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Solution Manual for Precalculus, 11th Edition

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    S OLUTIONS M ANUAL T IM B RITT Jackson State Community College P RECALCULUS E LEVENTH E DITION Michael Sullivan Chicago State University

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    Table of Contents Chapter 1 Graphs 1.1 The Distance and Midpoint Formulas ......................................................................................... 1 1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ................................................ 13 1.3 Lines ......................................................................................................................................... 26 1.4 Circles ....................................................................................................................................... 43 Chapter Review................................................................................................................................ 56 Chapter Test ..................................................................................................................................... 62 Chapter Projects ............................................................................................................................... 64 Chapter 2 Functions and Their Graphs 2.1 Functions................................................................................................................................... 65 2.2 The Graph of a Function ........................................................................................................... 83 2.3 Properties of Functions ............................................................................................................. 92 2.4 Library of Functions; Piecewise-defined Functions ............................................................... 109 2.5 Graphing Techniques: Transformations ................................................................................. 121 2.6 Mathematical Models: Building Functions ............................................................................. 139 Chapter Review.............................................................................................................................. 147 Chapter Test ................................................................................................................................... 154 Cumulative Review ....................................................................................................................... 157 Chapter Projects ............................................................................................................................. 161 Chapter 3 Linear and Quadratic Functions 3.1 Properties of Linear Functions and Linear Models ................................................................. 163 3.2 Building Linear Functions from Data ..................................................................................... 174 3.3 Quadratic Functions and Their Properties .............................................................................. 180 3.4 Build Quadratic Models from Verbal Descriptions and from Data ........................................ 203 3.5 Inequalities Involving Quadratic Functions ............................................................................ 211 Chapter Review.............................................................................................................................. 231 Chapter Test ................................................................................................................................... 239 Cumulative Review........................................................................................................................ 241 Chapter Projects ............................................................................................................................. 244 Chapter 4 Polynomial and Rational Functions 4.1 Polynomial Functions ............................................................................................................. 247 4.2 Graphing Polynomial Functions; Models ............................................................................... 257 4.3 Properties of Rational Functions............................................................................................. 273 4.4 The Graph of a Rational Function .......................................................................................... 283 4.5 Polynomial and Rational Inequalities ..................................................................................... 339 4.6 The Real Zeros of a Polynomial Function .............................................................................. 360 4.7 Complex Zeros; Fundamental Theorem of Algebra ............................................................... 391 Chapter Review.............................................................................................................................. 400 Chapter Test ................................................................................................................................... 415 Cumulative Review........................................................................................................................ 419 Chapter Projects ............................................................................................................................. 424

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    Chapter 5 Exponential and Logarithmic Functions 5.1 Composite Functions .............................................................................................................. 426 5.2 One-to-One Functions; Inverse Functions .............................................................................. 444 5.3 Exponential Functions ............................................................................................................ 466 5.4 Logarithmic Functions ............................................................................................................ 487 5.5 Properties of Logarithms ........................................................................................................ 509 5.6 Logarithmic and Exponential Equations ................................................................................. 518 5.7 Financial Models .................................................................................................................... 538 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models .................................................................................................................... 546 5.9 Building Exponential, Logarithmic, and Logistic Models from Data..................................... 555 Chapter Review.............................................................................................................................. 561 Chapter Test ................................................................................................................................... 573 Cumulative Review........................................................................................................................ 577 Chapter Projects ............................................................................................................................. 580 Chapter 6 Trigonometric Functions 6.1 Angles, Arc Length, and Circular Motion .............................................................................. 583 6.2 Trigonometric Functions: Unit Circle Approach .................................................................... 592 6.3 Properties of the Trigonometric Functions ............................................................................. 610 6.4 Graphs of the Sine and Cosine Functions ............................................................................... 624 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions...................................... 645 6.6 Phase Shift; Sinusoidal Curve Fitting ..................................................................................... 655 Chapter Review.............................................................................................................................. 666 Chapter Test ................................................................................................................................... 674 Cumulative Review........................................................................................................................ 678 Chapter Projects ............................................................................................................................. 682 Chapter 7 Analytic Trigonometry 7.1 The Inverse Sine, Cosine, and Tangent Functions .................................................................. 685 7.2 The Inverse Trigonometric Functions (Continued) ................................................................ 698 7.3 Trigonometric Equations ........................................................................................................ 710 7.4 Trigonometric Identities ......................................................................................................... 731 7.5 Sum and Difference Formulas ................................................................................................ 744 7.6 Double-angle and Half-angle Formulas .................................................................................. 769 7.7 Product-to-Sum and Sum-to-Product Formulas...................................................................... 795 Chapter Review.............................................................................................................................. 808 Chapter Test ................................................................................................................................... 823 Cumulative Review........................................................................................................................ 828 Chapter Projects ............................................................................................................................. 834 Chapter 8 Applications of Trigonometric Functions 8.1 Right Triangle Trigonometry; Applications ........................................................................... 838 8.2 The Law of Sines .................................................................................................................... 852 8.3 The Law of Cosines ................................................................................................................ 867 8.4 Area of a Triangle ................................................................................................................... 879 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves ........................................... 889 Chapter Review.............................................................................................................................. 899 Chapter Test ................................................................................................................................... 905 Cumulative Review........................................................................................................................ 909 Chapter Projects ............................................................................................................................. 915

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    Chapter 9 Polar Coordinates; Vectors 9.1 Polar Coordinates.................................................................................................................... 919 9.2 Polar Equations and Graphs .................................................................................................... 928 9.3 The Complex Plane; De Moivre’s Theorem ........................................................................... 958 9.4 Vectors .................................................................................................................................... 971 9.5 The Dot Product ...................................................................................................................... 984 9.6 Vectors in Space ..................................................................................................................... 990 9.7 The Cross Product................................................................................................................... 996 Chapter Review............................................................................................................................ 1007 Chapter Test ................................................................................................................................. 1016 Cumulative Review...................................................................................................................... 1020 Chapter Projects ........................................................................................................................... 1023 Chapter 10 Analytic Geometry 10.2 The Parabola ....................................................................................................................... 1026 10.3 The Ellipse .......................................................................................................................... 1041 10.4 The Hyperbola .................................................................................................................... 1058 10.5 Rotation of Axes; General Form of a Conic ....................................................................... 1078 10.6 Polar Equations of Conics ................................................................................................... 1091 10.7 Plane Curves and Parametric Equations ............................................................................. 1100 Chapter Review............................................................................................................................ 1115 Chapter Test ................................................................................................................................. 1124 Cumulative Review...................................................................................................................... 1129 Chapter Projects ........................................................................................................................... 1131 Chapter 11 Systems of Equations and Inequalities 11.1 Systems of Linear Equations: Substitution and Elimination ............................................... 1135 11.2 Systems of Linear Equations: Matrices .............................................................................. 1158 11.3 Systems of Linear Equations: Determinants ....................................................................... 1183 11.4 Matrix Algebra .................................................................................................................... 1197 11.5 Partial Fraction Decomposition .......................................................................................... 1216 11.6 Systems of Nonlinear Equations ......................................................................................... 1235 11.7 Systems of Inequalities ....................................................................................................... 1263 11.8 Linear Programming ........................................................................................................... 1279 Chapter Review............................................................................................................................ 1292 Chapter Test ................................................................................................................................. 1307 Cumulative Review...................................................................................................................... 1316 Chapter Projects ........................................................................................................................... 1319 Chapter 12 Sequences; Induction; the Binomial Theorem 12.1 Sequences............................................................................................................................ 1322 12.2 Arithmetic Sequences ......................................................................................................... 1332 12.3 Geometric Sequences; Geometric Series ............................................................................ 1341 12.4 Mathematical Induction ...................................................................................................... 1353 12.5 The Binomial Theorem ....................................................................................................... 1362 Chapter Review............................................................................................................................ 1369 Chapter Test ................................................................................................................................. 1373 Cumulative Review...................................................................................................................... 1376 Chapter Projects ........................................................................................................................... 1379

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    Chapter 13 Counting and Probability 13.1 Counting.............................................................................................................................. 1382 13.2 Permutations and Combinations ......................................................................................... 1385 13.3 Probability ........................................................................................................................... 1390 Chapter Review............................................................................................................................ 1397 Chapter Test ................................................................................................................................. 1399 Cumulative Review...................................................................................................................... 1400 Chapter Projects ........................................................................................................................... 1403 Chapter 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function 14.1 Investigating Limits Using Tables and Graphs ................................................................... 1406 14.2 Algebraic Techniques for Finding Limits ........................................................................... 1412 14.3 One-sided Limits; Continuity.............................................................................................. 1416 14.4 The Tangent Problem; The Derivative................................................................................ 1423 14.5 The Area Problem; The Integral ......................................................................................... 1432 Chapter Review............................................................................................................................ 1446 Chapter Test ................................................................................................................................. 1453 Chapter Projects ........................................................................................................................... 1456 Appendix A Review A.1 Algebra Essentials ................................................................................................................ 1462 A.2 Geometry Essentials ............................................................................................................. 1467 A.3 Polynomials.......................................................................................................................... 1473 A.4 Synthetic Division ................................................................................................................ 1481 A.5 Rational Expressions ............................................................................................................ 1483 A.6 Solving Equations ................................................................................................................ 1488 A.7 Complex Numbers; Quadratic Equations in the Complex Number System ........................ 1502 A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications ..... 1508 A.9 Interval Notation; Solving Inequalities ................................................................................ 1515 A.10 n th Roots; Rational Exponents ............................................................................................ 1527 Appendix B Graphing Utilities B.1 The Viewing Rectangle ........................................................................................................ 1537 B.2 Using a Graphing Utility to Graph Equations ...................................................................... 1538 B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ............................ 1543 B.5 Square Screens ..................................................................................................................... 1545

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    1 Chapter 1 Graphs Section 1.1 1. 0 2.   5 3 8 8     3. 2 2 3 4 25 5    4. 2 2 2 11 60 121 3600 3721 61      Since the sum of the squares of two of the sides of the triangle equals the square of the third side, the triangle is a right triangle. 5. 1 2 bh 6. true 7. x- coordinate or abscissa; y -coordinate or ordinate 8. quadrants 9. midpoint 10. False; the distance between two points is never negative. 11. False; points that lie in Quadrant IV will have a positive x -coordinate and a negative y -coordinate. The point   1, 4  lies in Quadrant II. 12. True; 1 2 1 2 , 2 2 x x y y M          13. b 14. a 15. (a) Quadrant II (b) x -axis (c) Quadrant III (d) Quadrant I (e) y -axis (f) Quadrant IV 16. (a) Quadrant I (b) Quadrant III (c) Quadrant II (d) Quadrant I (e) y -axis (f) x -axis 17. The points will be on a vertical line that is two units to the right of the y -axis.

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    Chapter 1: Graphs 2 18. The points will be on a horizontal line that is three units above the x -axis. 19. 2 2 1 2 2 2 ( , ) (2 0) (1 0) 2 1 4 1 5 d P P          20. 2 2 1 2 2 2 ( , ) ( 2 0) (1 0) ( 2) 1 4 1 5 d P P            21. 2 2 1 2 2 2 ( , ) ( 2 1) (2 1) ( 3) 1 9 1 10 d P P            22.   2 2 1 2 2 2 ( , ) 2 ( 1) (2 1) 3 1 9 1 10 d P P           23.       2 2 1 2 2 2 ( , ) (5 3) 4 4 2 8 4 64 68 2 17 d P P            24.         2 2 1 2 2 2 ( , ) 2 1 4 0 3 4 9 16 25 5 d P P            25.   2 2 1 2 2 2 ( , ) 4 ( 7) (0 3) 11 ( 3) 121 9 130            d P P 26.     2 2 1 2 2 2 ( , ) 4 2 2 ( 3) 2 5 4 25 29 d P P           27.   2 2 1 2 2 2 ( , ) (6 5) 1 ( 2) 1 3 1 9 10           d P P 28.     2 2 1 2 2 2 ( , ) 6 ( 4) 2 ( 3) 10 5 100 25 125 5 5 d P P             29.     2 2 1 2 2 2 ( , ) 2.3 ( 0.2) 1.1 (0.3) 2.5 0.8 6.25 0.64 6.89 2.62            d P P 30.     2 2 1 2 2 2 ( , ) 0.3 1.2 1.1 2.3 ( 1.5) ( 1.2) 2.25 1.44 3.69 1.92              d P P 31. 2 2 1 2 2 2 2 2 ( , ) (0 ) (0 ) ( ) ( ) d P P a b a b a b           32. 2 2 1 2 2 2 2 2 2 ( , ) (0 ) (0 ) ( ) ( ) 2 2 d P P a a a a a a a a             33. ( 2,5), (1,3), ( 1, 0) A B C            2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 1 ( 2) (3 5) 3 ( 2) 9 4 13 ( , ) 1 1 (0 3) ( 2) ( 3) 4 9 13 ( , ) 1 ( 2) (0 5) 1 ( 5) 1 25 26 d A B d B C d A C                                   

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    Section 1.1: The Distance and Midpoint Formulas 3 Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:             2 2 2 2 2 2 ( , ) ( , ) ( , ) 13 13 26 13 13 26 26 26 d A B d B C d A C        The area of a triangle is 1 2 A bh   . In this problem,     1 ( , ) ( , ) 2 1 1 13 13 13 2 2 13 square units 2 A d A B d B C          34. ( 2, 5), (12, 3), (10, 11) A B C            2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 12 ( 2) (3 5) 14 ( 2) 196 4 200 10 2 ( , ) 10 12 ( 11 3) ( 2) ( 14) 4 196 200 10 2 ( , ) 10 ( 2) ( 11 5) 12 ( 16) 144 256 400 20 d A B d B C d A C                                       Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:             2 2 2 2 2 2 ( , ) ( , ) ( , ) 10 2 10 2 20 200 200 400 400 400 d A B d B C d A C        The area of a triangle is 1 2 A bh  . In this problem,     1 ( , ) ( , ) 2 1 10 2 10 2 2 1 100 2 100 square units 2 A d A B d B C           35. ( 5,3), (6, 0), (5,5) A B C           2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 6 ( 5) (0 3) 11 ( 3) 121 9 130 ( , ) 5 6 (5 0) ( 1) 5 1 25 26 ( , ) 5 ( 5) (5 3) 10 2 100 4 104 2 26 d A B d B C d A C                                 Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:

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    Chapter 1: Graphs 4             2 2 2 2 2 2 ( , ) ( , ) ( , ) 104 26 130 104 26 130 130 130 d A C d B C d A B        The area of a triangle is 1 2 A bh  . In this problem,     1 ( , ) ( , ) 2 1 104 26 2 1 2 26 26 2 1 2 26 2 26 square units A d A C d B C              36. ( 6, 3), (3, 5), ( 1, 5) A B C             2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 3 ( 6) ( 5 3) 9 ( 8) 81 64 145 ( , ) 1 3 (5 ( 5)) ( 4) 10 16 100 116 2 29 ( , ) 1 ( 6) (5 3) 5 2 25 4 29 d A B d B C d A C                                     Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:             2 2 2 2 2 2 ( , ) ( , ) ( , ) 29 2 29 145 29 4 29 145 29 116 145 145 145 d A C d B C d A B           The area of a triangle is 1 2 A bh  . In this problem,     1 ( , ) ( , ) 2 1 29 2 29 2 1 2 29 2 29 square units A d A C d B C           37. (4, 3), (0, 3), (4, 2) A B C              2 2 2 2 2 2 2 2 2 2 2 2 ( , ) (0 4) 3 ( 3) ( 4) 0 16 0 16 4 ( , ) 4 0 2 ( 3) 4 5 16 25 41 ( , ) (4 4) 2 ( 3) 0 5 0 25 25 5 d A B d B C d A C                                   Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:

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    Section 1.1: The Distance and Midpoint Formulas 5         2 2 2 2 2 2 ( , ) ( , ) ( , ) 4 5 41 16 25 41 41 41 d A B d A C d B C        The area of a triangle is 1 2 A bh  . In this problem,     1 ( , ) ( , ) 2 1 4 5 2 10 square units A d A B d A C        38. (4, 3), (4, 1), (2, 1) A B C             2 2 2 2 2 2 2 2 2 2 2 2 ( , ) (4 4) 1 ( 3) 0 4 0 16 16 4 ( , ) 2 4 1 1 ( 2) 0 4 0 4 2 ( , ) (2 4) 1 ( 3) ( 2) 4 4 16 20 2 5 d A B d B C d A C                                   Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:         2 2 2 2 2 2 ( , ) ( , ) ( , ) 4 2 2 5 16 4 20 20 20 d A B d B C d A C        The area of a triangle is 1 2 A bh  . In this problem,     1 ( , ) ( , ) 2 1 4 2 2 4 square units A d A B d B C        39. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 4 4 3 5 , 2 2 8 0 , 2 2 (4, 0) x x y y x y                            40. The coordinates of the midpoint are:   1 2 1 2 ( , ) , 2 2 2 2 0 4 , 2 2 0 4 , 2 2 0, 2 x x y y x y                            41. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 1 8 4 0 , 2 2 7 4 , 2 2 7 , 2 2                                  x x y y x y

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    Chapter 1: Graphs 6 42. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 2 4 3 2 , 2 2 6 1 , 2 2 1 3, 2 x x y y x y                                    43. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 7 9 5 1 , 2 2 4 16 , 2 2 (8, 2)                              x x y y x y 44. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 4 2 3 2 , 2 2 2 1 , 2 2 1 1, 2 x x y y x y                                       45. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 0 0 , 2 2 , 2 2 x x y y x y a b a b                          46. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 0 0 , 2 2 , 2 2 x x y y x y a a a a                          47. The x coordinate would be 2 3 5   and the y coordinate would be 5 2 3   . Thus the new point would be   5,3 . 48. The new x coordinate would be 1 2 3     and the new y coordinate would be 6 4 10   . Thus the new point would be   3,10  49. a. If we use a right triangle to solve the problem, we know the hypotenuse is 13 units in length. One of the legs of the triangle will be 2+3=5. Thus the other leg will be: 2 2 2 2 2 5 13 25 169 144 12 b b b b       Thus the coordinates will have an y value of 1 12 13     and 1 12 11    . So the points are   3,11 and   3, 13  . b. Consider points of the form   3, y that are a distance of 13 units from the point   2, 1   .             2 2 2 1 2 1 2 2 2 2 2 2 3 ( 2) 1 5 1 25 1 2 2 26 d x x y y y y y y y y                           2 2 2 2 2 2 13 2 26 13 2 26 169 2 26 0 2 143 0 11 13 y y y y y y y y y y                11 0 11 y y    or 13 0 13 y y     Thus, the points   3,11 and   3, 13  are a distance of 13 units from the point   2, 1   .

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    Section 1.1: The Distance and Midpoint Formulas 7 50. a. If we use a right triangle to solve the problem, we know the hypotenuse is 17 units in length. One of the legs of the triangle will be 2+6=8. Thus the other leg will be: 2 2 2 2 2 8 17 64 289 225 15 b b b b       Thus the coordinates will have an x value of 1 15 14    and 1 15 16   . So the points are   14, 6   and   16, 6  . b. Consider points of the form   , 6 x  that are a distance of 17 units from the point   1, 2 .             2 2 2 1 2 1 2 2 2 2 2 2 1 2 6 2 1 8 2 1 64 2 65 d x x y y x x x x x x x                          2 2 2 2 2 2 17 2 65 17 2 65 289 2 65 0 2 224 0 14 16 x x x x x x x x x x                14 0 14 x x     or 16 0 16 x x    Thus, the points   14, 6   and   16, 6  are a distance of 13 units from the point   1, 2 . 51. Points on the x -axis have a y -coordinate of 0. Thus, we consider points of the form   , 0 x that are a distance of 6 units from the point   4, 3  .           2 2 2 1 2 1 2 2 2 2 2 2 4 3 0 16 8 3 16 8 9 8 25 d x x y y x x x x x x x                        2 2 2 2 2 2 2 6 8 25 6 8 25 36 8 25 0 8 11 ( 8) ( 8) 4(1)( 11) 2(1) 8 64 44 8 108 2 2 8 6 3 4 3 3 2 x x x x x x x x x                             4 3 3 x   or 4 3 3 x   Thus, the points   4 3 3, 0  and   4 3 3, 0  are on the x -axis and a distance of 6 units from the point   4, 3  . 52. Points on the y -axis have an x -coordinate of 0. Thus, we consider points of the form   0, y that are a distance of 6 units from the point   4, 3  .         2 2 2 1 2 1 2 2 2 2 2 2 4 0 3 4 9 6 16 9 6 6 25 d x x y y y y y y y y y                    

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    Chapter 1: Graphs 8   2 2 2 2 2 2 2 6 6 25 6 6 25 36 6 25 0 6 11 ( 6) (6) 4(1)( 11) 2(1) 6 36 44 6 80 2 2 6 4 5 3 2 5 2 y y y y y y y y y                               3 2 5 y    or 3 2 5 y    Thus, the points   0, 3 2 5   and   0, 3 2 5   are on the y- axis and a distance of 6 units from the point   4, 3  . 53. a. To shift 3 units left and 4 units down, we subtract 3 from the x -coordinate and subtract 4 from the y -coordinate.     2 3,5 4 1,1     b. To shift left 2 units and up 8 units, we subtract 2 from the x -coordinate and add 8 to the y -coordinate.     2 2,5 8 0,13    54. Let the coordinates of point B be   , x y . Using the midpoint formula, we can write   1 8 2,3 , 2 2 x y           . This leads to two equations we can solve. 1 2 2 1 4 5 x x x        8 3 2 8 6 2 y y y       Point B has coordinates   5, 2  . 55.   1 2 1 2 , , 2 2 x x y y M x y           .   1 1 1 , ( 3, 6) P x y    and ( , ) ( 1, 4) x y   , so 1 2 2 2 2 2 3 1 2 2 3 1 x x x x x x            and 1 2 2 2 2 2 6 4 2 8 6 2 y y y y y y        Thus, 2 (1, 2) P  . 56.   1 2 1 2 , , 2 2 x x y y M x y           .   2 2 2 , (7, 2) P x y    and ( , ) (5, 4) x y   , so 1 2 1 1 1 2 7 5 2 10 7 3 x x x x x x        and 1 2 1 1 1 2 ( 2) 4 2 8 ( 2) 6 y y y y y y             Thus, 1 (3, 6) P   . 57. The midpoint of AB is:   0 6 0 0 , 2 2 3, 0 D           The midpoint of AC is:   0 4 0 4 , 2 2 2, 2 E           The midpoint of BC is:   6 4 0 4 , 2 2 5, 2 F             2 2 2 2 ( , ) 0 4 (3 4) ( 4) ( 1) 16 1 17 d C D              2 2 2 2 ( , ) 2 6 (2 0) ( 4) 2 16 4 20 2 5 d B E            2 2 2 2 ( , ) (2 0) (5 0) 2 5 4 25 29 d A F         

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    Section 1.1: The Distance and Midpoint Formulas 9 58. Let 1 2 (0, 0), (0, 4), ( , ) P P P x y          2 2 1 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 , (0 0) (4 0) 16 4 , ( 0) ( 0) 4 16 , ( 0) ( 4) ( 4) 4 ( 4) 16 d P P d P P x y x y x y d P P x y x y x y                             Therefore,   2 2 2 2 4 8 16 8 16 2 y y y y y y y        which gives 2 2 2 2 16 12 2 3 x x x      Two triangles are possible. The third vertex is     2 3, 2 or 2 3, 2  . 59. 2 2 1 2 2 2 ( , ) ( 4 2) (1 1) ( 6) 0 36 6 d P P             2 2 2 3 2 2 ( , ) 4 ( 4) ( 3 1) 0 ( 4) 16 4 d P P             2 2 1 3 2 2 ( , ) ( 4 2) ( 3 1) ( 6) ( 4) 36 16 52 2 13 d P P               Since       2 2 2 1 2 2 3 1 3 ( , ) ( , ) ( , ) d P P d P P d P P   , the triangle is a right triangle. 60.   2 2 1 2 2 2 ( , ) 6 ( 1) (2 4) 7 ( 2) 49 4 53 d P P              2 2 2 3 2 2 ( , ) 4 6 ( 5 2) ( 2) ( 7) 4 49 53 d P P               2 2 1 3 2 2 ( , ) 4 ( 1) ( 5 4) 5 ( 9) 25 81 106 d P P             Since       2 2 2 1 2 2 3 1 3 ( , ) ( , ) ( , ) d P P d P P d P P   , the triangle is a right triangle. Since     1 2 2 3 , , d P P d P P  , the triangle is isosceles. Therefore, the triangle is an isosceles right triangle. 61.     2 2 1 2 2 2 ( , ) 0 ( 2) 7 ( 1) 2 8 4 64 68 2 17 d P P               2 2 2 3 2 2 ( , ) 3 0 (2 7) 3 ( 5) 9 25 34 d P P               2 2 1 3 2 2 ( , ) 3 ( 2) 2 ( 1) 5 3 25 9 34 d P P            Since 2 3 1 3 ( , ) ( , ) d P P d P P  , the triangle is isosceles. Since       2 2 2 1 3 2 3 1 2 ( , ) ( , ) ( , ) d P P d P P d P P   , the triangle is also a right triangle. Therefore, the triangle is an isosceles right triangle.

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    Chapter 1: Graphs 10 62.     2 2 1 2 2 2 ( , ) 4 7 0 2 ( 11) ( 2) 121 4 125 5 5 d P P                2 2 2 3 2 2 ( , ) 4 ( 4) (6 0) 8 6 64 36 100 10 d P P                2 2 1 3 2 2 ( , ) 4 7 6 2 ( 3) 4 9 16 25 5 d P P            Since       2 2 2 1 3 2 3 1 2 ( , ) ( , ) ( , ) d P P d P P d P P   , the triangle is a right triangle. 63. Using the Pythagorean Theorem: 2 2 2 2 2 90 90 8100 8100 16200 16200 90 2 127.28 feet d d d d         90 90 90 90 d 64. Using the Pythagorean Theorem: 2 2 2 2 2 60 60 3600 3600 7200 7200 60 2 84.85 feet d d d d          60 60 60 60 d 65. a. First: (90, 0), Second: (90, 90), Third: (0, 90) (0,0) (0,90) (90,0) (90,90) X Y b. Using the distance formula: 2 2 2 2 (310 90) (15 90) 220 ( 75) 54025 5 2161 232.43 feet d           c. Using the distance formula: 2 2 2 2 (300 0) (300 90) 300 210 134100 30 149 366.20 feet d          66. a. First: (60, 0), Second: (60, 60) Third: (0, 60) (0,0) (0,60) (60,0) (60,60) x y b. Using the distance formula: 2 2 2 2 (180 60) (20 60) 120 ( 40) 16000 40 10 126.49 feet d           c. Using the distance formula: 2 2 2 2 (220 0) (220 60) 220 160 74000 20 185 272.03 feet d         

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