’ S
SOLUTIONS M ANUAL
T IM BRITT
Jackson State Community College
P RECALCULUS
NINTH EDITION
Michael Sullivan
Chicago State University

Table of Contents
Preface
Chapter 1 Graphs
1.1 The Distance and Midpoint Formulas......................................................................................... 1
1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ................................................ 12
1.3 Lines ......................................................................................................................................... 24
1.4 Circles ....................................................................................................................................... 41
Chapter Review ............................................................................................................................... 52
Chapter Test ..................................................................................................................................... 62
Chapter Projects ............................................................................................................................... 64
Chapter 2 Functions and Their Graphs
2.1 Functions .................................................................................................................................. 66
2.2 The Graph of a Function ........................................................................................................... 80
2.3 Properties of Functions ............................................................................................................. 87
2.4 Library of Functions; Piecewise-defined Functions ............................................................... 102
2.5 Graphing Techniques: Transformations ................................................................................. 113
2.6 Mathematical Models: Building Functions ............................................................................. 129
Chapter Review ............................................................................................................................. 135
Chapter Test ................................................................................................................................... 148
Cumulative Review ....................................................................................................................... 152
Chapter Projects ............................................................................................................................. 155
Chapter 3 Linear and Quadratic Functions
3.1 Linear Functions and Their Properties.................................................................................... 157
3.2 Linear Models: Building Linear Functions from Data ........................................................... 167
3.3 Quadratic Functions and Their Properties .............................................................................. 172
3.4 Build Quadratic Models from Verbal Descriptions and from Data ........................................ 190
3.5 Inequalities Involving Quadratic Functions ............................................................................ 198
Chapter Review ............................................................................................................................. 216
Chapter Test ................................................................................................................................... 229
Cumulative Review........................................................................................................................ 230
Chapter Projects ............................................................................................................................. 233
Chapter 4 Polynomial and Rational Functions
4.1 Polynomial Functions and Models.......................................................................................... 236
4.2 Properties of Rational Functions............................................................................................. 258
4.3 The Graph of a Rational Function .......................................................................................... 265
4.4 Polynomial and Rational Inequalities ..................................................................................... 312
4.5 The Real Zeros of a Polynomial Function .............................................................................. 328
4.6 Complex Zeros; Fundamental Theorem of Algebra ............................................................... 355
Chapter Review ............................................................................................................................. 362
Chapter Test ................................................................................................................................... 391
Cumulative Review........................................................................................................................ 395
Chapter Projects ............................................................................................................................. 400

Chapter 5 Exponential and Logarithmic Functions
5.1 Composite Functions .............................................................................................................. 402
5.2 One-to-One Functions; Inverse Functions .............................................................................. 418
5.3 Exponential Functions ............................................................................................................ 437
5.4 Logarithmic Functions ............................................................................................................ 456
5.5 Properties of Logarithms ........................................................................................................ 476
5.6 Logarithmic and Exponential Equations................................................................................. 484
5.7 Financial Models .................................................................................................................... 501
5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth
and Decay Models .................................................................................................................... 508
5.9 Building Exponential, Logarithmic, and Logistic Models from Data .................................... 515
Chapter Review ............................................................................................................................. 520
Chapter Test ................................................................................................................................... 540
Cumulative Review........................................................................................................................ 544
Chapter Projects ............................................................................................................................. 548
Chapter 6 Trigonometric Functions
6.1 Angles and Their Measure ...................................................................................................... 550
6.2 Trigonometric Functions: Unit Circle Approach .................................................................... 557
6.3 Properties of the Trigonometric Functions ............................................................................. 572
6.4 Graphs of the Sine and Cosine Functions ............................................................................... 584
6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions...................................... 603
6.6 Phase Shift; Sinusoidal Curve Fitting ..................................................................................... 611
Chapter Review ............................................................................................................................. 622
Chapter Test ................................................................................................................................... 637
Cumulative Review........................................................................................................................ 640
Chapter Projects ............................................................................................................................. 644
Chapter 7 Analytic Trigonometry
7.1 The Inverse Sine, Cosine, and Tangent Functions .................................................................. 648
7.2 The Inverse Trigonometric Functions (Continued) ................................................................ 658
7.3 Trigonometric Equations ........................................................................................................ 669
7.4 Trigonometric Identities ......................................................................................................... 688
7.5 Sum and Difference Formulas ................................................................................................ 699
7.6 Double-angle and Half-angle Formulas .................................................................................. 721
7.7 Product-to-Sum and Sum-to-Product Formulas...................................................................... 744
Chapter Review ............................................................................................................................. 754
Chapter Test ................................................................................................................................... 778
Cumulative Review........................................................................................................................ 782
Chapter Projects ............................................................................................................................. 787
Chapter 8 Applications of Trigonometric Functions
8.1 Right Triangle Trigonometry; Applications ........................................................................... 791
8.2 The Law of Sines .................................................................................................................... 803
8.3 The Law of Cosines ................................................................................................................ 816
8.4 Area of a Triangle ................................................................................................................... 825
8.5 Simple Harmonic Motion; Damped Motion; Combining Waves ........................................... 832
Chapter Review ............................................................................................................................. 840
Chapter Test ................................................................................................................................... 852
Cumulative Review........................................................................................................................ 855
Chapter Projects ............................................................................................................................. 861

Chapter 9 Polar Coordinates; Vectors
9.1 Polar Coordinates.................................................................................................................... 865
9.2 Polar Equations and Graphs.................................................................................................... 872
9.3 The Complex Plane; DeMoivre’s Theorem ............................................................................ 900
9.4 Vectors .................................................................................................................................... 910
9.5 The Dot Product...................................................................................................................... 920
9.6 Vectors in Space ..................................................................................................................... 924
9.7 The Cross Product................................................................................................................... 930
Chapter Review ............................................................................................................................. 940
Chapter Test ................................................................................................................................... 958
Cumulative Review........................................................................................................................ 962
Chapter Projects ............................................................................................................................. 964
Chapter 10 Analytic Geometry
10.2 The Parabola ......................................................................................................................... 968
10.3 The Ellipse ............................................................................................................................ 982
10.4 The Hyperbola ...................................................................................................................... 997
10.5 Rotation of Axes; General Form of a Conic ....................................................................... 1015
10.6 Polar Equations of Conics................................................................................................... 1027
10.7 Plane Curves and Parametric Equations ............................................................................. 1034
Chapter Review ........................................................................................................................... 1045
Chapter Test ................................................................................................................................. 1063
Cumulative Review...................................................................................................................... 1068
Chapter Projects ........................................................................................................................... 1070
Chapter 11 Systems of Equations and Inequalities
11.1 Systems of Linear Equations: Substitution and Elimination............................................... 1074
11.2 Systems of Linear Equations: Matrices .............................................................................. 1093
11.3 Systems of Linear Equations: Determinants ....................................................................... 1115
11.4 Matrix Algebra.................................................................................................................... 1127
11.5 Partial Fraction Decomposition .......................................................................................... 1144
11.6 Systems of Nonlinear Equations ......................................................................................... 1157
11.7 Systems of Inequalities ....................................................................................................... 1184
11.8 Linear Programming ........................................................................................................... 1198
Chapter Review ........................................................................................................................... 1209
Chapter Test ................................................................................................................................. 1235
Cumulative Review...................................................................................................................... 1244
Chapter Projects ........................................................................................................................... 1248
Chapter 12 Sequences; Induction; the Binomial Theorem
12.1 Sequences ........................................................................................................................... 1250
12.2 Arithmetic Sequences ......................................................................................................... 1259
12.3 Geometric Sequences; Geometric Series ............................................................................ 1265
12.4 Mathematical Induction ...................................................................................................... 1275
12.5 The Binomial Theorem ....................................................................................................... 1282
Chapter Review ........................................................................................................................... 1287
Chapter Test ................................................................................................................................. 1294
Cumulative Review...................................................................................................................... 1297
Chapter Projects ........................................................................................................................... 1300

Chapter 13 Counting and Probability
13.1 Counting ............................................................................................................................. 1303
13.2 Permutations and Combinations ......................................................................................... 1304
13.3 Probability ........................................................................................................................... 1308
Chapter Review ........................................................................................................................... 1313
Chapter Test ................................................................................................................................. 1315
Cumulative Review...................................................................................................................... 1317
Chapter Projects ........................................................................................................................... 1320
Chapter 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function
14.1 Finding Limits Using Tables and Graphs ........................................................................... 1322
14.2 Algebra Techniques for Finding Limits .............................................................................. 1327
14.3 One-sided Limits; Continuous Functions ........................................................................... 1331
14.4 The Tangent Problem; The Derivative................................................................................ 1339
14.5 The Area Problem; The Integral ......................................................................................... 1348
Chapter Review ........................................................................................................................... 1362
Chapter Test ................................................................................................................................. 1374
Chapter Projects ........................................................................................................................... 1378
Appendix A Review
A.1 Algebra Essentials ................................................................................................................ 1383
A.2 Geometry Essentials............................................................................................................. 1388
A.3 Polynomials ......................................................................................................................... 1393
A.4 Synthetic Division ................................................................................................................ 1402
A.5 Rational Expressions............................................................................................................ 1404
A.6 Solving Equations ................................................................................................................ 1409
A.7 Complex Numbers; Quadratic Equations in the Complex Number System ........................ 1423
A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications ..... 1428
A.9 Interval Notation; Solving Inequalities ................................................................................ 1436
A.10 nth Roots; Rational Exponents............................................................................................ 1447
Appendix B Graphing Utilities
B.1 The Viewing Rectangle ........................................................................................................ 1456
B.2 Using a Graphing Utility to Graph Equations ...................................................................... 1457
B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ............................ 1461
B.5 Square Screens ..................................................................................................................... 1463

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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; y-coordinate
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. (a) Quadrant II
(b) x-axis
(c) Quadrant III
(d) Quadrant I
(e) y-axis
(f) Quadrant IV
14. (a) Quadrant I
(b) Quadrant III
(c) Quadrant II
(d) Quadrant I
(e) y-axis
(f) x-axis
15. 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
16. The points will be on a horizontal line that is
three units above the x-axis.
17. 2 2
1 2
2 2
( , ) (2 0) (1 0)
2 1 4 1 5
d P P = − + −
= + = + =
18. 2 2
1 2
2 2
( , ) ( 2 0) (1 0)
( 2) 1 4 1 5
d P P = − − + −
= − + = + =
19. 2 2
1 2
2 2
( , ) ( 2 1) (2 1)
( 3) 1 9 1 10
d P P = − − + −
= − + = + =
20. ( )2 2
1 2
2 2
( , ) 2 ( 1) (2 1)
3 1 9 1 10
d P P = − − + −
= + = + =
21. ( )( )
( )
22
1 2
22
( , ) (5 3) 4 4
2 8 4 64 68 2 17
d P P = − + − −
= + = + = =
22. ( )( ) ( )
( )
2 2
1 2
2 2
( , ) 2 1 4 0
3 4 9 16 25 5
d P P = − − + −
= + = + = =
23. ( )2 2
1 2
2 2
( , ) 6 ( 3) (0 2)
9 ( 2) 81 4 85
d P P = − − + −
= + − = + =
24. ( ) ( )2 2
1 2
2 2
( , ) 4 2 2 ( 3)
2 5 4 25 29
d P P = − + − −
= + = + =
25. ( )22
1 2
2 2
( , ) (6 4) 4 ( 3)
2 7 4 49 53
d P P = − + − −
= + = + =
26. ( ) ( )2 2
1 2
2 2
( , ) 6 ( 4) 2 ( 3)
10 5 100 25
125 5 5
d P P = − − + − −
= + = +
= =
27. 2 2
1 2
2 2 2 2
( , ) (0 ) (0 )
( ) ( )
d P P a b
a b a b
= − + −
= − + − = +
28. 2 2
1 2
2 2
2 2 2
( , ) (0 ) (0 )
( ) ( )
2 2
d P P a a
a a
a a a a
= − + −
= − + −
= + =

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Section 1.1: The Distance and Midpoint Formulas
3
[ ] [ ]1 ( , ) ( , )
2
1 113 13 13
2 2
13 square units2
A d A B d B C= ⋅⋅
= ⋅ = ⋅⋅
=
30. ( 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= ⋅ ⋅
= ⋅ ⋅
= ⋅ ⋅
=
31. ( 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 1: Graphs
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= ⋅ ⋅
= ⋅ ⋅
= ⋅ ⋅
= ⋅ ⋅
=
32. ( 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= ⋅ ⋅
= ⋅ ⋅
= ⋅ ⋅
=
33. (4, 3), (0, 3), (4, 2)A B C= − = − =
( )
( ) ( )
( )
22
2 2
2 2
2 2
22
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:
[ ] [ ] [ ]
( )
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

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Section 1.1: The Distance and Midpoint Formulas
5
problem,
[ ] [ ]1 ( , ) ( , )
2
1 4 5
2
10 square units
A d A B d A C= ⋅ ⋅
= ⋅ ⋅
=
34. (4, 3), (4, 1), (2, 1)A B C= − = =
( )
( ) ( )
( )
22
2 2
2 2
2 2
22
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= ⋅ ⋅
= ⋅ ⋅
=
35. The coordinates of the midpoint are:
1 2 1 2
( , ) ,
2 2
4 43 5 ,
2 2
8 0
,
2 2
(4, 0)
x x y y
x y + + 
=  
 
− ++ 
=  
 
 
=  
 
=
36. 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 + + 
=  
 
− + + 
=  
 
 
=  
 
=
37. 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 + + 
=  
 
− + + 
=  
 
 
=  
 
 
=  
 
38. 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 + + 
=  
 
+ − + 
=  
 
− 
=  
 
 
= − 
 

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Chapter 1: Graphs
6
39. The coordinates of the midpoint are:
1 2 1 2
( , ) ,
2 2
4 6 3 1
,
2 2
210 ,
2 2
(5, 1)
x x y y
x y + + 
=  
 
+ − + 
=  
 
− 
=  
 
= −
40. 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 + + 
=  
 
− + − + 
=  
 
− − 
=  
 
 
= − − 
 
41. 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
+ + 
=  
 
+ + 
=  
 
 
=  
 
42. 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
+ + 
=  
 
+ + 
=  
 
 
=  
 
43. 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 .
44. 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−
45. 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

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Section 1.1: The Distance and Midpoint Formulas
7
b. Consider points of the form ( ), 6x − that are
a distance of 17 units from the point ( )1, 2 .
( ) ( )
( ) ( )( )
( )
2 2
2 1 2 1
22
22
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 .
47. Points on the x-axis have a y-coordinate of 0. Thus,
we consider points of the form ( ), 0x that are a
distance of 6 units from the point ( )4, 3− .
( ) ( )
( ) ( )
( )
2 2
2 1 2 1
2 2
22
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 3x = + or 4 3 3x = −
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− .
48. Points on the y-axis have an x-coordinate of 0.
Thus, we consider points of the form (

Loading page 13...

Chapter 1: Graphs
8
are on the y-axis and a distance of 6 units from the
point ( )4, 3− .
49. ( ) 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 = .
50. ( ) 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 = − .
51. 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

Loading page 14...

Section 1.1: The Distance and Midpoint Formulas
9
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.
54. Let ( )1 0, 0P = , ( )2 , 0P 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

Loading page 15...

Chapter 1: Graphs
10
56. ( )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 22
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.
57. ( ) ( )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.
58. ( ) ( )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 = − − +

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Solution Manual for Precalculus, 8th Edition

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