266 CHAPTER 2 Graphs and Functions If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. 75. 1-1, 42, 1-2, -12, 11, 142 76. 10, -72, 1-3, 52, 12, -152 77. 1-1, -32, 1-5, 122, 11, -112 78. 10, 92, 1-3, -72, 12, 192 Relating Concepts For individual or collaborative investigation (Exercises 79–85) In this section we state that two lines, neither of which is vertical, are perpendicular if and only if their slopes have a product of -1. In Exercises 79–85, we outline a partial proof of this for the case where the two lines intersect at the origin. Work these exercises in order, and refer to the figure as needed. By the converse of the Pythagorean theorem, if 3d1O, P242 + 3d1O, Q242 = 3d1P, Q242, then triangle POQ is a right triangle with right angle at O. 79. Find an expression for the distance d1O, P2. 80. Find an expression for the distance d1O, Q2. 81. Find an expression for the distance d1P, Q2. 82. Use the results from Exercises 79–81, and substitute into the equation from the Pythagorean theorem. Simplify to show that this leads to the equation -2m1m2x1x2 - 2x1x2 = 0. 83. Factor -2x1x2 from the final form of the equation in Exercise 82. 84. Use the property that if ab = 0 then a = 0 or b = 0 to solve the equation in Exercise 83, showing that m1m2 = -1. 85. State a conclusion based on the results of Exercises 79–84. y2 = m2x Q(x2, m2x2) P(x 1, m1x1) y1= m1x x y O These summary exercises provide practice with some of the concepts covered so far in this chapter. For the points P and Q, find (a) the distance d1P, Q2, (b) the coordinates of the midpoint of the segment PQ, and (c) an equation for the line through the two points. Write the equation in slope-intercept form if possible. 1. P13, 52, Q12, -32 2. P1-1, 02, Q14, -22 3. P1-2, 22, Q13, 22 4. PA222, 22B, QA 22, 322B 5. P15, -12, Q15, 12 6. P11, 12, Q1-3, -32 7. PA223, 325B, QA623, 325B 8. P10, -42, Q13, 12 Summary Exercises on Graphs, Circles, Functions, and Equations
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