28. x = -5 29. RS = 13.5 30. 70 units (a - 0) 2 + (b -
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) Exercises 1. perpendicular bisector 3. 25.9 5. 21.9 7. 38° 9. y - 1 = x + 2 11. y - 2 = 4 __ 3 (x + 3) 13. 26.5 15. 1.3 17. 54° 19. y + 3 = - 1 __ 2 (x + 2) 21. y + 3 = 5 __ 2 (x - 2) 23. 38 25. 38 27. 24 29. Possible answer: C (3, 2) 31. 1. ̶̶ SR ⊥ 2. ∠QPS ≅ ∠RPS (Def. of ∠ bisector) 3. ∠SQP and ∠SRP are rt. . (Def. of ⊥) 4. ∠SQP ≅ ∠SRP (Rt. ∠ ≅ Thm.) 5. 6. △PQS ≅ △PRS (AAS) ̶̶ 7. SR (CPCTC) 8. SQ = SR (Def. of ≅ segs.) 33a. y = - 3 __ 4 x + 2 b. 2 c. 6.4 mi 35. D 39. the lines y = x and y = -x 43. parallel 45. perpendicular 47. y = - 1 __ 2 x - 10 ̶̶ PS (Reflex
. 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 point M. Exercises 1. They do not intersect at a single point. 3. 5.64 5. 3.95 7. (2, 6) 9. 42.1 11. The largest possible 〇 in the int. of the △ is its inscribed 〇, and the center of the inscribed 〇 is the incenter. Draw the △ and its ∠ bisectors. Center the 〇 at E, the pt. of concurrency of the ∠ bisectors. 13. 63.9 15. 63.9 17. (-1.5, 9.5) 19. 55° 23. perpendicular bisector 25. angle bisector 27. neither 29. S 31. N 33. (4, 3) 35a. ∠ Bisector Thm. b. the bisector of ∠B c. PX = PZ 37a. (4, - 7 __ 6 ) b. outside c. 4.2 mi 41. F 45. t = 4 47. y = 120 49. 35° 51. yes 53. no 5-3 Check It Out! 1a. 21 1b. 5.4 2. 3; 4; possible answer: the x-coordinate of the centroid is the average of the x-coordinates of the vertices of the △, and the y-coordinate of the centroid is the average of the y-coordinates of the vertices of the △. 3. Possible answer: An equation of the altitude ̶̶ JK is y = - 1 __ 2 x + 3. It is true that to 4 = - 1 __ 2 (-2) + 3, so (-2, 4) is a solution of this equation. Therefore this altitude passes through the orthocenter. Exercises 1. centroid 3.
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 ̶̶ ̶̶ ST = c ____ RT = 0 ; slope of b - a ̶̶ RS, slope of ℓ = - b __ c. b. Since ℓ ⊥ ̶̶ ____ ST, slope of m = - b - a Since m ⊥ c ̶̶ ____ = a - b RT, n is a vertical c line, and its slope is undefined. c. An equation of ℓ is y - 0 = - b __ c (x - a), or y = - b __ c x + ab __ c. An equation of m is y - 0 ____ ____ = a - b (x - 0), or y = a - b x. c c An equation of n is x = b. d. (b, ab - b 2 ______ c of line n is x = b and the xcoordinate of P is b, P lies on n. f. Lines ℓ, m, and n are concurrent at P. 47. F 49. 14.0 51. 108° ) e. Since the equation. Since n ⊥ 5-4 Check It Out! 1. M (1, 1) ; N (3, 4) ; ̶̶ ̶̶̶ RS = 3 __ 2 ; MN = 3 __ 2 ; slope of slope of since the slopes are the same, ̶̶̶ MN ǁ = 2 √  13 ; the length of the length of ̶̶ RS. MN = √  13 ; RS = √  52 ̶̶̶ MN is half ̶̶ RS. 2a. 72 2b. 48.5 Selected Answers S97 S97 � 2c. 102° 3. 775 m
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 __ 2 ) 51. (4, -2), (8, -1), (5, -4) 53. 6 55. 9 n = 2 6 - n 49. 2.25% 5-5 Check It Out! 1. Possible answer: Given: △RST Prove: △RST cannot have 2 rt. . Proof: Assume that △RST has 2 rt. . Let ∠R and ∠S be the rt. . By the def. of rt. ∠, m∠R = 90° and m∠S = 90°. By the △ Sum Thm., m∠R + m∠S + m∠T = 180°. But then 90° + 90° + m∠T = 180° by subst., so m∠T = 0°. However, a △ cannot have an ∠ with a measure of 0°. So there is no △RST, which contradicts the given information. This means the assumption is false, and △RST cannot have 2 rt. . ̶̶ DF, 2a. ∠B, ∠A, ∠C 2b. 3a. No; 8 + 13 = 21, which is not greater than the third side length. 3b. Yes; the sum of each pair of 2 lengths is greater than the third length. 3c. Yes; when t = 4, the value of t - 2 is 2, the value of 4t is 16, and the value of t 2 + 1 is 17. The sum of each pair of 2 lengths is greater than the third length. 4. greater than 5 in. and less than 39 in. 5. 28
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 Selected Answers and m∠R = 90°. By the △ Sum Thm., m∠P + m∠Q + m∠R = 180°. By subst., 90° + m∠Q + 90° = 180°, so m∠Q = 0°. However, a △ cannot have an ∠ with a measure of 0°. So there is no △PQR, which contradicts the given information. This means the assumption is false, and therefore △PQR cannot have a base ∠ that is rt. ̶̶ ̶̶ XY 7. no 9. no 11. yes YZ, ̶̶ XZ, ̶̶ ST, ̶̶ EF, ̶̶ RS, ̶̶ DE, ̶̶ AB ⊥ 5. 13. greater than 0 ft and less than 32 ft 15a. the path from the refrigerator to the stove b. no ̶̶ 19. RT 21. no 23. yes 25. no 27. greater than 5 km and less than 51 km 29. greater than 1.18 m and less than 4.96 m 31. greater than 2 2 __ 3 ft and less than 10 1 __ 3 ft 33. a > 7.5, where a is the ̶̶ length of a leg. 35. DF 37. m∠Y < 90°, and ∠Y is an obtuse ̶̶ ̶̶ angle. 39. AB ǁ BC, and 41. x is a multiple of 4, and x is prime. 43. < 45. = 47. > 49. > 51. < 53. = 55. ∠L,
∠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. 5-6 Check It Out! 1a. m∠EGF > m∠EGH 1b. BC > AB 2. The ∠ of the swing at full speed is greater than the ∠ at low speed. 3a. 1. C is the mdpt. of ̶̶ BD. m∠1 = ̶̶ BC ≅ m∠2, m∠3 > m∠4 (Given) ̶̶ 2. DC (Def. of mdpt.) 3. ∠1 ≅ ∠2 (Def. of ≅ ) 4. Thm.) 5. AB > ED (Hinge Thm.) 3b. 1. ∠SRT ≅ ∠STR, TU > RU ̶̶ EC (Conv. of Isosc. △ ̶̶ AC ≅ ̶̶ SR (Conv. of Isosc. △ (Given) ̶̶ 2. ST ≅ Thm.) ̶̶ ̶̶ 3. SU (Reflex. Prop. of ≅ ) SU ≅ 4. m∠TSU > m∠RSU (Conv. of Hinge Thm.) Exercises 1. AC < XZ 3. KL > KN 5. 1.2 < x < 3 7. the second position 9. m∠DCA > m∠BCA 11. TU > SV 13. -3.5 < z < 32.5 15. the second position 17. BC = YZ 19. > 21. =
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 nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2. 3c. 2.6; no; 2.4 and 2.6 are not whole numbers. 3d. 34; yes; the 3 side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2. 4a. yes; obtuse 4b. no 4c. yes, acute Exercises 1. no 3. x = 6 √  2 5. width: 14.8 in.; height: 11.9 in. 7. 16; yes 9. triangle; acute 11. triangle; right 13. triangle; acute 15. x = 10 17. x = 24 19. 6; no 21. 3 √  5 ; no 23. not a triangle 25. triangle; right 27. triangle; acute 29. B 31. x = 8 + √  13 33. x = 4 √  6 35. x = 6 √  13 39. perimeter: 16 + 4 √  7 units; area: 12 √  7 square units 41. perimeter: 14 + 2 √  13 units; area: 18 square units 43. perimeter: 22 units; area: 26 square units 47a. King City b. m∠SRM > 90° 49. B 51a. PA = √  2 ; PB = √  3 ; PC = √  4 ; PD = √  5 ; PE = √  6 ; PF = √  7 55a.
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 = 48; y = 24 √  3 15. x = 2 √  3 ____ 3 ; y = 4 √  3 ____ 3 17. perimeter: (12 + 12 √  2 ) in.; area: 36 in 2 19. perimeter: 36 √  2 m; area: 162 m 2 21. perimeter: 60 √  3 yd; area: 300 √  3 yd 2 23. no 25. (10, 3) 27. (5, 10 - 12 √  3 ) 29a. 640 m b. 453 m c. 234 m 31. F 33. 443.4 35. x = 32 __ 9 39. y = (x - 5) 2 - 27; x = 5 41. obtuse 43. right 45. 132° SGR ̶̶ AB and ̶̶ AB, ̶̶ CP, P is on the ̶̶ AP ⊥ ̶̶ AP ⊥ 1. equidistant 2. midsegment 3. incenter 4. locus 5. 7.4 6. 13.4 7. 5.8 8. 52° 9. y = x - 1 10. y - 6 = -0.25 (x - 4) 11. No; to apply the Conv. of the ∠ Bisector Thm., you need to know that ̶̶ ̶̶ CB. 12. Yes; because CP ⊥ ̶̶ ̶̶ ̶
̶ 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) ; ̶̶ ̶̶̶ GJ = 2 __ 7 ; VW = 2 __ 7 ; slope of slope of since the slopes are the same, ̶̶ ̶̶̶ GJ. VW = √  53 ; GJ = 2 √  53 ; VW ǁ since √  53 = 1 __ 2 (2 √  53 ), VW = 1 __ 2 GJ. 37. 39. greater than 9 cm and less than 18 cm 40. Yes; possible answer: the sum of each pair of 2 lengths is greater than the third length. 41. No; possible answer: when z = 5, the value of 3z is 15. So the 3 lengths are 5, 5, and 15. The sum of 5 and 5 is 10, which is not greater than 15. By the △ Inequality Thm., a △ cannot have these side lengths. 43. PS < RS 44. m∠BCA < m∠DCA 45. -1.4 < n < 3 46. 2.75 < n < 12.5 47. x = 2 √  10 48. x = 2 √  33 49. 6; the lengths do not form a Pythagorean triple because 4.5 and 7.5 are not whole numbers. 50. 40; the lengths do form a Pythagorean triple because they are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2. 51. triangle; ̶̶ AB 38. ∠F, ∠H, ∠G ̶̶ BC,
̶̶ 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. not a polygon 2a. regular, convex 2b. irregular, concave 3a. 2340° 3b. 144° 4a. 30° 4b. r = 15 5. 45° Exercises 3. not a polygon 5. not a polygon 7. irregular, concave 9. m∠A = m∠D = 81°; m∠B = 108°; m∠C = m∠E = 135° 11. 3240° 13. 72° 15. m∠Q = m∠S = 135° 17. not a polygon 19. irregular, concave 21. irregular, convex 23. 160° 25. 40° 27. 120° 29. 61.5 31. 72 33. 10 35. pentagon 37. dodecagon 39. 3; 60° 41. 10; 144° 43. A 45a. heptagon b. 900° c. 140° 53. A 55. D 57. x = 36; y = 36; z = 72 59. Yes, if you allow for ∠ measures greater than 180° 61. x = -3 or x = 4 63. 0 < x < 8 65. 4 < x < 10 67. 5 √  3 6-2 Check It Out! 1a. 28 in. 1b. 74° 1c. 13 in. 2a. 12 2b. 18 3. (7,6) 4. 1. GHJN and JKLM are . (Given) 2. ∠N and ∠HJN are supp. ∠K and ∠MJK are supp. ( → cons.  supp
.) 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 supp. to ∠1 ( → cons.  supp.); ∠4 is supp. to ∠1 ( → cons.  supp.); ∠5 is supp. to ∠1 ( → cons.  supp.); ∠7 is supp. to ∠1 (Subst.) 33. ∠KMP ( → ̶̶̶ KM ( → opp. sides ̶̶ RP (Def. of ) 39. ∠RTP opp.  ≅) 35. ≅) 37. (Vert.  Thm.) 41. x = 119; y = 61; z = 119 43. x = 24; y = 50; z = 50 47. x = 5; y = 8 49a. no b. no 51. A 53. 26.4 55. (2, 4), (4, -6), (-6, -2) 59. no correlation 61. alt. ext.  63. corr.  65. 8 sides; 45° 6-3 ̶̶ PQ ǁ ̶̶ PQ ≅ Check It Out! 1. PQ = RS = 16.8, ̶̶ RS. m∠Q = 74°, and m∠R so = 106°, so ∠Q and ∠R are supp., ̶
̶ 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 . 2b. No; 2 pairs of cons. sides are ≅. None of the sets of conditions for a  are met. ̶̶ 3. Possible answer: slope of KL = ̶̶̶ ̶̶̶ MN = - 7 __ 2 ; slope of LM = slope of ̶̶ NK = - 1 __ 4 ; both pairs of slope of opp. sides have the same slope, ̶̶ ̶̶̶ NK ; by def., MN and so KLMN is a . 4. Possible answer: Since ABRS is a , it is always ̶̶ RS. Since true that ̶̶ RS also remains vert. no vert., matter how the frame is adjusted. Therefore the viewing ∠ never changes. ̶̶ AB stays ̶̶̶ LM ǁ ̶̶ AB ǁ ̶̶ KL ǁ ̶̶ UR = 0; ̶̶ FH. EJ ̶̶ FH ̶̶ ST and ̶̶ ST ǁ Exercises 1. FJ = HJ = 10, so ̶̶ ̶̶ ̶̶ EG bisects FJ ≅ HJ. Thus ̶̶ ̶̶ = GJ = 18, so GJ. Thus EJ ≅ ̶̶ EG. So the diags. of EFGH bisects bisect each other. By Thm. 6-3-5, EFGH is a . 3. yes 5. yes ̶̶ 7. Possible answer: slope of ST = ̶̶ UR have slope of ̶̶ UR ; ST = the same slope, so UR = 6; 1 pair of opp. sides are \|\| and ≅; by Thm.
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 ; PQ = RS = √  34 ; 1 pair of opp. ǁ sides are \|\| and ≅; by Thm. 6-3-1, PQRS is a . 17. no 19. yes 21. a = 16.5; b = 23.2 23. a = 8.4; b = 20 27a. ∠Q b. ∠S. c. 35 B 37. no 39. (3, 1) ; (-6, -3.5) 41. ̶̶ RS e.  ̶̶ SP d. x -5 -2 0 0.5 y -38 -17 -3 0.5 43. x y -5 -2 77 14 0 2 0.5 2.75 47. 12 49. 12 6-4 Check It Out! 1a. 48 in. 1b. 61.6 in. 2a. 42.5 2b. 17° 3. SV ̶̶̶ = TW = √  TW. Slope ̶̶̶ TW = -11, of ̶̶ SV ≅ 122, so ̶̶ SV = 1 __ 11, and slope of ̶̶ SV ⊥ so mdpt. of ̶̶ SV and so So the diags. of STVW are ≅ ⊥ bisectors of each other. 4. Possible answer: ̶̶̶ TW. The coordinates of the ̶̶ ̶̶̶ TW are ( 1 __ 2, - 7 __ 2 ), SV and ̶̶̶ TW bisect each other. 1. PQTS is a rhombus. (Given) ̶̶ 2. PT
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. Possible answer: 1. RECT is a rect. (Given) ̶̶ 2. XY ≅ 3. RX = TY, XY = XY (Def. of ≅ segs.) 4. RX + XY = TY + XY (Add. Prop. of =) 5. RX + XY = RY, TY + XY = TX (Seg. Add. Post.) 6. RY = TX (Subst.) ̶̶ 7. TX (Def. of ≅ segs.) 8. ∠R and ∠T are rt. . (Def. of rect.) 9. ∠R ≅ ∠T (Rt. ∠ ≅ Thm.) 10. RECT is a . (Rect. → ) 11. 12. △REY ≅ △TCX (SAS) ̶̶ TC ( → opp. sides ≅) ̶̶ RE ≅ ̶̶ RY ≅ S100 S100 Selected Answers 11. 25 13. 14 1 __ 2 15. m∠VWX = 132°; m∠WYX = 66° 17. Possible answer: ̶̶ HB is a ̶̶̶ MH ≅ ̶̶ HB bisects ∠RHM. 1. RHMB is a rhombus. diag. of RHMB. (Given) ̶̶ 2. RH (Def. of rhombus) 3. (Rhombus → each diag. bisects opp. ) 4. ∠MHX �
� ∠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. →  ̶̶̶ b. HG c. Reflex. Prop. of ≅ d. Def. of rect. e. ∠GHE f. SAS g. CPCTC 41. 28 √  2 in. ≈ 39.60 in.; 98 in 2 45. D 47. H 51. 45 53. T 55. no 6-5 Check It Out! 1. Both pairs of opp. sides of WXYZ are ≅, so WXYZ is a . The contractor can use the carpenter’s square to see if 1 ∠ of WXYZ is a rt. ∠. If 1 ∠ is a rt. ∠, then by Thm. 6-5-1 the frame is a rect. 2. Not valid; by Thm. 6-5-1, if 1 ∠ of a  is a rt. ∠, then the  is a rect. To apply this thm., you need to know that ABCD is a . 3a. rect., rhombus, square 3b. rhombus ̶̶ QS ̶̶ PR ≅ Exercises 3. valid 5. rhombus 7. valid 9. square, rect., rhombus 11. , rect. 13. , rect., rhombus, square 15.
, 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.) 33b.  c. square 39. A 41a. 15x = 13x + 12; x = 6 b. yes c. not necessarily d. yes 43b. no c. no 45. linear 47. linear 49. 31 + √  61 ≈ 38.8 51. y = 4 ̶̶ BD. c. ABCD is a rhombus, ̶̶ AC ⊥ 6-6 Check It Out! 1. about 191.2 in.; 3 packages 2a. 51° 2b. 110° 2c. 62° 3a. 131° 3b. 25.4 4. x = 4 or x = -4 5. 8 ̶̶ QT ̶̶ RS and ̶̶ PV ; ̶̶ VS ; midsegment: Exercises 1. bases: ̶̶ PR and legs: 3. about 20.1 in.; 3 sun catchers 5. 63° 7. 106° 9. z = 2 or z = -2 11. 14 13. about 56.6 in.; about 418.3 in. 15. 122° 17. 62° 19. ±4 √  5 21. 3.6 23. S 25. N 27. m∠1 = 82°; m∠2 = 128° 29. m∠1 = 51°; m∠2 = 16° 31. m∠1 = 120° 33a. EF = FG = √  17, and GH = HE ̶̶ ̶̶ = √  29, so HE. FG, and Thus EFGH is a kite, since it has exactly 2 pairs of ≅ cons. sides. b. m∠E = m∠
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; △ 7. polygon; dodecagon 8. irregular; concave 9. irregular; convex 10. reg.; convex 11. 1800° 12. 162° 13. 90° 14. m∠A = m∠D = 144°; m∠B = m∠E = 126°; m∠C = m∠F = 90° 15. 37.5 16. 62.4 17. 37.5 18. 79° 19. 101° 20. 101° 21. 9.5 22. 9.5 23. 54° 24. 126° 25. 54° 26. 126° 27. T (6, - 5) 28. 1. GHLM is a . ∠L ≅ ∠JMG ̶̶ GJ ≅ (Given) 2. ∠G ≅ ∠L ( → opp.  ≅) 3. ∠G ≅ ∠JMG (Trans. Prop. of ≅) 4. Thm.) 5. △GJM is isosc. (Def. of isosc. △) ̶̶ MJ (Conv. of Isosc. △ 29. m∠A = m∠E = 63°; m∠G = 117°; since 117° + 63° = 180°, ∠G is supp. to ∠A and to ∠E. So 1 ∠ of ACEG is supp. to both of its cons. . By Thm. 6-3-4, ACEG is a . ̶̶ 30. RS = QT = 25, so QT. m∠R
= 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 the quad. bisect each other. By Thm. 6-3-5, the quad. is a . 32. No; a pair of alt. int.  are ≅, so 1 pair of opp. sides are \|\|. A different pair of opp. sides are ≅. None of the conditions ̶̶ for a  are met. 33. slope of BD FH = 1 __ ̶̶ ̶̶ = slope of BH = ̶̶ DF = -6; both pairs of slope of opp. sides have the same slope, so BDFH is a . 34. 18 35. 39.6 36. 39.6 37. 19.8 38. 25.5 39. 10.5 40. 25.5 41. 21 42. 41° 43. 49° 44. 82° 45. 98° 46. m∠1 = 57°; m∠2 = 66°; m∠3 = 33°; m∠4 = 114°; m∠5 = 57° 47. m∠1 = 37°; m∠2 = 53°; m∠3 = 90°; m∠4 = 37°; m∠5 = 53° ̶̶ 48. RT = SU = 2 √  10, so RT ≅ ̶̶ RT = -3, and slope of Slope of ̶̶ = 1 __ 3, so RT ⊥ of the mdpt. of ̶̶ (-4, -3), so RT and other. So 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 the diags. of a  are ⊥, then the  is a rhombus. If a  is both a rect. and a rhombus, then the  is a square. To apply this chain of reasoning, you must first know that EFRS is a . 51. valid 52. valid 53. rhombus 54. rect. 55. rect., rhombus, square 56. 64° 57. 25° 58. 65° 59. 123° 60. m∠R = 126°; m∠S = 54° 61. 51.6 62. 48.5 63. 3.5 64. n = 3 or n = -3 65. kite 66. trap. 67. isosc. trap. ̶̶ EG = -1, and slope of ̶̶ EG ⊥ ̶̶ ̶̶ RT and SU are ̶̶ SU bisect each ̶̶ SU. The coordinates ̶̶ FH. The coordinates ̶̶ FH. ̶̶ FH ̶̶ EG and ̶̶ FH are Chapter 7 7-1 Check It Out! 1. 1 __ 4 2. 9°; 54°; 117° 3a. x = 21 3b. y = ± 3 3c. d = 9 3d. x = 3 or -9 4. 4 : 5 5. 1527.2 m b. 800 in., Exercises 1. means: 3 and 2; extremes: 1 and 6 3. 1 __ 2 5. - 2 __ 3 7. 95° 9. y = 9 11. y = ± 9 13. x = 0 or x = 3 15. 2 :
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 ___ 36 = 1080 36. Finally, simplify: x = 30. You must assume that x ≠ 0. 49. Given a __ = c __, add 1 to both sides d b + d __ = c __ + b __ of the eqn. as shown: a __. d d b b Adding the fractions on both sides of the eqn. gives a + b = c + d ____ ____. b d 51. x + 3 ____ x - 6, where x ≠ ±6 53. 1 55. 96° 57. acute 59. right 7-2 = AC ___ JH = BC ___ JK = DA ___ LH = BC ___ GH Check It Out! 1. ∠A ≅ ∠J; ∠B ≅ ∠G; ∠C ≅ ∠H; AB ___ = 2 JG 2. yes; 5 __ 2 ; △LMJ ∼ △PNS 3. 5 in. Exercises 3. ∠A ≅ ∠H; ∠B ≅ ∠J; ∠C ≅ ∠K; ∠D ≅ ∠L; AB ___ HJ = 2 __ 3 5. yes; 2 __ 3 ; △RMP ∼ = CD ___ KL △XWU 7. ∠J ≅ ∠S; ∠K ≅ ∠T; ∠L ≅ ∠U; ∠M ≅ ∠V; JK ___ = MJ ___ = LM ___ UV VS ST = 5 __ 6 9. yes; 7 __ 8 ; △RSQ ∼ △UZX 11. 14 ft 13. S 15. N 17. S 19. 5 23. ∠O; ∠Q 27. C 29. The ratios of
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 ∼ △DEF by AA ∼. 2. ∠TXU ≅ ∠VXW by the Vert.  Thm. TX ___ = VX = 15 __ 20 = 3 __ 4. Therefore 12 __ 16 = 3 __ 4, and XU ___ XW △TXU ∼ △VXW by SAS ∼. 3. It is given that ∠RSV ≅ ∠T. By the Reflex. Prop. of ≅, ∠R ≅ ∠R. Therefore △RSV ∼ △RTU by AA ∼. RT = 15. 4. 1. M is the mdpt. of ̶̶ JK, N is the ̶̶ KL, and P is the mdpt. mdpt. of ̶̶ JL. (Given) of 2. MP = 1 __ 2 KL, MN = 1 __ 2 JL, NP = 1 __ 2 KJ (△Midsegs. Thm.) 3. MP ___ = NP ___ = MN ___ KJ JL KL of =) 4. △JKL ∼ △NPM (SSS ∼ Step 3) 5. 5 = 1 __ 2 (Div. Prop. = EF ___ KL Exercises 1. By the △ Sum Thm., m∠A = 47°. So by the def. of ≅, ∠A ≅ ∠F, and ∠C ≅ ∠H. Therefore △ABC ∼ △FGH by AA ∼. 3. DF ___ JL = DE ___ JK SSS ∼. 5. It is given that ∠AED
≅ ∠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 ∼ △JKG by AA ∼. 13. ∠K ≅ ∠K by = 3 __ 2. the Reflex. Prop. of ≅. KL ___ KN Therefore △KLM ∼ △KNL by SAS ∼. 15. It is given that ∠ABD ≅ ∠C. ∠A ≅ ∠A by the Reflex. Prop. of ≅. Therefore △ABD ∼ △ACB by AA ∼. AB = 8 = KM ___ KL 17. 1. CD = 3AC, CE = 3BC (Given) = 3, CE ___ BC = 3 (Div. Prop. 2. CD ___ AC of =) 3. ∠ACB ≅ ∠DCE (Vert.  Thm.) 4. △ABC ≅ △DEC (SAS ∼ Steps 2, 3) 19. 1.5 in. 21. yes; SSS ∼ 23. x = 3 25a. Pyramids A and C are ∼ because the ratios of their corr. side lengths are =. b. 5 __ 4 27. 2 ft; 4 ft 31a. The  are ∼ by AA ∼ if you assume that the camera is ǁ to the hurricane (that is, b. △YWZ ∼ △BCZ, and △XWZ ∼ △ACZ, also by AA ∼. c. 105 mi 35. J 37. 30 41. 94 43. Possible answer: (0, k), (2k, k), (2k, 0
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. 4 __ 5 27. 0.3 ft by 1.2 ft 29. 20 in.; 12 in. 31a. 1 __ 24 b. 1 ___ 576 c. 24 ft 2 33. 1 __ 9 cm 35. 1 cm : 5 m; since each centimeter will equal 5 m, this drawing will be 1 __ 5 the size of the drawing with a scale of 1 cm : 1 m. 39. D 41. C 43a. 150 m b. 1.28 cm 47. x = -4 or x = 10 49. x ≈ 0.65 or x ≈ -4.65 51. The slopes of ̶̶ KL and of ̶̶ JK and ̶̶ JM = -1. Since both ̶̶̶ LM = 1. The slopes S102 S102 Selected Answers pairs of opp. sides have the same ̶̶ JK ǁ slope, def., JKLM is a . ̶̶̶ LM, and ̶̶ JM. By ̶̶ KL ǁ 7-6 = RT ___ RV = 1 __ 3. ∠R ≅ ∠R by Check It Out! 1. The photo should have vertices A′ (0, 0), B′ (0, 2), C′ (1.5, 2), and D′ (1.5, 0). 2. N (0, -20) ; 2 __ 3 3. RS = √  2, RU = 3 √  2, RT = √  5, and RV = 3 √  5, so RS ___ RU the Reflex. Prop. of ≅. So △RST ∼ △RUV by SAS ∼. 4. Check students’
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 1. dilation 5. S (0, -8) ; 5 __ 2 7. JK = 2 √  5, JM = 3 √  5, JL = 2 √  5, and JN = 3 √  5, so JK ___ = 2 __ 3. ∠J ≅ ∠J by the JM Reflex. Prop. of ≅. So △JKL ∼ △JMN by SAS ∼. 9. The image of △RST has vertices R′ (-3, 3), S′ (3, 6), and T ′ (0, -3). RS = 2 √  5, RT = 2 √  5, and ST = 2 √  10. R′S′ = 3 √  5, R′T ′ = 3 √  5, and S′T ′ = 3 √  10. R′S′ ___ RS △R′S′T ′ by SSS ∼. 11. X (-24, 0) ; 8 __ 3 13. DE = 2 √  5, DG = 3 √  5, DF = = DF ___ 4 √  2, and DH = 6 √  2, so DE ___ DH DG = 2 __ 3. ∠D ≅ ∠D by the Reflex. Prop. of �
�. 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 25. 12 31. 5 33. 12 35. 6 = L′K′ ___ LK = J′L′ ___ JL = 3. 19. 1. JM 2. JL __ = 1 __ 3, JK ___ = 1 __ 3 JN (Div. Prop. of =) = JK ___ 3. JL __ (Trans. Prop. of =) JM JN 4. ∠J ≅ ∠J (Reflex. Prop. of ≅) 5. △JKL ∼ △JMN (SAS ∼ Steps 3, 4) ̶̶ QR ǁ ̶̶ ST (Given) 2. ∠RQP ≅ ∠STP (Alt. Int. ∠ Thm.) 3. ∠RPQ ≅ ∠SPT (Vert.  Thm.) 4. △PQR ∼ △PTS (AA ∼ Steps 2, 3) ̶̶ BC ǁ ̶̶ CE (Given) 20. 1. EA = BD ___ CE (Def. of ∼ polygons) 2. ∠ABD ≅ ∠C (Corr. ∠ Post.) 3. ∠ADB ≅ ∠E (Corr. ∠ Post.) 4. △ABD ∼ △ACE (AA ∼ Steps 2, 3) 5. AB ___ AC 6. AB (CE) = AC (BD) (Cross Products Prop.) = JL __ 21. 10 22. 3 1 __ 3 23. JK ___ = 1 __ 2. J
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 √  2, RT = √  10, and RV = 2 √  10. = 1 __ 2. ∠R ≅ ∠R by the RS ___ RU Reflex. Prop. of ≅. So △RST ∼ △RUV by SAS ∼. 31. By the Dist. Formula, JK = √  5, JM = 4 √  5, JL = 2, and JN = 8. JK ___ = 1 __ 4. ∠J ≅ ∠J JM by the Reflex. Prop. of ≅. So △JKL ∼ △JMN by SAS ∼. 32. (0, -6) ; 2 __ 3 33. The image of △KLM has vertices K′ (0, 9), L′ (0, 0), and M′ (12, 0). By the Dist. Formula, KL = 3, LM = 4, KM = 5, K′L′ = 9, L′M′ = 12, and K′M′ = 15. K′L′ ___ ____ = K′M′ KM KL △KLM ∼ △K′L′M′ by SSS ∼. = 3 __ 1. Therefore ____ = L′M′ LM = RT ___ RV = JL __ JN SGR 1. proportion 2. dilation 3. means 4. ratio 5. 1 __ 5 6. - 1 __ 2 7. 3 __ 2 8. 54 9. 17.5; 30; 17.5; 30 10. y = 21 11. s = 10 12. x = ±6 13. z = 13 or z = -11 14. x = ±8 15
. 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. 20 41. B 43. By 11. 2 √  15 ; 2 √  6 ; 2 √  10 13. 12; 4 √  13 ; 8 15. △MPN ∼ △PQN ∼ △MQP 17. △RSU ∼ △RTS ∼ △STU 19. 3 √  5 21. 2 √  5 23. 3 √  5 ____ 10 25. 20 √  3 ; 10 √  21 ; 20 √  7 27. 1670 ft 29. 10 __ 3, or 3 1 __ 3 31. x + y 33. z 35. x 37. 4 √  5 39. √  10 ____ 2 Corollary 8-1-3, a 2 = x (x + y), and b 2 = y (x + y). So a 2 + b 2 = x (x + y) + y (x + y). By the Distrib. Prop., this expression simplifies to (x + y) (x + y) = (x + y) 2 = c 2. So a 2 + b 2 = c 2. 47. D 49. A 51. 7; √  35 ; 2 √  15 53. AC ≈ 15.26 cm; AB ≈ 8.53 cm 55. -4; 2 57. 6 59. 4 61. 39° 8-2 Check It Out!
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. 15 __ 8 ≈ 1.88 25. 15 __ 17 ≈ 0.88 27. 15 __ 17 ≈ 0.88 29. 1 __ 2 31. 1.23 33. 0.22 35. 0.82 37. 3.58 cm 39. 19.67 ft 41. 5.27 ft 43. 6.10 m 45. sine; cosine 47. 60° 49. 1.2 ft 2 + ( √  3 ___ 2 ) 53. 0.6 55. 753 ft 59. ( 1 __ 2 ) = 1 __ 4 + 3 __ 4 = 1 61a. sin A = a __ c ; cos A = b __ c b. (sin A) 2 + (cos A) 2 = ( a __ __ = c 2 __ + b 2 ___ ______ c 2 c 2 c 2 c 2 63. 18.64 cm; 16.00 cm 2 65. 22.60 in.; 14.69 in 2 69. H 71. x ≈ 5; AB ≈ 20; BC ≈ 18; AC ≈ 27 75. 1.25 77. 0.75 79. Possible answers: (-2, 11) ; (0, 10) ; (2, 9) 81. Trans. Prop. of ≅ 83. Sym. Prop. of ≅ 85. 12 + ( b __ c ) = 1 2 2 8-3 Check It Out! 1a. ∠2 1b. ∠1 2a. 37° 2b. 87° 2c. 42° 3. DF ≈ 16.51; EF ≈
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 ≈ 63° 21. ∠2 23. ∠1 25. ∠2 27. 18° 29. 37° 31. 57° 33. JK ≈ 2.88; LK ≈ 1.40; m∠L = 64° 35. QR ≈ 4.90; m∠P ≈ 36°; m∠R ≈ 54° 37. MN = NP = 4; MP ≈ 5.66; m∠N = 90°; m∠M = m∠P = 45° 39. 74° 41. cos 43. 0.93 47a. 5° b. 85° c. 31 ft 1 in. 49. 23°; 67° 51. The acute ∠ measure changes from about 58° to about 73°, an increase by a factor of 1.26. 53a. AB = √  50 ; BC = 2 √  10 ; AC = √  10 b. AC 2 + BC 2 = AB 2, so △ABC is a rt. △, and ∠C is the rt. ∠. c. m∠A = 63°; m∠B = 27° 55. 35° 57. 62° 59. 72° 61. 39° 65. D 67. A 69. 58° 71. 34° 73. x 77. F 79
. 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 ft 27a. 2080 ft b. 14 s 29. J 31. 98 m 33. 1318 ft 35. 6 min 37. rhombus and square 39. rectangle, rhombus, and square 41. 4 43. 16 __ 3 8-5 Check It Out! 1a. -0.09 1b. -0.03 1c. 0.34 2a. 34.9 2b. 29° 2c. 26° 2d. 17.7 3a. 6.5 3b. 30° 3c. 7.0 3d. 65° 4. 68.6 m; 54° Exercises 1. 0.98 3. -28.64 5. -0.68 7. 0.54 9. -0.91 11. 43° 13. 44° 15. 17.3 17. -0.09 19. -1.88 21. 0.99 23. -0.87 25. 0.79 27. 20.6 29. 10.4 31. 65° 33. 33° 35. 8.4 37. 21° 39. 8.2 cm 41. 50° 43. no 45. 63° 47. 41.2 ft 49. 42°; 138° 51. Law of Sines 53. Law of Sines 55. BC ≈ 10.73; AB ≈ 10.34; m∠ABC ≈ 50° 57a. y 2 + h 2 b. b 2 c. a 2 = c 2 - 2cx + x 2 + h 2 d. a 2 = c 2 + b 2 - 2cx e. b cos A f. Subst. 59. A 61. C 63. 31° 65. 3n 67. 2n + 2 69. Alt. Int.  Thm. 71. Alt. Ext.  Thm. 73. ∠
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° E 19. 〈-3.5, 5.5〉 21. 2.0 23. 4.3 25. 36° 27.  DE = 31. 190.1 km/h; 54°, or N 36° E 33. 〈2, 2〉 35. 〈6, 2〉 37a. 98° b. 68.9 mi/h c. 36° d. N 81° E 39. 〈7.1, 1.1〉 41. 〈2.2, 5.4〉 43a. 1 __ 12 b. 1 __ 6 45. 4; 0° 47. 3.6; 56° 49. 〈0, 10〉, 〈10, 0〉; 〈10, 10〉; the magnitude of the resultant is 10 √ �
� 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〉 = 〈-x, -y〉 61. G 63. 8.2 65. 180° 67. 6.4 mi/h at a bearing of N 58° E 69. (2, 7) 71. (6, -1) 73. 54 cm 2 75. 73°  v.  v is the  v = -1  v. SGR 1. component form 2. equal vectors 3. geometric mean 4. angle of elevation 5. trigonometric ratio 6. △PRQ ∼ △RSQ ∼ △PSR 7. 5 8. √  51 9. x = √  35 ; y = 2 √  15 ; z = 2 √  21 10. x = 3 11. x = 5; y = √  5 ; z = √  30 12. 11.17 m 13. 6.30 m 14. 10.32 cm 15. 1.31 cm 16. m∠C = 68°; AB ≈ 4.82; AC ≈ 1.95 17. m∠H ≈ 53°; m∠G ≈ 37°; HG ≈ 5.86 18. m∠S = 40°; RS ≈ 42.43; RT ≈ 27.27 19. m∠Q ≈ 41°; m∠N ≈ 49°; QN ≈ 13.11 Selected Answers S103 S103 �� 20. angle of depression 21. angle of elevation 22. 36 ft 23. 458 m 24. 22° 25. 31.4 26
. 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 x 2 - 12x + 9) π m 2 2. C ≈ 31.4 in.; C ≈ 37.7 in.; C ≈ 44.0 in. 3. A ≈ 77.3 cm 2 Exercises 1. Draw a segment perpendicular to a side with one endpoint at the center. The apothem is 1 __ 2 s. 3. A = 9 x 2 π in 2 5. A ≈ 50.3 in 2 ; A ≈ 78.5 in 2 ; A ≈ 113.1 in 2 7. A ≈ 32.7 cm 2 9. A ≈ 279.9 m 2 11. C = 5π 13. A ≈ 962.1 ft 2 ; A ≈ 1963.5 f t 2 ; A ≈ 3421.2 ft 2 15. A ≈ 13.3 ft 2 17. A ≈ 14.5 ft 2 19. 90° 21. 60° 23. 45° 25. 36° 27. A ≈ 84.3 cm 2 29. A ≈ 46.8 m 2 31. A ≈ 90.8 ft 2 35. 20 √  π ___ π ; 10 √  π ___ π ; 20 √  π 37. 36; 18; 324π 39a. A ≈ 745.6 in 2 b. A ≈ 1073.6 in 2 c. 44% 43. B 45. B 47. A = C 2 ___ 4π 49. y = 3x - 13 51. m∠B = 124° 53. d 2 = 1.4 cm 9-1 9-3 Check It Out! 1. b
= 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 units 2 ; P = 12 units; A = 5 units 2 ; P = 12 units 11. A ≈ 43.5 units 2 13. rhombus; P = 4 √  29 units; A = 20 units 2 15. isosceles trapezoid; P = (8 + 2 √  29 ) units; A = 20 units 2 17. A = 53 units 2 19. P = (6 + 3 √  2 ) units; A = 4.5 units 2 21a. A = 20 mi 2 b. A ≈ 150 mi 2. The area represents the distance the boat traveled in 5 h. 23a. A = 6 units 2 b. Possible answer: C (2, 1) and H (8, 2) 25. J 27. A ≈ 10.5 units 2 29. A ≈ 17.5 units 2 31. P = 8 √  A = 2 √  2 units 2 33. -2 < a < 3 37. d = 22 ft 2 - √  2 units; 9-5 Check It Out! 1. The area is tripled. 2. The perimeter is tripled, and the area is multiplied by 9. 3. The side length is multiplied by 1 ___. 4. Possible answer: √  2 29a. h = 31.2 in. Exercises 1. A = 120 cm 2 3. P = 52 cm 5. b = 13 in. 7. A = 336 in 2 9. d 2 = 8x y 2 cm 11. h = 1.25 m 13. A = (21 x 2 + 32x -
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 __ 2 (a + b) 2 = 1 __ 2 ab + 1 __ 2 ab + 1 __ 47a. Possible answers: A: 4.2 cm 2 ; B: 3.8 cm 2 ; C: 4.3 cm 2 b. C has the greatest area. 49. 23 cases 53. H 55. H 57. h = 4 in. 59. b = (7x + 5) cm; h = (6x + 3) cm 61a. A = x (12 - x) b. D: 0 < x < 12; R: 0 < y < 36 c. 6 ft by 6 ft d. Solve the area S104 S104 Selected Answers Exercises 3. A ≈ 16.3 ft 2 5. A = 17.5 m 2 7. A ≈ 4.5 in 2 9. A = 49.5 mm 2 11. A ≈ 2.3 m 2 13. 7 qt 15. A ≈ 9 m 2 17. A = 540 in 2 19. A = (25 √  3 + 75π ___ 2 ) in 2 21. Possible answer: 35,000 mi 2 23a. A = 675 in 2 b. c. 675 in 2 25. A = (26 + 2π) in 2 27. A = 2 29. Possible answer: A ≈ 10 c m 2 31. A 33. C 37. 15.96 39. 1.4 41. A ≈ 3.9 c m 2 9-4 Check It Out! 1. A ≈ 38 units 2 2. parallelogram; P ≈ 20.8 units 2 ; A
= 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 multiplied by √  3. 15. The area is multiplied by 64. 17. The area is multiplied by �� 28. 19. The area is divided by 16. 21. The area is multiplied by 4. 23. 800,000 acres 25a. The area is multiplied by 3. b. The area is multiplied by 3 c. The area is multiplied by 9. 27a. The area is multiplied by 3. b. The area is multiplied by 3. c. The area is multiplied by 9. 29a. 8 √  2 in. b. 4 √  2 in. 31. G 33. 36 35. A = (9π x 2 + 54πx + 81π) in 2 37. t __ 2 = 36 39. 128.6°; 51.4° 41. 154.3°; 25.7° 43. A = 32 units 2 9-6 Check It Out! 1. 2 __ 3 2. 1 __ 2 3. 1 __ 2 4. 0.71 Exercises 3. 1 __ 2 5. 7 __ 10 7. 9 times 9. 3 __ 8 11. 5 __ 12 13. 0.08 15. 0.79 17. 0.78 19. 0.46 21. 0.62 23. 1 __ 2 25. 3 __ 4 27. 0.5 29. 0.11 31. A 33. 0.84 35. 0.13 37. 0.77 39–41. Possible answers given. 39. The point lies on AC. 41. The point lies in the blue triangle or the green triangle. 43. 1 __ 2 ; it does not matter which regions are shaded because they all have the same area. ____ 4 45. A 47. D 49. 4 - π ≈ 0.21 53
. 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. A = 252 yd 2 9. d 2 = 42 xy 4 in. 10. A = 288 m 2 11. C = 2 ft 12. A ≈ 153.9 yd 2 13. d = 16x m 14. A ≈ 172.0 ft 2 15. A ≈ 6.9 in 2 16. A ≈ 309.0 cm 2 17. A = 72 m 2 18. A ≈ 200.9 ft 2 19. A = 192 cm 2 20. A ≈ 21.4 mm 2 21. A ≈ 49.5 units 2 22. A ≈ 44 units 2 23. square; P = 12 √  2 units; A = 18 units 2 24. right triangle; P = (12 + √  74 ) units; A = 17.5 units 2 25. isosceles trapezoid; P = (12 + 4 √  5 ) units; A = 24 units 2 26. parallelogram; P = (8 + 2 √  13 ) units; A = 12 units 2 27. A = 30.5 units 2 28. A = 17.5 units 2 29. A = 12 units 2 30. A = 16 units 2 31. The perimeter is multiplied by 3. The area is multiplied by 9. 32. The perimeter is doubled. The area is multiplied by 4. 33. The circumference is multiplied by 1 __ 2. The area is multiplied by 1 __ 4. 34. The perimeter is multiplied by 4. The area is multiplied by 16. 35. 7 __ 13 36. 8 __ 13 37. 12 __ 13 38. 6 __ 13 39. 0.17 40. 0.05 41. 0.17 42
. 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. cylinder 3a. hexagon 3b. triangle 4. Cut through the midpoints of 3 edges that meet at 1 vertex. ̶̶ JD, ̶̶ GF, ̶̶ HE, ̶̶ ST, ̶̶ XY, ̶̶ GK, ̶̶ DE, ̶̶̶ GH, ̶̶ CD, ̶̶̶ ̶̶ ̶̶ UV, SW, VS, ̶̶̶ ZW ; bases: Exercises 1. cylinder 3. rectangular prism; vertices: C, D, ̶̶ HJ, E, F, G, H, J, K; edges: ̶̶ ̶̶ ̶̶ ̶̶ EF ; FC, KC, JK, bases: GHJK, CDEF 5. rectangular prism 7. cube 9. pentagon 11. Cut parallel to the bases. 13. cube; vertices: S, T, U, V, W, X, ̶̶ ̶̶ TU, TX, Y, Z; edges: ̶̶ ̶̶̶ ̶̶ ̶̶ UY, YZ, WX, VZ, STUV, WXYZ 15. cylinder; vertices: none; edges: none; bases: ⊙R, ⊙Q 17. triangular pyramid 19. square 21. rectangle 23. Cut perpendicular to the ground. 25. rectangular prism 27. hexagonal prism 29. The figure is a cylinder whose bases each have a radius of 12 ft. The height of the cylinder is 9 ft. 31. 39. 41. D 43. B 45. 47. 49. 51a. A and B, C and F, D and G, E and H b.
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 Answers 43. The first number is 20, and the second number is 10. 45. The first number is 0, and the second number is 5. 47. 2 49. 2 pentagons and 5 parallelograms 51. 4 triangles 10-3 Check It Out! 1a. V = 6; E = 12; F = 8; 6 - 12 + 8 = 2 1b. V = 7; E = 12; F = 7; 7 - 12 + 7 = 2 2. 5 √  3 ≈ 8.7 m 3. 4a. d ≈ 12.9 units; M (3, 4.5, 8.5) 4b. d ≈ 11.4 units; M (8.5, 12, 18) 5. 36.2 ft Exercises 1. because the bases are circles, which are not polygons 3. V = 6; E = 10; F = 6; 6 - 10 + 6 = 2 5. 15.0 ft 7. 17 in. 9. 11. d ≈ 14.4 units; M (2.5, 4.5, 5) 13. d ≈ 9.3 units; M (6.5, 9, 12.5) 15. V = 8; E = 12; F = 6; 8 - 12 + 6 = 2 17. V = 11; E = 20; F = 11; 11 - 20 + 11 = 2 19. h ≈ 5.3 m �� 21. 23. 45. d ≈
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, AC = 11, and BC = 11 √  2, so △ABC is an isosc. rt. △. 57. C 59. B 61. AB = BC = 2 √  6, and AC = 4 √  6, so AB + BC = AC. The points are collinear. 65. 0–9 yr old 67. A = 1 __ 2 h ( 1 __ 2 b 1 + b 2 ) 69. cone 71. ⊙C 10-4 Check It Out! 1. L = 256 cm 2 ; S = 384 cm 2 2. L = 196π in 2 ; S = 294π in 2 3. 239.7 cm 2 4. The surface area is multiplied by 1 __ 4. 5. It will melt at about the same rate as the half cylinder. Exercises 1. 5 3. L = 24 cm 2 ; S = 36 cm 2 5. L = 24π ft 2 ; S = 42π ft 2 7. L = 80π m 2 ; S = 208π m 2 9. S ≈ 2855.0 ft 2 11. The surface area is multiplied by 4 __ 9. 13. L = 200 cm 2 ; S = 250 cm 2 15. L = 336 ft 2 ; S ≈ 391.4 ft 2 17. L = 184π cm 2 ; S = 216π cm 2 19. S ≈ 352.0 cm 2 21. The surface area is multiplied by 9. 23. the cell that measures 35 µm by 7 µm by 10 µm 25. h = 3.
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 cm 2 3. The surface area is multiplied by 4 __ 9. 4. S ≈ 28.9 yd 2 5. 9 in. Exercises 1. the vertex and the center of the base 3. L = 544 ft 2 ; S = 800 ft 2 5. L = 175π in 2 ; S = 224π in 2 7. L = 48π m 2 ; S = 84π m 2 9. The surface area is multiplied by 9. 11. S = 1056π m 2 13. L = 60 ft 2 ; S = 96 ft 2 15. L = 315 ft 2 ; S ≈ 442.3 ft 2 17. L = 444π in 2 ; S = 588π in 2 19. The surface area is divided by 9. 21. S = 287π in 2 23. 6 in. 25. 4 √  3 m 2 27. 3π ft 2 29. r = 8 m 31. P = 24 cm 33. S = 330 cm 2 35. Possible answer: 526,000 ft 2 39. F 41a. S = 500π cm 2 b. L = 100π cm 2 c. B = 25π cm 2 d. S = 500π - 100π + 25π = 425π cm 2 43a. c = 2πr b. C = 2πℓ c. c __ C larger circle is A = π ℓ 2. The lateral surface area is c __ = r _ times the area ℓ C of the circle, so L = π ℓ 2 ( r _ ) = πrℓ 45. yes 47. 0.25 49. 0.21 51. S = 700 cm
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 �� Exercises 1. the same length as 3. V ≈ 748.2 m 3 5. 2552 gal; 12,071 lb 7. V = 45π m 3 ≈ 141.4 m 3 9. The volume is multiplied by 1 __ 64. 11. V ≈ 1209.1 ft 3 13. V = 810 yd 3 15. V = 245 ft 3 17. V = 1764π cm 3 ≈ 5541.8 cm 3 19. V = 384π cm 3 ≈ 1206.4 cm 3 21. The volume is multiplied by 27 ___ 25a. 235.6 in 2 25b. 0.04 27. h = 11 ft 29. V = 392π m 3 31. 576 in 3, or 1 __ 3 ft 3 33. 2,468,729 gal 37. A 39. B 41. V = x 3 + x 2 - 2x 43 __________ 4 49. 8 51. S ≈ 181.9 cm 2 125. 23. V ≈ 242.3 ft 3 45. 139 47. 50° 10-7 Check It Out! 1. V = 36 cm 3 2. 107,800 yd 3 or 2,910,600 ft 3 3. V = 216π m 3 ≈ 678.6 m 3 4. The volume is multiplied by 8. 5. V = 3000 ft 3 Exercises 1. perpendicular 3. V = 96 cm 3 5. V ≈ 65 mm 3 7. V = 1440π in 3 ≈ 4523.9 in 3 9
. 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 = 17.5 units 3 39. 3 : 2 41a. 33.5 in 3 b. 134.0 in 3 c. $5; the large size holds 4 times as much. 43. H 45. 9 47. V = 2π ___ 3 ft 3 49. V = 1000 √  2 ______ c m 3 51. 38 and 14 3 53. 79 and 118 55. AA; PQ = 6 57. 10.0; (0.5, 0, -2) 59. 7; (-2, 2, 3.5) 10-8 Check It Out! 1. r = 12 ft 2. about 72.3 times as great 3. S = 2500π cm 2 4. The surface area is divided by 9. 5. S = 57π ft 2 ; V = 27π ft 3 Exercises 1. One endpoint is the center of the sphere, and the other is a point on the sphere. 3. V = 4π ___ 3 m 3 5. about 8 times as great 7. S = 196π cm 2 9. The surface area is multiplied by 4. S108 S108 Selected Answers m 3 units 3 11. S = 36π ft 2 ; V = 92π ___ 3 ft 3 13. V = 972π cm 3 15. d = 36 in. 17. S = 1764π in 2 19. V = 15,625π ______ 6 21. The volume is multiplied by 216. 23. S ≈ 1332.0 mm 2 ; V ≈ 1440.9 mm 3 25. C = 2π √  15 in. _____ 27. S = 196π units 2 ; V = 1372π 3 29. 5.28 in.; 8.87 in 2 ; 2.48 in 3 31. 7.
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 radius of x, so the area is ). 41a. 33.5 in 3 b. 44.6 in 3 43. H 45. 1 in. 47. The volume of the cylinder is 1.5 times the volume of the sphere. 49. y = x 2 + 1 51. 4.6 in 2 53. The volume is multiplied by 27 __ 64. r 2 - x 2, so its area is A = SGR 2. cross section 3. cone; vertex: M; edges: none; base: ⊙L 4. rectangular pyramid; vertices: N, P, Q, R, S; ̶̶ ̶̶ QR, NR, edges: ̶̶ ̶̶ SP ; base: PQRS 5. cylinder RS, 6. square pyramid 7. ̶̶̶ NQ, ̶̶ PQ, ̶̶ NP, ̶̶ NS, 8. 9. 10. 11. yes 12. no 13. V = 9; E = 16; F = 9; 9 - 16 + 9 = 2 14. V = 8; E = 12; F = 6; 8 - 12 + 6 = 2 15. d ≈ 7.7; M (4.5, 3.5, 2.5) 16. d ≈ 10.2; M (2.5, 5, 4) 17. d ≈ 2.4; M (8, 1.5, 5.5) 18. d ≈ 7.5; M (4, 4.5, 6) 19. L ≈ 628.3 yd 2 ; S ≈ 785.4 yd 2 20. L = 100 ft 2 ; S = 150 ft 2 21. L = 126 m 2 ; S ≈ 157.2 m 2 22. L = 160 cm 2 ; S ≈ 215.1 cm 2 23. L = 630 ft 2 ; S = 855 ft 2 24. L = 175π m 2
; 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 ≈ 338.3 cm 2 ; V ≈ 293.5 cm 3 42. S ≈ 245.0 ft 2 ; V ≈ 84.8 ft 3 m 3 Chapter 11 11-1 ̶̶ PQ, ̶̶ QR, ̶̶ Check It Out! 1. chords: ST ; secant:   ST ; tangent:   UV ; diam.: ̶̶ PS 2. radius of ⊙C: radii: 1; radius of ⊙D: 3; pt. of tangency: (2, -1) ; eqn. of tangent line: y = -1 3. 171 mi 4a. 2.1 4b. 7 ̶̶ PT, ̶̶ ST ; ̶̶ PV, ̶̶ PQ, ̶̶̶ VW ; radii: ̶̶̶ PW 13. radius Exercises 1. secant 3. congruent ̶̶ 5. chord: QS ; secant:   QS ; tangent: ̶̶ ̶̶ ̶̶   ST ; diam.: PS PR, QS ; radii: 7. radius of ⊙R: 2; radius of ⊙S: 2; pt. of tangency: (1, 2) ; eqn. of tangent line: x = 1 9. 19 11. chords: ̶̶ ̶̶̶ RS, VW ; secant:   VW ; tangent: ℓ; diam.: of ⊙
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. is 360°. Thus BCDE has 4 rt.  and is a rect. b. 17 in.; 2 in. c. 17.1 in. 39. G 43. 18.6 in. 45. 3 __ 4 47. 13 __ 20 ̶̶ AC 25. �� 11-2 Check It Out! 1a. 108° 1b. 270° 1c. 36° 2a. 140° 2b. 295° 3a. 12 3b. 100° 4. 34.6 Exercises 1. semicircle 3. major arc 5. 162° 7. 61.2° 9. 39.6° 11. 129° 13. 108° 15. 24 17. 24.0 19. 136.3° 21. 136.3° 23. 223.7° 25. 152° 27. 155° 29. 147° 31. 6.6 33. F 35. T 37. 45°; 60°; 75° 39. 108° 41. 1. ⁀ BC ≅ ⁀ DE (Given) 2. m ⁀ BC = m ⁀ DE (Def. of ≅ arcs) 3. m∠BAC = m∠DAE (Def. of arc measures) 4. ∠BAC ≅ ∠DAE (Def. of ≅ ) ̶̶ JK is the ⊥ bisector of ̶̶̶ GH. 43. 1. (Given) 2. A is equidistant from G and H. (Def. of center of ⊙) 3. A lies on the ⊥ bisector of (⊥ Bisector Thm.) 4. (Def
. 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 4b. 3π cm; 9.42 cm Exercises 1. seg. 3. 24π cm 2 ; 75.40 cm 2 5. 12 mi 2 7. 36.23 m 2 9. 4π ft; 12.57 ft 11. 2 __ 3 π in; 2.09 in. 13. 45 __ 2 π in 2 ; 70.69 in 2 15. 628 in 2 17. 15.35 in 2 19. 25 __ 18 π mm; 4.36 mm 21. 1 __ 10 π ft; 0.31 ft 23. N 25. A 27. 12 29a. 3.9 ft b. 103° 33. G 35. 7 __ 3 π 37a. 1 __ 8 b. 3 __ 8 c. 1 __ 2 39. neither 41. 36π cm 3 43. 122° 45. 302° 11-4 Check It Out! 1a. 270° 1b. 38° 2. 43°; 120° 3a. 12 3b. 39° 4. 51°; 129°; 72°; 108° Exercises 1. inscribed 3. 58° 5. 26° 7. 112.5 9. 46° 11. 70°; 110°; 115°; 65° 13. 47.5° 15. 47.6° 17. ±6 19. 100° 21. 100°; 39°, 80°; 141° 23. A 25. S 27. 115° 29a. 30° b. 120° c. Rt.; ∠FBC is inscribed in a semicircle, so it must be a rt
. ∠; 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! 1a. 83° 1b. 142° 2a. 51° 2b. 22° 3. 33 4. 45° 5. 72° ̶̶ AB is ̶̶ AB is a diam. Exercises 1. 70° 3. 122° 5. 67° 7. 94° 9. 58 11. 142° 13. 96° 15. 116° 17. 124° 19. 260° 21. 107.5° 23. 57.5 25. 18 27. 45° 29. 90° 31. 2x° 33. (360 - 2x) ° 39. 150°; 30°; 35° 41a. 60° b. 120° c. obtuse isosceles 43. J 45. Case 1: Assume of the circle. Then m ⁀ AB = 180°, and ∠ABC is a rt. ∠. Thus m∠ABC = 1 __ 2 m ⁀ AB. Case 2: Assume not a diam. of the ⊙. Let X be the ̶̶ center of the ⊙ and draw radii XA ̶̶ ̶̶ XB. Since they are radii, XA and ̶̶ ≅ XB, so △AXB is isosceles. Thus ∠XAB ≅ ∠XBA and 2m∠XBA + m∠AXB = 180. This means that m∠XBA = 90 - 1 __ 2 m∠AXB. By Thm. 11-1-1, ∠XBC is a rt. ∠, so m∠XBA + m∠ABC = 90 or m∠ABC = 90 - m∠XBA. By subst.,
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 Exercises 1. tangent seg. 3. x = 9; AB = 13; CD = 12 5. 51 1 __ 4 ft 7. y = 10.6; PR = 15.6; PT = 13 9. 4 11. √  33 13. x = 4.2; JL = 14.2; MN = 13 15. ≈ 1770 ft 17. y = 14.3; HL = 24.3; NL = 27 19. 2 √  21 21. 4 √  10 23a. 6 in. b. 12 in. 25. x = 8; y = 6 √  3 27. Solution B 33. B 35. CE = ED = 6 and by the Chord-Chord Product Thm., 6 · 6 = 3 · EF. So EF = 12, FB = 15, and the radius AB must be 7.5. 37. 3.2 in. 39. 7.44 41. 72% 43. 45. 22π ft 2 ; 69.12 ft 2 47. 45°  CD 11-7 Check It Out! 1a. x 2 + (y + 3) 2 2 = 64 1b. (x - 2) 2a. + (y + 1) 2 = 16 2b. 3. (2, - 1) Exercises 1. (x - 3) 2 + (y + 5) 2 = 144 3. (x - 4) 2 + y 2 = 4 5. 7. 9a. (-2, 3) b. 10 ft 11. (x - 1.5) 2 + (y +
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) 2 + (z - 3) 2 = 69 b. 15; if 2 segs. are tangent to a ⊙ or sphere from the same ext. pt., then the segs. are ≅. 47. (x - 3) 2 + (y - 4) 2 = 5 49. 9a + 2 51. 8 53. 196° Selected Answers S109 S109 �� 9. 11. 13. no 15. yes 17. 19. 21. 23. 27. 29. 31. (5, 2) → (5, -2) 33. (0, 12) → (0, -12) 35. (0, -5) → (-5, 0) 37a. no b. (7, 4) c. (6, 3.5) 39. y = x 41. 47. J 49. (4, 4) 51. (-1, 5) 53. Use the fact that the reflection of a seg. is ≅ to the preimage and the def. of ≅ segs. 55. Use the fact that the reflection of a seg. is ≅ to the preimage to prove △ABC ≅ △A′B′C′ by SSS. 59. 25 __ 36 61. 11 cm 63. 13.2 cm 65. 41° 12-2 Check It Out! 1a. Yes 1b. no 2. 3. 4. (16, -24) Exercises 1. no 3. yes 7. 9. 11. yes 13. no SGR
̶̶ 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. 81° 12. 210° 13. 99° 14. 279° 15. 17.0 16. 8.7 17. 12π in 2 ; 37.70 in 2 18. π __ 4 m 2 ; 0.79 m 2 19. 16π cm; 50.27 cm 20. 3π ft; 9.42 ft 21. 164° 22. 32° 23. 26 24. 39° 25. 82° 26. 79° 27. 67° 28. 90° 29. 11 2 __ 3 ; DE = 12; BC = 14 2 __ 3 30. 12; RQ = 22; ST = 23 31. 7; JG = 10; JL = 12 32. 8 1 __ 2 ; AC = 12 1 __ 2 ; AE = 10 33. (x + 4) 2 + (y + 3) 2 = 9 34. (x + 2) 2 + y 2 = 4 35. (x - 1) 2 + (y + 1) 2 = 16 36. Chapter 12 12-1 Check It Out! 1a. no 1b. yes 2. ̶̶ AX and ̶̶ BX would be ≅. 3. 4. Exercises 1. They are ≅. 3. no 5. no 7. S110 S110 Selected Answers �� Exercises 1. yes 3. no 5. 7. 9
. 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 seg. is ≅ to the preimage and the def. of ≅ segs. 49. Use the fact that the rotation of a seg. is ≅ to the preimage to prove △ABC ≅ △A′B′C′ by SSS. 51. If A, B, and C are collinear, then one pt. is between the other two. Case 1: If C is between A and B, then AC + BC = AB. Use the fact that the rotation of a seg. is ≅ to the preimage to prove A′C′ + B′C′ = A′B′. Then C′ is between A′ and B′, so A′, B′, and C′ are collinear. Prove the other two cases similarly. 53. - 3 __ 2, 4 55. 94° 57. 〈4, -9〉 59. 〈3, 2〉 12-4 Check It Out! 1. 21. 17. 19. 21. 23a. 1 __ 4 b. 1 __ 2. c. 0 27. No; there are no fixed pts. because, by def. of a translation, every pt. must move by the same distance. 29. 〈4, 0〉, (-3, 2) → (1, 2) 31. (-3, -2), (3, -1), (0, -3) 33. (-3, 1), (3, -1) → (0, 0) 39. A 41. C 43a. the vector b. 3.46 cm 45. Use the fact that the reflection of a seg. is ≅ to the preimage and the def. of ≅ segs. 47. Use the fact that the translation of a seg. is ≅ to the preimage to prove �
�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. Then reflect the image across a line. 3. 3. ̶̶ ST 31a. 72° 27. T 29. 31b. (4.9, 5.9) 33. 4. (20.9, 64.2) 35a. 90° 35b. 6 hours 37. No; although all pts. are rotated 5. a rotation of 100° about the pt. of intersection of the lines Selected Answers S111 S111 �� 7. 9. 11a. The move is a horiz. or vert. translation by 2 spaces followed by a vert. or horiz. translation by 1 space. 11b. 1b. yes; 1 line of symmetry 1c. yes; 1 line of symmetry 35. line symmetry; x = 2 37a. no b. yes; 180°; 2. c. Yes; if color is not taken into account the ∠ of rotational symmetry is 90. 39. parallelogram 41. square 43. 15° 45. It has rotational symmetry of order 3, with an ∠ of rotational symmetry of 120°. 47. 2a. yes; 120°; order: 3 2b. yes; 180°; order: 2° 2c. no 3a. line symmetry and rotational symmetry; 72°; order: 49. 51. A 53. C 55. 72 57. x = -4 59. x = 0 61. 7 63. $246.40 65
. 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. neither 23. isosc. 25. scalene 27. 0 29. line symmetry 17. never 19. always 23. A 25. C 29. yes 31. 6.4 33. 8 35. N′ (1, 3) 12-5 31. line symmetry Check It Out! 1a. yes; 2 lines of symmetry 33. rotational symmetry of order 4 S112 S112 Selected Answers Check It Out! 1a. translation symmetry 1b. translation symmetry and glide reflection symmetry 2. 3a. regular 3b. neither. 3c. semiregular 4a. yes 4b. no Exercises 3. translation symmetry and glide reflection symmetry 5. translation symmetry and glide reflection symmetry 9. regular 11. semiregular 13. yes; possible answer 15. translation symmetry 17. translation symmetry 19. 21. neither 23. neither 25. no 27. translation symmetry and glide reflection symmetry �� 29. translation, reflection, rotation 31. always 33. always 35. never 41. The tessellation has translation symmetry, reflection symmetry, and order 3 rotation symmetry. 43. 21. 23. 11. 12. 47. H 51. yes 53. 7.5% 55. (x + 2) 2 + (y - 3) 2 = 5 57. - (x - 5) 2 + (y + 3) 2 = 20 59. angle of rotational symmetry: 72°; order: 5 12-7 Check It Out! 1a. no 1b. yes 2. 3. 1600 in 2 4. Exercises 1. The center is the origin; the scale factor is 3. 3. yes 5. yes 7. 9. 11. 13. yes 15. no 19. 108 25. ABCDE ∼
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. 19. 20. 10. 21. yes 22. yes 23. no 24. no 25. Selected Answers S113 S113 �� 26. 27. 28. 29. 30. 40. 41. neither 42. semiregular 43. yes 44. yes 45. 31. yes 32. yes 46. 33. yes; 120°; 3 34. no 35. yes; 120°; 3 36. yes; 180°; 2 37. 38. 39. S114 S114 Selected Answers �� Glossary/Glosario AA A ENGLISH acute angle (p. 21) An angle that measures greater than 0° and less than 90°. SPANISH ángulo agudo Ángulo que mide más de 0° y menos de 90°. EXAMPLES acute triangle (p. 216) A triangle with three acute angles. triángulo acutángulo Triángulo con tres ángulos agudos. adjacent angles (p. 28) Two angles in the same plane with a common vertex and a common side, but no common interior points. ángulos adyacentes Dos ángulos en el mismo plano que tienen un vértice y un lado común per
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 sides of the transversal and outside the other two lines. ángulos alternos externos Dadas dos rectas cortadas por una transversal, par de ángulos no adyacentes ubicados en los lados opuestos de la transversal y fuera de las otras dos rectas. alternate interior angles (p. 147) For two lines intersected by a transversal, a pair of nonadjacent angles that lie on opposite sides of the transversal and between the other two lines. ángulos alternos internos Dadas dos rectas cortadas por una transversal, par de ángulos no adyacentes ubicados en los lados opuestos de la transversal y entre las otras dos rectas. altitude of a cone (p. 690) A segment from the vertex to the plane of the base that is perpendicular to the plane of the base. altura de un cono Segmento que se extiende desde el vértice hasta el plano de la base y es perpendicular al plano de la base. altitude of a cylinder (p. 680) A segment with its endpoints on the planes of the bases that is perpendicular to the planes of the bases. altura de un cilindro Segmento con sus extremos en los planos de las bases que es perpendicular a los planos de las bases. ∠1 and ∠2 are adjacent angles. ⁀ RS and ⁀ ST are adjacent arcs. ∠4 and ∠5 are alternate exterior angles. ∠3 and ∠6 are alternate interior angles. Glossary/Glosario S115 S115 �� EXAMPLES ENGLISH altitude of a prism (p. 680) A segment with its endpoints on the planes of the bases that is perpendicular to the planes of the bases. altitude of a pyramid (p. 689
) 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 y es perpendicular al plano de la base. altitude of a triangle (p. 316) A perpendicular segment from a vertex to the line containing the opposite side. altura de un triángulo Segmento perpendicular que se extiende desde un vértice hasta la recta que forma el lado opuesto. ambiguous case of the Law of Sines (p. 556) If two sides and a nonincluded angle of a triangle are given in order to solve the triangle using the Law of Sines, it is possible to have two different answers. angle (p. 20) A figure formed by two rays with a common endpoint. caso ambiguo de la Ley de los senos Si se conocen dos lados y un ángulo no incluido de un triángulo y se quiere resolver el triángulo aplicando la Ley de los senos, es posible obtener dos respuestas diferentes. ángulo Figura formada por dos rayos con un extremo común. angle bisector (p. 23) A ray that divides an angle into two congruent angles. bisectriz de un ángulo Rayo que divide un ángulo en dos ángulos congruentes. JK is an angle bisector of ∠LJM.  angle of depression (p. 544) The angle formed by a horizontal line and a line of sight to a point below. ángulo de depresión Ángulo formado por una recta horizontal y una línea visual a un punto inferior. angle of elevation (p. 544) The angle formed by a horizontal line and a line of sight to a point above. ángulo de elevación Ángulo formado por una recta horizontal y una línea visual a un punto
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 symmetry (p. 857) The smallest angle through which a figure with rotational symmetry can be rotated to coincide with itself. ángulo de simetría de rotación El ángulo más pequeño alrededor del cual se puede rotar una figura con simetría de rotación para que coincida consigo misma. The angle of rotation is 135°. S116 S116 Glossary/Glosario �� ENGLISH annulus (p. 612) The region between two concentric circles. SPANISH corona circular Región comprendida entre dos círculos concéntricos. EXAMPLES apothem (p. 601) The perpendicular distance from the center of a regular polygon to a side of the polygon. apotema Distancia perpendicular desde el centro de un polígono regular hasta un lado del polígono. arc (p. 756) An unbroken part of a circle consisting of two points on the circle, called the endpoints, and all the points on the circle between them. arco Parte continua de un círculo formada por dos puntos del círculo denominados extremos y todos los puntos del círculo comprendidos entre éstos. arc length (p. 766) The distance along an arc measured in linear units. longitud de arco Distancia a lo largo de un arco medida en unidades lineales. arc marks (p. 22) Marks used on a figure to indicate congruent angles. marcas de arco Marcas utilizadas en una figura para indicar ángulos congruentes. m ⁀ CD = 5π ft area (p. 36) The number of nonoverlapping unit
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 para describir una transformación. The area is 10 square units. auxiliary line (p. 223) A line drawn in a figure to aid in a proof. recta auxiliar Recta dibujada en una figura como ayuda en una demostración. axiom (p. 7) See postulate. axioma Ver postulado. axis of a cone (p. 690) The segment with endpoints at the vertex and the center of the base. eje de un cono Segmento cuyos extremos se encuentran en el vértice y en el centro de la base. Glossary/Glosario S117 S117 �� ENGLISH axis of a cylinder (p. 681) The segment with endpoints at the centers of the two bases. SPANISH eje de un cilindro Segmentos cuyos extremos se encuentran en los centros de las dos bases. EXAMPLES axis of symmetry (p. 858) A line that divides a plane figure or a graph into two congruent reflected halves. eje de simetría Línea que divide una figura plana o una gráfica en dos mitades reflejadas congruentes. B base angle of a trapezoid (p. 429) One of a pair of consecutive angles whose common side is a base of the trapezoid. ángulo base de un trapecio Uno de los dos ángulos consecutivos cuyo lado en común es la base del trapecio. base angle of an isosceles triangle (p. 273) One of the two angles that have the base of the triangle as a side. ángulo base de un triángulo isósceles Uno de los dos �
�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. 429) A side of a polygon; a face of a three-dimensional figure by which the figure is measured or classified. base de una figura geométrica Lado de un polígono; cara de una figura tridimensional por la cual se mide o clasifica la figura. base of a prism (p. 654) One of the two congruent parallel faces of the prism. base de un prisma Una de las dos caras paralelas y congruentes del prisma. base of a pyramid (p. 654) The face of the pyramid that is opposite the vertex. base de una pirámide Cara de la pirámide opuesta al vértice. base of a trapezoid (p. 429) One of the two parallel sides of the trapezoid. base de un trapecio Uno de los dos lados paralelos del trapecio. base of a triangle (p. 36) Any side of a triangle. base de un triángulo Cualquier lado de un triángulo. S118 S118 Glossary/Glosario �� ENGLISH base of an isosceles triangle (p. 273) The side opposite the vertex angle. SPANISH base de un triángulo isósceles Lado opuesto al ángulo del vértice. EXAMPLES bearing (p. 252) Indicates direction. The number of degrees in the angle whose initial side is a line due north and whose terminal side is determined by a clockwise rotation. rumbo Indica dirección. La cantidad de grados en el ángulo cuyo lado inicial es una línea rect
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 sólo si los tres puntos se encuentran en la misma línea y AB + BC = AC. biconditional statement (p. 96) A statement that can be written in the form “p if and only if q.” enunciado bicondicional Enunciado que puede expresarse en la forma “p si y sólo si q”. A figure is a triangle if and only if it is a three-sided polygon. bisect (p. 15) To divide into two congruent parts. trazar una bisectriz Dividir en dos partes congruentes. JK bisects ∠LJM.  C Cartesian coordinate system (p. 808) See coordinate plane. sistema de coordenadas cartesianas Ver plano cartesiano. center of a circle (p. 600) The point inside a circle that is the same distance from every point on the circle. centro de un círculo Punto dentro de un círculo que se encuentra a la misma distancia de todos los puntos del círculo. center of a regular polygon (p. 601) The point that is equidistant from all vertices of the regular polygon. centro de un polígono regular Punto equidistante de todos los vértices del polígono regular. center of a sphere (p. 714) The point inside a sphere that is the same distance from every point on the sphere. centro de una esfera Punto dentro de una esfera que está a la misma distancia de cualquier punto de la esfera. center of dilation (p. 873) The intersection of
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 rotation (p. 840) The point around which a figure is rotated. SPANISH centro de rotación Punto alrededor del cual rota una figura. EXAMPLES central angle of a circle (p. 756) An angle whose vertex is the center of a circle. ángulo central de un círculo Ángulo cuyo vértice es el centro de un círculo. central angle of a regular polygon (p. 601) An angle whose vertex is the center of the regular polygon and whose sides pass through consecutive vertices. ángulo central de un polígono regular Ángulo cuyo vértice es el centro del polígono regular y cuyos lados pasan por vértices consecutivos. centroid of a triangle (p. 314) The point of concurrency of the three medians of a triangle. Also known as the center of gravity. centroide de un triángulo Punto donde se encuentran las tres medianas de un triángulo. También conocido como centro de gravedad. chord (p. 746) A segment whose endpoints lie on a circle. cuerda Segmento cuyos extremos se encuentran en un círculo. The centroid is P. circle (p. 600) The set of points in a plane that are a fixed distance from a given point called the center of the circle. circle graph (p. 755) A way to display data by using a circle divided into non-overlapping sectors. círculo Conjunto de puntos en un plano que se encuentran a una distancia fija de un punto determinado denominado centro del círculo. gráfica circular Forma de mostrar datos mediante un cí
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 Distancia alrededor del círculo. The circumcenter is P. S120 S120 Glossary/Glosario �� ENGLISH circumscribed circle (p. 308) Every vertex of the polygon lies on the circle. SPANISH círculo circunscrito Todos los vértices del polígono se encuentran sobre el círculo. EXAMPLES circumscribed polygon (p. 599) Each side of the polygon is tangent to the circle. polígono circunscrito Todos los lados del polígono son tangentes al círculo. coincide (p. 221) To correspond exactly; to be identical. coincidir Corresponder exactamente, ser idéntico. collinear (p. 6) Points that lie on the same line. colineal Puntos que se encuentran sobre la misma línea. common tangent (p. 748) A line that is tangent to two circles. tangente común Recta que es tangente a dos círculos. K, L, and M are collinear points. complement of an angle (p. 29) The sum of the measures of an angle and its complement is 90°. complemento de un ángulo La suma de las medidas de un ángulo y su complemento es 90°. The complement of a 53° angle is a 37° angle. complement of an event (p. 628) All outcomes in the sample space that are not in an ̶ E. event E, denoted complemento de un suceso Todos los resultados en el espacio muestral que no están en el ̶ E. suceso
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 vertical and horizontal change from the initial point to the terminal point. forma de componente Forma de un vector que muestra el cambio horizontal y vertical desde el punto inicial hasta el punto terminal. The component form of  CD is 〈2, 3〉. composite figure (p. 600) A plane figure made up of triangles, rectangles, trapezoids, circles, and other simple shapes, or a three-dimensional figure made up of prisms, cones, pyramids, cylinders, and other simple threedimensional figures. figura compuesta Figura plana compuesta por triángulos, rectángulos, trapecios, círculos y otras formas simples, o figura tridimensional compuesta por prismas, conos, pirámides, cilindros y otras figuras tridimensionales simples. Glossary/Glosario S121 S121 �� ENGLISH SPANISH EXAMPLES composition of transformations (p. 848) One transformation followed by another transformation. composición de transformaciones Una transformación seguida de otra transformación. compound statement (p. 128) Two statements that are connected by the word and or or. enunciado compuesto Dos enunciados unidos por la palabra y u o. The sky is blue and the grass is green. I will drive to school or I will take the bus. concave polygon (p. 383) A polygon in which a diagonal can be drawn such that part of the diagonal contains points in the exterior of the polygon. polígono cóncavo Polígono en el cual se puede trazar una diagonal tal que parte de la diagonal contiene puntos ubicados fu
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 + 1 = 5, then x = 4. Conclusion concurrent (p. 307) Three or more lines that intersect at one point. concurrente Tres o más líneas rectas que se cortan en un punto. conditional statement (p. 81) A statement that can be written in the form “if p, then q,” where p is the hypothesis and q is the conclusion. enunciado condicional Enunciado que se puede expresar como “si p, entonces q”, donde p es la hipótesis y q es la conclusión. cone (p. 654) A three-dimensional figure with a circular base and a curved lateral surface that connects the base to a point called the vertex. cono Figura tridimensional con una base circular y una superficie lateral curva que conecta la base con un punto denominado vértice. congruence statement (p. 231) A statement that indicates that two polygons are congruent by listing the vertices in the order of correspondence. enunciado de congruencia Enunciado que indica que dos polígonos son congruentes enumerando los vértices en orden de correspondencia. congruence transformation (p. 824) See isometry. transformación de congruencia Ver isometría. congruent (p. 13) Having the same size and shape, denoted by ≅. congruente Que tiene el mismo tamaño y forma, expresado por ≅. S122 S122 Glossary/Glosario If x + 1 = 5, then x = 4. Hypothesis Conclusion △HKL ≅ △YWK ̶̶ PQ ≅ ̶̶ SR �� ENGLISH congru
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 congruentes y que tienen la misma medida. congruent circles (p. 747) Two circles that have congruent radii. círculos congruentes Dos círculos que tienen radios congruentes. congruent polygons (p. 231) Two polygons whose corresponding sides and angles are congruent. polígonos congruentes Dos polígonos cuyos lados y ángulos correspondientes son congruentes. congruent segments (p. 13) Two segments that have the same length. segmentos congruentes Dos segmentos que tienen la misma longitud. conjecture (p. 74) A statement that is believed to be true. conjetura Enunciado que se supone verdadero. ̶̶ PQ ≅ ̶̶ SR A sequence begins with the terms 2, 4, 6, 8, 10. A reasonable conjecture is that the next term in the sequence is 12. conjunction (p. 128) A compound statement that uses the word and. conjunción Enunciado compuesto que contiene la palabra y. 3 is less than 5 AND greater than 0. consecutive interior angles (p. 147) See same-side interior angles. ángulos internos consecutivos Ver ángulos internos del mismo lado. construction (p. 14) A method of creating a figure that is considered to be mathematically precise. Figures may be constructed by using a compass and straightedge, geometry software, or paper folding. construcción Método para crear una figura que es considerado matemáticamente preciso. Se pueden construir figuras utilizando un compás y una regla, un programa de
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 conclusión de un enunciado condicional. Statement: If n + 1 = 3, then n = 2 Contrapositive: If n ≠ 2, then n + 1 ≠ 3 converse (p. 83) The statement formed by exchanging the hypothesis and conclusion of a conditional statement. expresión recíproca Enunciado que se forma intercambiando la hipótesis y la conclusión de un enunciado condicional. Statement: If n + 1 = 3, then n = 2 Converse: If n = 2, then n + 1 = 3 Glossary/Glosario S123 S123 �� ENGLISH convex polygon (p. 383) A polygon in which no diagonal contains points in the exterior of the polygon. SPANISH EXAMPLES polígono convexo Polígono en el cual ninguna diagonal contiene puntos fuera del polígono. coordinate (p. 13) A number used to identify the location of a point. On a number line, one coordinate is used. On a coordinate plane, two coordinates are used, called the x-coordinate and the y-coordinate. In space, three coordinates are used, called the x-coordinate, the y-coordinate, and the z-coordinate. coordenada Número utilizado para identificar la ubicación de un punto. En una recta numérica se utiliza una coordenada. En un plano cartesiano se utilizan dos coordenadas, denominadas coordenada x y coordenada y. En el espacio se utilizan tres coordenadas, denominadas coordenada x, coordenada y y coordenada z. coordinate plane (p. 43) A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y
-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 Tipo de demostración que utiliza geometría de coordenadas y álgebra. coplanar (p. 6) Points that lie in the same plane. coplanar Puntos que se encuentran en el mismo plano. corollary (p. 224) A theorem whose proof follows directly from another theorem. corolario Teorema cuya demostración proviene directamente de otro teorema. corresponding angles of lines intersected by a transversal (p. 147) For two lines intersected by a transversal, a pair of angles that lie on the same side of the transversal and on the same sides of the other two lines. ángulos correspondientes de líneas cortadas por una transversal Dadas dos rectas cortadas por una transversal, el par de ángulos ubicados en el mismo lado de la transversal y en los mismos lados de las otras dos rectas. ∠1 and ∠3 are corresponding. corresponding angles of polygons (p. 231) Angles in the same position in two different polygons that have the same number of angles. ángulos correspondientes de los polígonos Ángulos que tienen la misma posición en dos polígonos diferentes que tienen el mismo número de ángulos. corresponding sides of polygons (p. 231) Sides in the same position in two different polygons that have the same number of sides. lados correspondientes de los polígonos Lados que tienen la misma posición en dos polígonos diferentes que tienen el mismo número de lados. ∠A and ∠D are corresponding angles. ̶̶ AB and ̶̶ DE are
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 function. cosecante En un triángulo rectángulo, la cosecante del ángulo A es la razón entre la longitud de la hipotenusa y la longitud del cateto opuesto a A. Es la inversa de la función seno. cosine (p. 525) In a right triangle, the cosine of angle A is the ratio of the length of the leg adjacent to angle A to the length of the hypotenuse. It is the reciprocal of the secant function. coseno En un triángulo rectángulo, el coseno del ángulo A es la razón entre la longitud del cateto adyacente al ángulo A y la longitud de la hipotenusa. Es la inversa de la función secante. cotangent (p. 532) In a right triangle, the cotangent of angle A is the ratio of the length of the side adjacent to A to the length of the side opposite A. It is the reciprocal of the tangent function. cotangente En un triángulo rectángulo, la cotangente del ángulo A es la razón entre la longitud del cateto adyacente a A y la longitud del cateto opuesto a A. Es la inversa de la función tangente. counterexample (p. 75) An example that proves that a conjecture or statement is false. contraejemplo Ejemplo que demuestra que una conjetura o enunciado es falso. CPCTC (p. 260) An abbreviation for “Corresponding Parts of Congruent Triangles are Congruent,” which can be used as a justification in a proof after two triangles are proven congruent. PCTCC Ab
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 indica la cantidad de partes iguales que hay en un entero. The denominator of 3 _ 7 is 7. diagonal of a polygon (p. 382) A segment connecting two nonconsecutive vertices of a polygon. diagonal de un polígono Segmento que conecta dos vértices no consecutivos de un polígono. diagonal of a polyhedron (p. 671) A segment whose endpoints are vertices of two different faces of a polyhedron. diagonal de un poliedro Segmento cuyos extremos son vértices de dos caras diferentes de un poliedro. diameter (p. 37) A segment that has endpoints on the circle and that passes through the center of the circle; also the length of that segment. diámetro Segmento que atraviesa el centro de un círculo y cuyos extremos están sobre el círculo; longitud de dicho segmento. dilation (p. 495) A transformation in which the lines connecting every point P with its preimage P′ all intersect at a point C known as the center of dilation, and CP′ ___ CP is the same for every point P; a transformation that changes the size of a figure but not its shape. dilatación Transformación en la cual las rectas que conectan cada punto P con su imagen original P′ se cruzan en un punto C conocido como centro de dilatación, y CP′ ___ CP para cada punto P; transformación que cambia el tamaño de una figura pero no su forma. es igual direct reasoning (p. 332) The process of reasoning that begins with a true hypothesis and builds a logical argument to show that a conclusion is true. razonamiento
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 be written in the form y = kx, where k is a nonzero constant. direction of a vector (p. 560) The orientation of a vector, which is determined by the angle the vector makes with a horizontal line. variación directa Relación lineal entre dos variables, x e y, que puede expresarse en la forma y = kx, donde k es una constante distinta de cero. dirección de un vector Orientación de un vector, determinada por el ángulo que forma el vector con una recta horizontal. disjunction (p. 128) A compound statement that uses the word or. disyunción Enunciado compuesto que contiene la palabra o. John will walk to work or he will stay home. distance between two points (p. 13) The absolute value of the difference of the coordinates of the points. distancia entre dos puntos Valor absoluto de la diferencia entre las coordenadas de los puntos. distance from a point to a line (p. 172) The length of the perpendicular segment from the point to the line. distancia desde un punto hasta una línea Longitud del segmento perpendicular desde el punto hasta la línea. dodecagon (p. 382) A 12-sided polygon. dodecágono Polígono de 12 lados. dodecahedron (p. 669) A polyhedron with 12 faces. The faces of a regular dodecahedron are regular pentagons, with three faces meeting at each vertex. dodecaedro Poliedro con 12 caras. Las caras de un dodecaedro regular son pentágonos regulares, con tres caras que concurren en cada vértice. E edge of a graph (
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 intersección de dos caras de la figura. endpoint (p. 7) A point at an end of a segment or the starting point of a ray. extremo Punto en el final de un segmento o punto de inicio de un rayo. enlargement (p. 873) A dilation with a scale factor greater than 1. In an enlargement, the image is larger than the preimage. agrandamiento Dilatación con un factor de escala mayor que 1. En un agrandamiento, la imagen es más grande que la imagen original. The distance from P to   AC is 5 units. Glossary/Glosario S127 S127 �� ENGLISH equal vectors (p. 561) Two vectors that have the same magnitude and the same direction. SPANISH EXAMPLES vectores iguales Dos vectores de la misma magnitud y con la misma dirección. equiangular polygon (p. 382) A polygon in which all angles are congruent. polígono equiangular Polígono cuyos ángulos son todos congruentes. equiangular triangle (p. 216) A triangle with three congruent angles. triángulo equiangular Triángulo con tres ángulos congruentes. equidistant (p. 300) The same distance from two or more objects. equidistante Igual distancia de dos o más objetos. equilateral polygon (p. 382) A polygon in which all sides are congruent. equilateral triangle (p. 217) A triangle with three congruent sides. polígono equilátero Polígono cuyos lados son todos congru
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. geometría euclidiana Sistema geométrico desarrollado por Euclides. Específicamente, el sistema de la geometría euclidiana cumple con el postulado de las paralelas, que establece que por un punto dado se puede trazar una única recta paralela a una recta dada. Euler line (p. 321) The line containing the circumcenter (U ), centroid (C ), and orthocenter (O ) of a triangle. recta de Euler Recta que contiene el circuncentro (U ), el centroide (C ) y el ortocentro (O ) de un triángulo. X is equidistant from A and B. event (p. 628) An outcome or set of outcomes in a probability experiment. suceso Resultado o conjunto de resultados en un experimento de probabilidades. In the experiement of rolling a number cube, the event “an odd number” consists of the outcomes 1, 3, 5. expansion (p. 873) See enlargement. expansión Ver agrandamiento. experiment (p. 628) An operation, process, or activity in which outcomes can be used to estimate probability. experimento Una operación, proceso o actividad en la que se usan los resultados para estimar una probabilidad. Tossing a coin 10 times and noting the number of heads. S128 S128 Glossary/Glosario �� ENGLISH exterior of a circle (p. 746) The set of all points outside a circle. SPANISH exterior de un círculo Conjunto de todos los puntos que se encuentran fuera de un círculo. EXAMPLES �� exterior of
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 angle of a polygon (p. 225) An angle formed by one side of a polygon and the extension of an adjacent side. ángulo externo de un polígono Ángulo formado por un lado de un polígono y la prolongación del lado adyacente. external secant segment (p. 793) A segment of a secant that lies in the exterior of the circle with one endpoint on the circle. segmento secante externo Segmento de una secante que se encuentra en el exterior del círculo y tiene un extremo sobre el círculo. extremes of a proportion (p. 455) In the proportion = c __ a __, a and d are the extremes. b d If the proportion is written as a:b = c:d, the extremes are in the first and last positions. valores extremos de una proporción En la proporción a __ = c __, a y d son los d b valores extremos. Si la proporción se expresa como a:b = c:d, los extremos están en la primera y última posición. F face of a polyhedron (p. 654) A flat surface of the polyhedron. cara de un poliedro Superficie plana de un poliedro. ∠4 is an exterior angle. ̶̶̶ NM is an external secant segment. fair (p. 628) When all outcomes of an experiment are equally likely. justo Cuando todos los resultados de un experimento son igualmente probables. When tossing a fair coin, heads and tails are equally likely. Each has a probability of 1 __. 2 Fibonacci sequence (p. 78) The infinite sequence of numbers beginning with 1, 1, … such
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 reflection. SPANISH inversión Ver reflexión. EXAMPLES flowchart proof (p. 118) A style of proof that uses boxes and arrows to show the structure of the proof. demostración con diagrama de flujo Tipo de demostración que se vale de cuadros y flechas para mostrar la estructura de la prueba. fractal (p. 882) A figure that is generated by iteration. fractal Figura generada por iteración. frieze pattern (p. 863) A pattern that has translation symmetry along a line. patrón de friso Patrón con simetría de traslación a lo largo de una línea. frustum of a cone (p. 668) A part of a cone with two parallel bases. tronco de cono Parte de un cono con dos bases paralelas. frustum of a pyramid (p. 696) A part of a pyramid with two parallel bases. tronco de pirámide Parte de una pirámide con dos bases paralelas. function (p. 389) A relation in which every input is paired with exactly one output. función Una relación en la que cada entrada corresponde exactamente a una salida. Function:  (0, 5), (1, 3), (2, 1), (3, 3)  Not a Function:  (0, 1), (0, 3), (2, 1), (2, 3)  G geometric mean (p. 519) For positive numbers a and b, the positive number x such that a __ x = x __. In a geometric sequence, b a term that
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 probability determined by a ratio of geometric measures such as lengths, areas, or volumes. probabilidad geométrica Método para calcular probabilidades basado en una medida geométrica como la longitud o el área. glide reflection (p. 848) A composition of a translation and a reflection across a line parallel to the translation vector. deslizamiento con inversión Composición de una traslación y una reflexión sobre una línea paralela al vector de traslación. S130 S130 Glossary/Glosario a x _ _ = x b x 2 = ab x = √  ab The probability of the pointer landing on red is 2 __. 9 �� ENGLISH SPANISH EXAMPLES glide reflection symmetry (p. 863) A pattern has glide reflection symmetry if it coincides with its image after a glide reflection. simetría de deslizamiento con inversión Un patrón tiene simetría de deslizamiento con inversión si coincide con su imagen después de un deslizamiento con inversión. golden ratio (p. 460) If a segment is divided into two parts so that the ratio of the lengths of the whole segment to the longer part equals the ratio of the lengths of the longer part to the shorter part, then that ratio is called the golden ratio. The golden ratio is equal to 1 + √  5 ______ 2 ≈ 1.618. razón áurea Si se divide un
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 (p. 460) A rectangle in which the ratio of the lengths of the longer side to the shorter side is the golden ratio. rectángulo áureo Rectángulo en el cual la razón entre la longitud del lado más largo y la longitud del lado más corto es la razón áurea. great circle (p. 714) A circle on a sphere that divides the sphere into two hemispheres. círculo máximo En una esfera, círculo que divide la esfera en dos hemisferios. H head-to-tail method (p. 561) A method of adding two vectors by placing the tail of the second vector on the head of the first vector; the sum is the vector drawn from the tail of the first vector to the head of the second vector. height of a figure (p. 36) The length of an altitude of the figure. método de cola a punta Método para sumar dos vectores colocando la cola del segundo vector en la punta del primer vector. La suma es el vector trazado desde la cola del primer vector hasta la punta del segundo vector. altura de una figura Longitud de la altura de la figura. height of a triangle (p. 36) A segment from a vertex that forms a right angle with a line containing the base. altura de un triángulo Segmento que se extiende desde el vértice y forma un ángulo recto con la línea de la base. hemisphere (p. 714) Half of a sphere. hemisferio Mitad de una esfera. Golden ratio = AC _
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 Polígono de seis lados. horizon (p. 662) The horizontal line in a perspective drawing that contains the vanishing point(s). horizonte Línea horizontal en un dibujo en perspectiva que contiene el punto de fuga o los puntos de fuga. hypotenuse (p. 45) The side opposite the right angle in a right triangle. hipotenusa Lado opuesto al ángulo recto de un triángulo rectángulo. hypothesis (p. 81) The part of a conditional statement following the word if. hipótesis La parte de un enunciado condicional que sigue a la palabra si. If x + 1 = 5, then x = 4. Hypothesis I icosahedron (p. 669) A polyhedron with 20 faces. A regular icosahedron has equilateral triangles as faces, with 5 faces meeting at each vertex. icosaedro Poliedro con 20 caras. Las caras de un icosaedro regular son triángulos equiláteros y cada vértice es compartido por 5 caras. identity (p. 531) An equation that is true for all values of the variables. identidad Ecuación verdadera para todos los valores de las variables. 3 = 3 2 (x - 1) = 2x - 2 image (p. 50) A shape that results from a transformation of a figure known as the preimage. imagen Forma resultante de la transformación de una figura conocida como imagen original. incenter of a triangle (p. 309) The point of concurrency of the three angle bisectors of a triangle. incentro de un triángulo Punto
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 ��HPGJMKL�� ENGLISH SPANISH EXAMPLES included side (p. 252) The common side of two consecutive angles of a polygon. lado incluido Lado común de dos ángulos consecutivos de un polígono. ̶̶ PQ is the included side between ∠P and ∠Q. indirect measurement (p. 488) A method of measurement that uses formulas, similar figures, and/or proportions. medición indirecta Método para medir objetos mediante fórmulas, figuras similares y/o proporciones. indirect proof (p. 332) A proof in which the statement to be proved is assumed to be false and a contradiction is shown. demostración indirecta Prueba en la que se supone que el enunciado a demostrar es falso y se muestra una contradicción. indirect reasoning (p. 332) See indirect proof. razonamiento indirecto Ver demostración indirecta. inductive reasoning (p. 74) The process of reasoning that a rule or statement is true because specific cases are true. razonamiento inductivo Proceso de razonamiento por el que se determina que una regla o enunciado son verdaderos porque ciertos casos específicos son verdaderos. inequality (p. 92) A statement that compares two expressions by using one of the following signs: <, >, ≤, ≥, or ≠. desigualdad Enunciado que compara dos expresiones utilizando uno de los siguientes signos: <, >, ≤, ≥ o ≠. x > 2 initial point of
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 estándar. inscribed angle (p. 772) An angle whose vertex is on a circle and whose sides contain chords of the circle. ángulo inscrito Ángulo cuyo vértice se encuentra sobre un círculo y cuyos lados contienen cuerdas del círculo. inscribed circle (p. 309) A circle in which each side of the polygon is tangent to the circle. círculo inscrito Círculo en el que cada lado del polígono es tangente al círculo. inscribed polygon (p. 599) A polygon in which every vertex of the polygon lies on the circle. polígono inscrito Polígono cuyos vértices se encuentran sobre el círculo. Glossary/Glosario S133 S133 ��A��B��v��yx��DEF ENGLISH integer (p. 559) A member of the set of whole numbers and their opposites. SPANISH EXAMPLES entero Miembro del conjunto de números cabales y sus opuestos. … -3, -2, -1, 0, 1, 2, 3, … intercepted arc (p. 772) An arc that consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between the endpoints. arco abarcado Arco cuyos extremos se encuentran en los lados de un ángulo inscrito y consta de todos los puntos del círculo ubicados entre dichos extremos. interior angle (p. 225) An angle formed by two sides of a polygon with a common vertex. ángulo intern
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) The set of all points between the sides of an angle. interior de un ángulo Conjunto de todos los puntos entre los lados de un ángulo. interior of a polygon (p. 225) The set of all points inside a polygon. interior de un polígono Conjunto de todos los puntos que se encuentran dentro de un polígono. inverse (p. 83) The statement formed by negating the hypothesis and conclusion of a conditional statement. inverse cosine (p. 534) The measure of an angle whose cosine ratio is known. inverse function (p. 533) The function that results from exchanging the input and output values of a one-to-one function. The inverse of f(x) is denoted f -1 (x). inverso Enunciado formado al negar la hipótesis y la conclusión de un enunciado condicional. coseno inverso Medida de un ángulo cuya razón coseno es conocida. Statement: If n + 1 = 3, then n = 2 Inverse: If n + 1 ≠ 3, then n ≠ 2 If cos A = x, then cos -1 x = m∠A. función inversa Función que resulta de intercambiar los valores de entrada y salida de una función uno a uno. La función inversa de f (x) se indica f -1 (x). S134 S134 Glossary/Glosario The function y = 1 _ 2 inverse of the function y = 2x + 4. x - 2 is the DEF�� ENGLISH inverse sine (p. 534) The measure of an angle whose sine
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. irrational number (p. 80) A real number that cannot be expressed as the ratio of two integers. número irracional Número real que no se puede expresar como una razón de dos enteros. √  2, π, e irregular polygon (p. 382) A polygon that is not regular. polígono irregular Polígono que no es regular. isometric drawing (p. 662) A way of drawing three-dimensional figures using isometric dot paper, which has equally spaced dots in a repeating triangular pattern. dibujo isométrico Forma de dibujar figuras tridimensionales utilizando papel punteado isométrico, que tiene puntos espaciados uniformemente en un patrón triangular que se repite. isometry (p. 824) A transformation that does not change the size or shape of a figure. isosceles trapezoid (p. 429) A trapezoid in which the legs are congruent. isometría Transformación que no cambia el tamaño ni la forma de una figura. Reflections, translations, and rotations are all examples of isometries. trapecio isósceles Trapecio cuyos lados no paralelos son congruentes. isosceles triangle (p. 217) A triangle with at least two congruent sides. triángulo isósceles Triángulo que tiene al menos dos lados congruentes. iteration (p. 882) The repetitive application of the same rule. iteración Aplicación repetitiva de la misma regla. K kite (p. 427) A quadrilateral with
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 formado a partir de un triángulo sustituyendo el tercio central de cada segmento por dos segmentos que forman un ángulo de 60°. Glossary/Glosario S135 S135 BADC�� ENGLISH SPANISH EXAMPLES L lateral area (p. 680) The sum of the areas of the lateral faces of a prism or pyramid, or the area of the lateral surface of a cylinder or cone. área lateral Suma de las áreas de las caras laterales de un prisma o pirámide, o área de la superficie lateral de un cilindro o cono. lateral edge (p. 680) An edge of a prism or pyramid that is not an edge of a base. borde lateral Borde de un prisma o pirámide que no es el borde de una base. Lateral area = (28) (12) = 336 cm 2 lateral face (p. 680) A face of a prism or a pyramid that is not a base. cara lateral Cara de un prisma o pirámide que no es la base. lateral surface (p. 681) The curved surface of a cylinder or cone. superficie lateral Superficie curva de un cilindro o cono. leg of a right triangle (p. 45) One of the two sides of the right triangle that form the right angle. leg of a trapezoid (p. 429) One of the two nonparallel sides of the trapezoid. cateto de un triángulo rectángulo Uno de los dos lados de un triángulo rectángulo que forman un ángulo recto. cateto de un trapecio Uno de los dos lados no paralelos del trapecio. leg of an isosceles triangle (p. 273) One of the two congruent sides of the
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 thickness and extends forever. línea Término indefinido en geometría; una línea es un trazo recto que no tiene grosor y se extiende infinitamente. S136 S136 Glossary/Glosario �� ENGLISH line of best fit (p. 198) The line that comes closest to all of the points in a data set. SPANISH línea de mejor ajuste Línea que más se acerca a todos los puntos de un conjunto de datos. EXAMPLES line of symmetry (p. 865) A line that divides a plane figure into two congruent reflected halves. eje de simetría Línea que divide una figura plana en dos mitades reflejas congruentes. line symmetry (p. 856) A figure that can be reflected across a line so that the image coincides with the preimage. simetría axial Figura que puede reflejarse sobre una línea de forma tal que la imagen coincida con la imagen original. linear pair (p. 28) A pair of adjacent angles whose noncommon sides are opposite rays. par lineal Par de ángulos adyacentes cuyos lados no comunes son rayos opuestos. literal equation (p. 588) An equation that contains two or more variables. locus (p. 300) A set of points that satisfies a given condition. ∠3 and ∠4 form a linear pair. ecuación literal Ecuación que contiene dos o más variables. d = rt ) lugar geométrico Conjunto de puntos que
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 se expresa ⎜  AB ⎟ o ⎜  v ⎟. major arc (p. 756) An arc of a circle whose points are on or in the exterior of a central angle. arco mayor Arco de un círculo cuyos puntos están sobre un ángulo central o en su exterior.  u ⎟ = 5 ⎜ ⁀ ADC is a major arc of the circle. Glossary/Glosario S137 S137 �� ENGLISH mapping (p. 50) An operation that matches each element of a set with another element, its image, in the same set. SPANISH EXAMPLES correspondencia Operación que establece una correlación entre cada elemento de un conjunto con otro elemento, su imagen, en el mismo conjunto. matrix (p. 846) A rectangular array of numbers. matriz Arreglo rectangular de números. ⎡ 1 ⎢ -2 7 ⎣ 0 2 -6 ⎤ 3 ⎥ -5 3 ⎦ means of a proportion (p. 455) In the proportion a __ = c __, b and c are d b the means. If the proportion is written as a:b = c:d, the means are in the two middle positions. valores medios de una proporción En la proporción a __ = c __, b y c son los d b valores medios. Si la proporción se expresa como a:b = c:d, los valores medios están en las dos posiciones del medio. measure of an angle (p. 20) Angles are
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 medida del arco menor asociado. measure of a minor arc (p. 756) The measure of its central angle. medida de un arco menor Medida de su ángulo central. m∠M = 26.8° m ⁀ ADC = 360° - x° m ⁀ AC = x° median of a triangle (p. 314) A segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. mediana de un triángulo Segmento cuyos extremos son un vértice del triángulo y el punto medio del lado opuesto. midpoint (p. 15) The point that divides a segment into two congruent segments. punto medio Punto que divide un segmento en dos segmentos congruentes. B is the midpoint of ̶̶ AC. midsegment of a trapezoid (p. 431) The segment whose endpoints are the midpoints of the legs of the trapezoid. segmento medio de un trapecio Segmento cuyos extremos son los puntos medios de los catetos del trapecio. midsegment of a triangle (p. 322) A segment that joins the midpoints of two sides of the triangle. segmento medio de un triángulo Segmento que une los puntos medios de dos lados del triángulo. S138 S138 Glossary/Glosario �� ENGLISH SPANISH EXAMPLES midsegment triangle (p. 322) The triangle formed by the three midsegments of a triangle. triángulo de segmentos medios Triángulo formado por los tres segmentos medios de un triángulo. minor arc (p. 756) An arc of a circle whose
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 p is “not p,” written as ∼p. negación La negación de un enunciado p es “no p”, que se escribe p. negation of a vector (p. 566) The vector obtained by negating each component of a given vector. negación de un vector Vector que se obtiene por la negación de cada componente de un vector dado. The negation of 〈3, -2〉 is 〈-3, 2〉. net (p. 655) A diagram of the faces of a three-dimensional figure arranged in such a way that the diagram can be folded to form the three-dimensional figure. plantilla Diagrama de las caras y superficies de una figura tridimensional que se puede plegar para formar la figura tridimensional. network (p. 95) A diagram of vertices and edges. red Diagrama de vértices y aristas. n-gon (p. 382) An n-sided polygon. nonagon (p. 382) A nine-sided polygon. n-ágono Polígono de n lados. nonágono Polígono de nueve lados. noncollinear (p. 6) Points that do not lie on the same line. no colineal Puntos que no se encuentran sobre la misma línea. non-Euclidean geometry (p. 726) A system of geometry in which the Parallel Postulate, which states that there is exactly one line through a given point parallel to a given line, does not hold. geometría no euclidiana Sistema de geometría en el cual no se cumple el postulado de las paralelas

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