794 CHAPTER 11 Systems of Equations and Inequalities 62. Geometry:Volume of aTetrahedron A tetrahedron (triangular pyramid) has vertices ( ) ( ) ( ) x y z x y z x y z , , , , , , , 1, 1 1 2 2 2 3 3 3 , and ( ) x y z , , . 4 4 4 The volume of the tetrahedron is given by the absolute value of D, where = D x y z x y z x y z x y z 1 6 1 1 1 1 . 1 1 1 2 2 2 3 3 3 4 4 4 Use this formula to find the volume of the tetrahedron with vertices ( ) 0, 0, 8 , ( ) 2, 8, 0 , ( ) 10, 4, 4 , and ( ) 4, 10, 6 . 63. Geometry: Equation of a Circle An equation of the circle containing the distinct points ( ) ( ) x y x y , , , , 1 1 2 2 and ( ) x y , 3 3 can be found using the following equation. + + + + = x x x x y y y y x y x y x y x y 1 1 1 1 0 1 2 3 1 2 3 2 2 1 2 1 2 2 2 2 2 3 2 3 2 Find the standard equation of the circle containing the points ( ) ( ) − 7, 5 , 3,3 ,and ( ) 6, 2 . 64. Show that ( )( )( ) = − − − x x y y z z y z x y x z 1 1 1 . 2 2 2 65. Complete the proof of Cramer’s Rule for two equations containing two variables. [Hint: In system (5), page 786, if = a 0, then ≠ b 0 and ≠ c 0, since = − ≠ D bc 0. Now show that equation (6) provides a solution of the system when = a 0. Then three cases remain: = = b c 0, 0, and = d 0.] 66. Challenge Problem Interchange columns 1 and 3 of a 3 by 3 determinant. Show that the value of the new determinant is −1 times the value of the original determinant. 67. Challenge Problem Multiply each entry in row 2 of a 3 by 3 determinant by the number ≠ k k , 0. Show that the value of the new determinant is k times the value of the original determinant. 68. Challenge Problem Prove that a 3 by 3 determinant in which the entries in column 1 equal those in column 3 has the value 0. 69. Challenge Problem If row 2 of a 3 by 3 determinant is multiplied by ≠ k k , 0, and the result is added to the entries in row 1, prove that there is no change in the value of the determinant. 57. Geometry: Equation of a Line An equation of the line containing the two points ( ) x y , 1 1 and ( ) x y , 2 2 may be expressed as the determinant x y x y x y 1 1 1 0 1 1 2 2 = Prove this result by expanding the determinant and comparing the result to the two-point form of the equation of a line. 58. Geometry: Collinear Points Using the result obtained in Problem 57, show that three distinct points ( ) ( ) x y x y , , , , 1 1 2 2 and ( ) x y , 3 3 are collinear (lie on the same line) if and only if = x y x y x y 1 1 1 0 1 1 2 2 3 3 59. Geometry: Area of a Triangle A triangle has vertices ( ) ( ) x y x y , , , , 1 1 2 2 and ( ) x y , . 3 3 The area of the triangle is given by the absolute value of D, where = D x x x y y y 1 2 1 1 1 . 1 2 3 1 2 3 Use this formula to find the area of a triangle with vertices ( ) 2, 3 , ( ) 5, 2 , and ( ) 6, 5 . 60. Geometry: Area of a Polygon The formula from Problem 59 can be used to find the area of a polygon.To do so, divide the polygon into non-overlapping triangular regions and find the sum of the areas. Use this approach to find the area of the given polygon. x y (8, 4) (6, 8) (1, 6) (21, 3) (6, 22) 61. Geometry: Area of a Polygon Another approach for finding the area of a polygon by using determinants is to use the formula = + + + + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ ⎟ … A x y x y x y x y x y x y x y x y 1 2 n n 1 1 2 2 2 2 3 3 3 3 4 4 1 1 where ( ) ( ) ( ) x y x y x y , , , , , , n n 1 1 2 2 … are the n corner points in counterclockwise order. Use this formula to compute the area of the polygon from Problem 60 again. Which method do you prefer? Retain Your Knowledge Problems 70–79 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 70. For the points ( ) = − P 4, 3 and ( ) = − Q 5, 1 write the vector v represented by the directed line segment PQ in the form + a b i j and find v . 71. List the potential rational zeros of the polynomial function ( ) = − + − P x x x x 2 5 10. 3 2 72. Graph ( ) ( ) = + − f x x 1 4 2 using transformations (shifting, compressing, stretching, and/or reflecting). 73. Find the exact value of ° − ° tan42 cot48 without using a calculator. 74. Express − + i 5 5 in polar form and in exponential form.
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