Parallel and Perpendicular Lines

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7.2 Parallel and Perpendicular Lines 7.2 OBJECTIVES 1. Determine whether two lines are parallel 2. Determine whether two lines are perpendicular 3. Find the slope of a line perpendicular to a given line For most ineperienced drivers, the most difficult driving maneuver to master is parallel parking. What is parallel parking? It is the act of backing into a curbside space so that the car s tires are parallel to the curb. How can ou tell that ou ve done a good job of parallel parking? Most people check to see that both the front tires and the back tires are the same distance (8 in. or so) from the curb. This is checking to be certain that the car is parallel to the curb. How can we tell that two equations represent parallel lines? Look at the sketch below. If two lines are parallel, the have the same slope. If their equations are in slopeintercept form, ou simpl compare the slopes. Eample 1 Determining That Two Lines Are Parallel Which two equations represent parallel lines? (a) 2 3 567

568 CHAPTER 7 GRAPHING AND INEQUALITIES (b) 1 2 5 (c) 2 1 2 (d) 2 9 Because (c) and (d) both have a slope of 2, the lines are parallel. CHECK YOURSELF 1 Which two equations represent parallel lines? (a) 5 5 (b) 1 5 5 (c) 5 1 (d) 5 9 2 More formall, we can state the following about parallel lines. Definitions: Slope of Parallel Lines NOTE This means that if the lines are parallel, then their slopes are equal. Conversel, if the slopes are equal, then the lines are parallel. For nonvertical lines L 1 and L 2, if line L 1 has slope m 1 and line L 2 has slope m 2, then L 1 is parallel to L 2 if and onl if m 1 m 2 Note: All vertical lines are parallel to each other. As we discovered in Chapter 6, we can find the slope of a line from an two points on the line. Eample 2 Parallel Lines Are lines L 1 through (2, 3) and (4, 6) and L 2 through ( 4, 2) and (0, 8) parallel, or do the intersect? NOTE Unless, of course, L 1 and L 2 are actuall the same line. In this case a quick sketch will show that the lines are distinct. m 1 6 3 4 2 3 2 m 2 8 2 0 ( 4) 6 4 3 2 Because the slopes of the lines are equal, the lines are parallel. The do not intersect.

PARALLEL AND PERPENDICULAR LINES SECTION 7.2 569 CHECK YOURSELF 2 Are lines L 1 through ( 2, 1) and (1, 4) and L 2 through ( 3, 4) and (0, 8) parallel, or do the intersect? Man important characteristics of lines are evident from a cit map. Note that Fourth Street and Fifth Street are parallel. Just as these streets never meet, it is true that two parallel lines will never meet. Recall that the point at which two lines meet is called their intersection. This is also true with two streets. We call the common area of the two streets the intersection. In this case, the two streets meet at right angles. When two lines meet at right angles, we sa that the are perpendicular. Definitions: Slope of Perpendicular Lines For nonvertical lines L 1 and L 2, if line L 1 has slope m 1 and line L 2 has slope m 2, then L 1 is perpendicular to L 2 if and onl if m 1 1 m 2 or equivalentl m 1 m 2 1 Note: Horizontal lines are perpendicular to vertical lines.

570 CHAPTER 7 GRAPHING AND INEQUALITIES Eample 3 Determining That Two Lines Are Perpendicular Which two equations represent perpendicular lines? (a) 2 3 (b) 1 2 5 (c) 2 1 2 (d) 2 9 Because the product of the slopes for (a) and (b) is 2 1 2 1 these two lines are perpendicular. Note that none of the other pairs of slopes have a product of 1. CHECK YOURSELF 3 Which two equations represent perpendicular lines? (a) 5 5 (b) 1 5 5 1 (c) 5 (d) 5 9 2 Eample 4 Perpendicular Lines Are lines L 1 through points ( 2, 3) and (1, 7) and L 2 through points (2, 4) and (6, 1) perpendicular? NOTE 4 3 3 4 1 m 1 7 3 1 ( 2) 4 3 m 2 1 4 6 2 3 4

PARALLEL AND PERPENDICULAR LINES SECTION 7.2 571 Because the slopes are negative reciprocals, the lines are perpendicular. (1, 7) 4 m 1 3 ( 2, 3) L 1 (2, 4) m 2 4 3 L 2 (6, 1) CHECK YOURSELF 4 Are lines L 1 through points (1, 3) and (4, 1) and L 2 through points ( 2, 4) and (2, 10) perpendicular? We can also use the slope-intercept form to determine whether the graphs of given equations will be parallel, intersecting, or perpendicular lines. Eample 5 Verifing That Two Lines Are Parallel Show that the graphs of 3 2 4 and 6 4 12 are parallel lines. First, we solve each equation for : 3 2 4 2 3 4 3 2 2 (1) NOTE Notice that the slopes are the same, but the intercepts are different. Therefore the lines are distinct. 6 4 12 4 6 12 3 2 3 Because the two lines have the same slope, here, the lines are parallel. 3 2 (2)

572 CHAPTER 7 GRAPHING AND INEQUALITIES CHECK YOURSELF 5 Show that the graphs of the equations 3 2 4 and 2 3 9 are perpendicular lines. CHECK YOURSELF ANSWERS 1. (a) and (c) 2. The lines intersect 3. (b) and (d) 4. The lines are perpendicular 5. 3 2 2 2 3 3 3 2 2 3 1

Name 7.2 Eercises Section Date In eercises 1 to 4, determine which two equations represent parallel lines. 1. (a) 4 5 (b) 4 5 (c) 1 (d) 4 9 4 5 ANSWERS 2. (a) 3 5 (b) 3 5 (c) 3 2 (d) 1 3 5 1. 2. 3. (a) 2 (b) 3 (c) 4 12 (d) 4 3 3 3 2 6 3. 4. 4. (a) 9 (b) 4 (c) 4 (d) 4 3 9 7 9 7 9 4 7 5. In eercises 5 to 8, determine which two equations represent perpendicular lines. 6. 7. 5. (a) 6 3 (b) 1 (c) 1 (d) 6 3 6 3 6. (a) 2 (b) (c) 2 (d) 3 8 2 3 6 3 6 7. (a) 1 (b) 3 9 (c) 1 (d) 3 9 3 9 8. (a) 5 (b) 6 5 (c) 1 (d) 6 5 9 6 9 9 Are the following pairs of lines parallel, perpendicular, or neither? 9. L 1 through ( 2, 3) and (4, 3) L 2 through (3, 5) and (5, 7) 10. L 1 through ( 2, 4) and (1, 8) L 2 through ( 1, 1) and ( 5, 2) 11. L 1 through (8, 5) and (3, 2) L 2 through ( 2, 4) and (4, 1) 1 6 3 3 2 6 1 3 9 1 6 5 9 8. 9. 10. 11. 12. 13. 14. 15. 12. L 1 through ( 2, 3) and (3, 1) L 2 through ( 3, 1) and (7, 5) 13. L 1 with equation 3 6 L 2 with equation 3 3 14. L 1 with equation 2 4 L 2 with equation 2 4 5 15. Find the slope of an line parallel to the line through points ( 2, 3) and (4, 5). 573

ANSWERS 16. 17. 18. 19. 20. 21. 22. 23. 16. Find the slope of an line perpendicular to the line through points (0, 5) and ( 3, 4). 17. A line passing through ( 1, 2) and (4, ) is parallel to a line with slope 2. What is the value of? 3 18. A line passing through (2, 3) and (5, ) is perpendicular to a line with slope. What 4 is the value of? In eercises 19 to 21, use the concept of slope to determine if the given figure is a parallelogram or a rectangle. 19. 20. 21. In eercises 22 to 24, use the concept of slope to determine whether the given figure is a right triangle (i.e., does the triangle contain a right angle?). 22. 23. 574

ANSWERS 24. 24. 25. 26. 27. a. b. c. In eercises 25 to 27, use the concept of slope to draw a line perpendicular to the given line segment, passing through the marked point. d. e. 25. 26. f. 27. Getting Read for Section 7.3 [Section 2.5] Solve the following equations for. (a) 3 4 12 (b) 5 (c) 2 6 4 (d) 5 (e) 5 2 10 (f) 3 5 575

Answers 1. a and d 3. c and d 5. a and c 7. b and d 9. Parallel 11. Neither 13. Perpendicular 15. 1 3 17. 12 19. Parallelogram 21. Rectangle 23. Yes 25. 27. a. 3 b. 5 c. 1 d. 5 2 3 4 3 2 e. 5 f. 3 5 2 5 576

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