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Solution Manual for Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry, 4th Edition

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    S OLUTIONS M ANUAL T IM B RITT Jackson State Community College P RECALCULUS : C ONCEPTS T HROUGH F UNCTIONS , A U NIT C IRCLE A PPROACH T O T RIGONOMETRY F OURTH EDITION Michael Sullivan Chicago State University Michael Sullivan, III Joliet Junior College

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    Table of Contents Chapter F: Foundations: A Prelude to functions F.1 The Distance and Midpoint Formulas ...........................................................................................1 F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ..................................................12 F.3 Lines ............................................................................................................................................29 F.4 Circles .........................................................................................................................................45 Chapter Project ...................................................................................................................................56 Chapter 1: Functions and Their Graphs 1.1 Functions .....................................................................................................................................57 1.2 The Graph of a Function .............................................................................................................72 1.3 Properties of Functions ................................................................................................................80 1.4 Library of Functions; Piecewise-defined Functions ....................................................................97 1.5 Graphing Techniques: Transformations ....................................................................................109 1.6 Mathematical Models: Building Functions ...............................................................................126 1.7 Building Mathematical Models Using Variation ......................................................................132 Chapter Review ................................................................................................................................136 Chapter Test .....................................................................................................................................143 Chapter Projects ...............................................................................................................................147 Chapter 2: Linear and Quadratic Functions 2.1 Properties of Linear Functions and Linear Models ...................................................................149 2.2 Building Linear Models from Data ...........................................................................................161 2.3 Quadratic Functions and Their Zeros ........................................................................................166 2.4 Properties of Quadratic Functions .............................................................................................185 2.5 Inequalities Involving Quadratic Functions ..............................................................................207 2.6 Building Quadratic Models from Verbal Descriptions and from Data .....................................226 2.7 Complex Zeros of a Quadratic Function ...................................................................................234 2.8 Equations and Inequalities Involving the Absolute Value Function .........................................239 Chapter Review ................................................................................................................................246 Chapter Test .....................................................................................................................................258 Cumulative Review ..........................................................................................................................262 Chapter Projects ...............................................................................................................................265 Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Models ............................................................................................269 3.2 The Real Zeros of a Polynomial Function.................................................................................293 3.3 Complex Zeros; Fundamental Theorem of Algebra..................................................................323 3.4 Properties of Rational Functions ...............................................................................................332 3.5 The Graph of a Rational Function .............................................................................................341 3.6 Polynomial and Rational Inequalities........................................................................................398 Chapter Review ................................................................................................................................421 Chapter Test .....................................................................................................................................437 Cumulative Review ..........................................................................................................................441 Chapter Projects ...............................................................................................................................446

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    Chapter 4: Exponential and Logarithmic Functions 4.1 Composite Functions .................................................................................................................448 4.2 One-to-One Functions; Inverse Functions ................................................................................465 4.3 Exponential Functions ...............................................................................................................487 4.4 Logarithmic Functions ..............................................................................................................507 4.5 Properties of Logarithms ...........................................................................................................528 4.6 Logarithmic and Exponential Equations ...................................................................................536 4.7 Financial Models .......................................................................................................................556 4.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models .............................................................................................................564 4.9 Building Exponential, Logarithmic, and Logistic Models from Data .......................................573 Chapter Review ................................................................................................................................579 Chapter Test .....................................................................................................................................591 Cumulative Review ..........................................................................................................................595 Chapter Projects ...............................................................................................................................599 Chapter 5: Trigonometric Functions 5.1 Angles and Their Measure ........................................................................................................602 5.2 Trigonometric Functions: Unit Circle Approach ......................................................................610 5.3 Properties of the Trigonometric Functions ................................................................................628 5.4 Graphs of the Sine and Cosine Functions .................................................................................641 5.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions ........................................661 5.6 Phase Shift; Sinusoidal Curve Fitting .......................................................................................670 Chapter Review ................................................................................................................................684 Chapter Test .....................................................................................................................................692 Cumulative Review ..........................................................................................................................696 Chapter Projects ...............................................................................................................................700 Chapter 6: Analytic Trigonometry 6.1 The Inverse Sine, Cosine, and Tangent Functions ....................................................................704 6.2 The Inverse Trigonometric Functions (Continued) ...................................................................716 6.3 Trigonometric Equations ...........................................................................................................728 6.4 Trigonometric Identities ............................................................................................................747 6.5 Sum and Difference Formulas ...................................................................................................760 6.6 Double-angle and Half-angle Formulas ....................................................................................784 6.7 Product-to-Sum and Sum-to-Product Formulas ........................................................................812 Chapter Review ................................................................................................................................821 Chapter Test .....................................................................................................................................836 Cumulative Review ..........................................................................................................................841 Chapter Projects ...............................................................................................................................846

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    Table of Contents Chapter 7: Applications of Trigonometric Functions 7.1 Right Triangle Trigonometry; Applications ..............................................................................850 7.2 The Law of Sines .......................................................................................................................862 7.3 The Law of Cosines ...................................................................................................................876 7.4 Area of a Triangle......................................................................................................................889 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves .............................................898 Chapter Review ................................................................................................................................907 Chapter Test .....................................................................................................................................914 Cumulative Review ..........................................................................................................................917 Chapter Projects ...............................................................................................................................924 Chapter 8: Polar Coordinates; Vectors 8.1 Polar Coordinates ......................................................................................................................928 8.2 Polar Equations and Graphs ......................................................................................................935 8.3 The Complex Plane; DeMoivre’s Theorem ..............................................................................964 8.4 Vectors.......................................................................................................................................975 8.5 The Dot Product ........................................................................................................................987 8.6 Vectors in Space ........................................................................................................................993 8.7 The Cross Product .....................................................................................................................999 Chapter Review ..............................................................................................................................1009 Chapter Test ...................................................................................................................................1018 Cumulative Review ........................................................................................................................1022 Chapter Projects .............................................................................................................................1024 Chapter 9: Analytic Geometry 9.2 The Parabola ............................................................................................................................1028 9.3 The Ellipse ...............................................................................................................................1043 9.4 The Hyperbola .........................................................................................................................1059 9.5 Rotation of Axes; General Form of a Conic............................................................................1078 9.6 Polar Equations of Conics .......................................................................................................1090 9.7 Plane Curves and Parametric Equations ..................................................................................1097 Chapter Review ..............................................................................................................................1110 Chapter Test ...................................................................................................................................1119 Cumulative Review ........................................................................................................................1124 Chapter Projects .............................................................................................................................1126 Chapter 10: Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Substitution and Elimination .................................................1130 10.2 Systems of Linear Equations: Matrices .................................................................................1151 10.3 Systems of Linear Equations: Determinants .........................................................................1175 10.4 Matrix Algebra ......................................................................................................................1189 10.5 Partial Fraction Decomposition .............................................................................................1207 10.6 Systems of Nonlinear Equations ...........................................................................................1224 10.7 Systems of Inequalities ..........................................................................................................1252 10.8 Linear Programming..............................................................................................................1266 Chapter Review ..............................................................................................................................1278 Chapter Test ...................................................................................................................................1294 Cumulative Review ........................................................................................................................1302 Chapter Projects .............................................................................................................................1306

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    Chapter 11: Sequence; Induction; the Binomial Theorem 11.1 Sequences ..............................................................................................................................1308 11.2 Arithmetic Sequences ............................................................................................................1317 11.3 Geometric Sequences; Geometric Series ..............................................................................1324 11.4 Mathematical Induction .........................................................................................................1335 11.5 The Binomial Theorem .........................................................................................................1343 Chapter Review ..............................................................................................................................1348 Chapter Test ...................................................................................................................................1353 Cumulative Review ........................................................................................................................1356 Chapter Projects .............................................................................................................................1359 Chapter 12: Counting and Probability 12.1 Counting ................................................................................................................................1361 12.2 Permutations and Combinations ............................................................................................1363 12.3 Probability .............................................................................................................................1367 Chapter Review ..............................................................................................................................1373 Chapter Test ...................................................................................................................................1374 Cumulative Review ........................................................................................................................1376 Chapter Projects .............................................................................................................................1379 Chapter 13: A Preview of Calculus: The Limit, Derivative, and Integral of a Function 13.1 Finding Limits Using Tables and Graphs..............................................................................1382 13.2 Algebra Techniques for Finding Limits ................................................................................1388 13.3 One-sided Limits; Continuous Functions ..............................................................................1393 13.4 The Tangent Problem; The Derivative ..................................................................................1400 13.5 The Area Problem; The Integral ............................................................................................1410 Chapter Review ..............................................................................................................................1424 Chapter Test ...................................................................................................................................1431 Chapter Projects .............................................................................................................................1434 Appendix A: Review A.1 Algebra Essentials ..................................................................................................................1440 A.2 Geometry Essentials ...............................................................................................................1445 A.3 Polynomials ............................................................................................................................1450 A.4 Factoring Polynomials ............................................................................................................1457 A.5 Synthetic Division ..................................................................................................................1462 A.6 Rational Expressions ..............................................................................................................1464 A.7 n th Roots; Rational Exponents ...............................................................................................1473 A.8 Solving Equations ...................................................................................................................1481 A.9 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications .......1491 A.10 Interval Notation; Solving Inequalities ................................................................................1499 A.11 Complex Numbers ................................................................................................................1509 Appendix B: Graphing Utilities B.1 The Viewing Rectangle ..........................................................................................................1515 B.2 Using a Graphing Utility to Graph Equations ........................................................................1516 B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ..............................1520 B.5 Square Screens ........................................................................................................................1522

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    1 Chapter F Foundations: A Prelude to Functions Section F.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. Distance is always a positive number. 11. False. The point would lie in quadrant II. 12. True 13. b 14. a 15. (a) Quadrant II (b) Positive x-axis (c) Quadrant III (d) Quadrant I (e) Negative y-axis (f) Quadrant IV 16. (a) Quadrant I (b) Quadrant III (c) Quadrant II (d) Quadrant I (e) Positive y-axis (f) Negative x-axis 17. The points will be on a vertical line that is two units to the right of the y-axis. 18. The points will be on a horizontal line that is three units above the x-axis.

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    Chapter F Foundations: A Prelude to Functions 2 19. 2 2 1 2 ( , ) (2 0) (1 0) 4 1 5 d P P        20. 2 2 1 2 ( , ) ( 2 0) (1 0) 4 1 5 d P P         21. 2 2 1 2 ( , ) ( 2 1) (2 1) 9 1 10 d P P         22.   2 2 1 2 ( , ) 2 ( 1) (2 1) 9 1 10 d P P         23.       2 2 2 2 1 2 ( , ) (5 3) 4 4 2 8 4 64 68 2 17 d P P            24.         2 2 2 2 1 2 ( , ) 2 1 4 0 3 4 9 16 25 5 d P P            25.   2 2 1 2 2 2 ( , ) 6 ( 3) (0 2) 9 ( 2) 81 4 85 d P P            26.     2 2 2 2 1 2 ( , ) 4 2 2 ( 3) 2 5 4 25 29 d P P           27.   2 2 1 2 2 2 ( , ) ( 2 4) 5 ( 2) ( 6) ( 3) 36 9 45 3 5 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 2 2 1 2 ( , ) (0 ) (0 ) d P P a b a b       32. 2 2 2 2 1 2 2 ( , ) (0 ) (0 ) 2 2 d P P 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                                    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,

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    Section F.1: The Distance and Midpoint Formulas 3     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 2 100 square units 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:             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

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    Chapter F Foundations: A Prelude to Functions 4 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 ( , ) ( , ) ( , ) 29 116 145 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 29 116 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 F.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 3 6 2 0 , 2 2 3 2 , 2 2 3 ,1 2 x x y y x y                                 

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    Chapter F Foundations: A Prelude to Functions 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 4 ( 2) 2 ( 5) , 2 2 2 7 , 2 2 7 1, 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 0.2 2.3 0.3 1.1 , 2 2 2.1 1.4 , 2 2 (1.05, 0.7) x x y y x y                            46. The coordinates of the midpoint are:   1 2 1 2 ( , ) , 2 2 1.2 ( 0.3) 2.3 1.1 , 2 2 0.9 3.4 , 2 2 0.45, 1.7 x x y y x y                            47. 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                          48. 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                          49. Consider points of the form   2, y that are a distance of 5 units from the point   2, 1   .             2 2 2 1 2 1 2 2 2 2 2 2 2 2 1 4 1 16 1 2 2 17 d x x y y y y y y y y                            2 2 2 2 2 2 5 2 17 5 2 17 25 2 17 0 2 8 0 4 2 y y y y y y y y y y                4 0 4 y y     or 2 0 2 y y   

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    Section F.1: The Distance and Midpoint Formulas 7 Thus, the points   2, 4  and   2, 2 are a distance of 5 units from the point   2, 1   . 50. Consider points of the form   , 3 x  that are a distance of 13 units from the point   1, 2 .             2 2 2 1 2 1 2 2 2 2 2 2 1 2 3 2 1 5 2 1 25 2 26 d x x y y x x x x x x x                          2 2 2 2 2 2 13 2 26 13 2 26 169 2 26 0 2 143 0 13 11 x x x x x x x x x x                13 0 13 x x    or 11 0 11 x x     Thus, the points   13, 3  and   11, 3   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 5 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 5 8 25 5 8 25 25 8 25 0 8 0 8 x x x x x x x x x x              0 x  or 8 0 8 x x    Thus, the points   0, 0 and   8, 0 are on the x- axis and a distance of 5 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 5 units from the point   4, 4 .         2 2 2 1 2 1 2 2 2 2 2 2 4 0 4 4 16 8 16 16 8 8 32 d x x y y y y y y y y y                         2 2 2 2 2 2 5 8 32 5 8 32 25 8 32 0 8 7 0 7 1 y y y y y y y y y y                7 0 7 y y    or 1 0 1 y y    Thus, the points   0, 7 and   0,1 are on the y- axis and a distance of 5 units from the point   4, 4 . 53. 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           

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    Chapter F Foundations: A Prelude to Functions 8   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          54. 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  . 55. Let   1 0, 0 P  ,   2 0, P s  ,   3 , 0 P s  , and   4 , P s s  . (0,s) (s,0) (0,0) (s,s) X Y The points 1 P and 4 P are endpoints of one diagonal and the points 2 P and 3 P are the endpoints of the other diagonal. 1,4 0 0 , , 2 2 2 2 s s s s M                 2,3 0 0 , , 2 2 2 2 s s s s M                 The midpoints of the diagonals are the same. Therefore, the diagonals of a square intersect at their midpoints. 56. Let   1 0, 0 P  ,   2 , 0 P a  , and 3 3 , 2 2 a a P          . To show that these vertices form an equilateral triangle, we need to show that the distance between any pair of points is the same constant value.           2 2 1 2 2 1 2 1 2 2 2 , 0 0 0 d P P x x y y a a a                 2 2 2 3 2 1 2 1 2 2 2 2 2 2 , 3 0 2 2 3 4 4 4 4 d P P x x y y a a a a a a a a                                  2 2 1 3 2 1 2 1 2 2 2 2 2 2 , 3 0 0 2 2 3 4 4 4 4 d P P x x y y a a a a a a a                            Since all three distances have the same constant value, the triangle is an equilateral triangle. Now find the midpoints:

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    Section F.1: The Distance and Midpoint Formulas 9 1 2 2 3 1 3 0 0 0 , , 0 2 2 2 3 3 3 0 , 2 2 , 4 4 2 2 3 0 0 3 2 2 , , 2 2 4 4 P P P P P P a a D M a a a a a E M a a a a F M                                                                      2 2 2 2 2 2 3 3 , 0 4 2 4 3 4 4 3 16 16 2 a a a d D E a a a a a                                        2 2 2 2 2 2 3 , 0 4 2 4 3 4 4 3 16 16 2 a a a d D F a a a a a                                         2 2 2 2 2 3 3 3 , 4 4 4 4 0 2 4 2 a a a a d E F a a a                             Since the sides are the same length, the triangle is equilateral. 57. 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. 58.   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.

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    Chapter F Foundations: A Prelude to Functions 10 59.     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. 60.     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. 61. 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 62. 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 63. 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          

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