2. Parallel and Perpendicular Lines By Lindsay Hojnowski (2014) Buffalo State College 04/2014L. Hojnowski © 20141 Click here to play tutorial introduction Parallel Lines Perpendicular Lines

3. Learning Objectives The learner will be able to put the equations in slope-intercept form to identify the slope 85% of the time. The learner will be able to identify what the parallel or perpendicular slope is 85% of the time. Student wills be able to use the point-slope formula to find parallel lines given a point and a line (using the given slope) 80% of the time. Students will be able to use the point-slope formula to find perpendicular lines given a point and a line (using the given slope) 80% of the time. 04/2014L. Hojnowski © 20142 Aim for the Target

4. Menu 04/2014L. Hojnowski © 20143 References Characteristics of Parallel Lines Parallel Lines- Steps Given a point and an equation Parallel Lines- Example 1 Perpendicular Lines- Example 1 Parallel Lines- Example 3 Characteristics of Perpendicular Lines Perpendicular Lines- Steps Given a point and an equation Parallel Lines- Example 2 Perpendicular Lines- Example 2 Perpendicular Lines- Example 3 Determine whether parallel, perpendicular, or neither- Steps Determine- Example 1 Determine- Example 2 Determine- Example 3 Quiz Question #1 Quiz Question #2 Quiz Question #7 Quiz Question #6 Quiz Question #5 Quiz Question #4 Quiz Question #3

5. Characteristics of Parallel Lines Parallel lines: 1)Are lines that do not intersect 2)Have different y-intercepts - Click on the picture below to see a video on how to write a parallel line to another line using point-slope form 04/2014L. Hojnowski © 20144 Parallel Lines- JMAP Video

6. Parallel Lines- Steps Given a point and an equation 04/2014L. Hojnowski © 20145 Steps to writing a parallel line STEPS: 1)Rewrite the given equation into slope- intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the parallel slope (found in step 1) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

7. Parallel Lines- Example 1 04/2014L. Hojnowski © 20146 Example 1: Write an equation in slope-intercept form for the line that passes through (-2, 2) and is parallel to y = 4x – 2. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = 43) y – 2 = 4 (x + 2) 2)y – y 1 = m (x – x 1 ) y – 2 = 4x + 8 y – 2 = 4 (x - - 2) +2 +2 y – 2 = 4 (x + 2) 4) y = 4x + 10 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the parallel slope (found in step 1) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

8. Parallel Lines- Example 2 04/2014L. Hojnowski © 20147 Example 2: Write an equation in slope-intercept form for the line that passes through (6, 4) and is parallel to y = (1/3)x + 1. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = 1/33) y – 4 = (1/3) (x - 6) 2)y – y 1 = m (x – x 1 ) y – 4 = (1/3)x - 2 y – 4 = (1/3) (x - 6) +4 +4 4) y = (1/3)x + 2 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the parallel slope (found in step 1) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

9. Parallel Lines- Example 3 04/2014L. Hojnowski © 20148 Example 3: Write an equation in slope-intercept form for the line that passes through (-1, 6) and is parallel to 3x + y = 12. **Use the point-slope formula** The equation is NOT in slope-intercept form, m = ? 3) y – 6 = -3 (x – -1) **In order to identify the slope, solve for y! y – 6 = -3 (x + 1) 3x + y = 12 y – 6 = -3x - 3 -3x y = -3x +12 m = -3 4) y – 6 = -3x - 3 +6 +6 2) y – y 1 = m (x – x 1 ) y = -3x +3 y – 6 = -3 (x – -1) Example of a given point and a line

10. Characteristics of Perpendicular Lines 04/2014L. Hojnowski © 20149 Perpendicular Lines: 1)Are lines that intersect at right angles 2)Have negative reciprocal slopes -Example: m = 2  m = -1/2 - Click on the picture below to see a video to review how to write a perpendicular line to another line using slope-intercept form (you can use point-slope formula just like parallel lines) Perpendicular Lines- JMAP Video

11. Perpendicular Lines- Steps Given a point and an equation 04/2014L. Hojnowski © 201410 Steps to writing a perpendicular l line STEPS: 1)Rewrite the given equation into slope- intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the perpendicular slope (negative reciprocal) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

12. Perpendicular Lines- Example 1 04/2014L. Hojnowski © 201411 Example 1: Write an equation in slope-intercept form for the line that passes through (4, 2) and is perpendicular to y = (1/2)x + 1. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = (1/2)3) y – 2 = -2 (x - 4) Perpendicular slope: -2 y – 2 = -2x + 8 2) y – y 1 = m (x – x 1 ) + 2 + 2 y – 2 = -2 (x - 4) 4) y= -2x + 10 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the perpendicular slope (negative reciprocal) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

13. Perpendicular Lines- Example 2 04/2014L. Hojnowski © 201412 Example 2: Write an equation in slope-intercept form for the line that passes through (-5, -1) and is perpendicular to y = (5/2)x - 3. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = (5/2)3) y + 1= (-2/5) (x + 5) Perpendicular slope: (-2/5) y + 1= (-2/5)x - 2 2)y – y 1 = m (x – x 1 ) - 1 - 1 y – -1 = (-2/5) (x - - 5) 4) y= (-2/5)x - 3 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the perpendicular slope (negative reciprocal) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

14. Example 3: Write an equation in slope-intercept form for the line that passes through (-4, 6) and is perpendicular to 2x + 3y = 12. **Use the point-slope formula** 1) The equation is NOT in slope-intercept form, m = ? **In order to identify the slope, solve for y! 2x + 3y = 12 2) y – y 1 = m (x – x 1 ) 3) y – 6 = (3/2)(x + 4) -2x -2x y – 6 = (3/2)(x – -4) y – 6 = (3/2)x + 6 3y = -2x + 12+ 6 + 6 3 y = (-2/3)x + 4 4) y = (3/2)x + 12 m = -2/3 Perpendicular slope: (3/2) Perpendicular Lines- Example 3 04/2014L. Hojnowski © 201413 Given a Point and a Line

15. Determine whether parallel, perpendicular, or neither- Steps 04/2014L. Hojnowski © 201414 STEPS: 1)Rewrite both equation into slope-intercept form (y = mx + b) and identify each slope 2)Compare the slopes to see if they are the same, negative reciprocal, or neither Example of Parallel Lines- Same Slope Example of Perpendicular l Lines- Negative Reciprocal Slope Example of Neither Parallel or Perpendicular Lines

16. Determine whether parallel, perpendicular, or neither- Example 1 04/2014L. Hojnowski © 201415 Example1: Determine whether the graphs of the pair of equations are parallel, perpendicular, or neither. 3x + 5y = 105x – 3y = -6 STEPS: 1)Rewrite both equation into slope-intercept form (y = mx + b) and identify each slope 2)Compare the slopes to see if they are the same, negative reciprocal, or neither 5x – 3y = -6 -5x -3y = -5x - 6 -3 -3 y = (-5/-3)x + 2 m = (5/3) 3x + 5y = 10 -3x 5y = -3x + 10 5 5 y = (-3/5)x + 2 m = (-3/5) PERPENDICULAR- they have negative reciprocal slopes

17. Determine whether parallel, perpendicular, or neither- Example 2 04/2014L. Hojnowski © 201416 Example 2: Determine whether the graphs of the pair of equations are parallel, perpendicular, or neither. 2x - 8y = -24 x – 4y = 4 STEPS: 1)Rewrite both equation into slope-intercept form (y = mx + b) and identify each slope 2)Compare the slopes to see if they are the same, negative reciprocal, or neither PARALLEL- they have the same slope x – 4y = 4 -x -4y = -x + 4 -4 -4 y = (-1/-4)x - 1 m = (1/4) 2x - 8y = -24 -2x -8y = -2x - 24 -8 -8 y = (-2/-8)x + 3 m = (1/4)

18. Determine whether parallel, perpendicular, or neither- Example 3 04/2014L. Hojnowski © 201417 Example 3: Determine whether the graphs of the pair of equations are parallel, perpendicular, or neither. -3x + 4y = 8 -4x + 3y = -6 NEITHER- they aren’t the same slope and are not negative reciprocals They are reciprocals but not negative reciprocals -4x + 3y = -6 +4x 3y = 4x - 6 3 3 y = (4/3)x - 2 m = (4/3) -3x + 4y = 8 +3x 4y = 3x + 8 4 4 y = (3/4)x + 2 m = (3/4)

19. Quiz Question #1 04/2014L. Hojnowski © 201418 1.What is the perpendicular slope of the line that passes through the line: y = (-3/4)x + 4 ? a. a. -4/3 b. 3/4 c. 4/3d. -3/4 b. c. d.

20. Try Again… 04/2014L. Hojnowski © 201419 This slope is the reciprocal of the slope given. Perpendicular slopes are the negative reciprocals. Quiz Question #1 Quiz Question #2 Try Again Perpendicular Lines

21. Try Again… 04/2014L. Hojnowski © 201420 Quiz Question #1 Quiz Question #2 Try Again This slope is the negative of the slope given. Perpendicular slopes are the negative reciprocals. Intersecting Lines

22. Correct!! 04/2014L. Hojnowski © 201421 Quiz Question #1 Quiz Question #2 Smile You are correct! Perpendicular slopes are the negative reciprocals. Negative Reciprocal Slope

23. Try Again… 04/2014L. Hojnowski © 201422 Quiz Question #1 Quiz Question #2 Try Again This slope is the same as the slope given. This would be correct if the question asked for the parallel slope. Perpendicular slopes are the negative reciprocals. Parallel Lines

24. Quiz Question # 2 04/2014L. Hojnowski © 201423 2. Which line is parallel to the line 4x + y = 3? a.a. y = (1/4) x – 1 b. y = 4x + 2 c. y = (-1/4) x – 6 d. y = -4x + 5b. c. d.

25. Try Again… 04/2014L. Hojnowski © 201424 If the question asked you to write the equation of the line in point-slope form, you would be correct. From this answer, you need to distribute and get y by itself. Quiz Question #1 Quiz Question #2 Try Again Quiz Question #3

26. Try Again… 04/2014L. Hojnowski © 201425 Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 If the question asked you to write the equation of the line in standard, you would be correct. You went one step too far.

27. Try Again… 04/2014L. Hojnowski © 201426 Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 There is a sign off in your equation. This is in slope-intercept form though! You are right about that.

28. Correct!! 04/2014L. Hojnowski © 201427 You plugged in and solved correctly! Slope-intercept form is y = mx + b; by just knowing this detail, answers #1 and #3 could be eliminated. Smile Quiz Question #1 Quiz Question #2 Quiz Question #3

29. 04/2014L. Hojnowski © 201428 3. What is the slope of the line 2x + 7y = -35? a.a. 2/7b. -2/7c. 7/2 d. -7/2b. c. d.

30. Try Again… 04/2014L. Hojnowski © 201429 When you start to solve for y, you subtract 2x from both sides. This leads the slope to be negative and not positive. Quiz Question #2 Quiz Question #1 Try Again Quiz Question #4 Quiz Question #3

31. Correct!! 04/2014L. Hojnowski © 201430 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 You solved correctly! The slope is -2/7! When solving for y, you subtracted 2x from both sides and divided by 7.

32. Try Again… 04/2014L. Hojnowski © 201431 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 This is the perpendicular slope of the equation you get after you solve. If the question asked for the perpendicular slope, you would be correct.

33. Try Again… 04/2014L. Hojnowski © 201432 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 This is the reciprocal slope of the equation you get after you solve.

34. Quiz Question # 4 04/2014L. Hojnowski © 201433 4. Write an equation in slope-intercept form for the line that passes through (0, 4) and is parallel to y = -4x + 5. a. a. y = -4xb. y = -4x - 4c. y = (1/4)x + 4d. y = -4x + 4b. c. d.

35. Try Again… 04/2014L. Hojnowski © 201434 Careful anything multiplied by zero is zero! Try again using this knowledge. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5

36. Try Again… 04/2014L. Hojnowski © 201435 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Careful with your signs and solving! To get the -4 to the other side, you must add 4 to both sides. You don’t subtract. Try again using this knowledge.

37. Try Again… 04/2014L. Hojnowski © 201436 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 This would be correct if the question asked for the perpendicular line. Remember parallel lines have the same slope. Try again using this knowledge.

38. Correct!! 04/2014L. Hojnowski © 201437 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 You plugged in and solved correctly! You found the parallel slope.

39. Quiz Question # 5 04/2014L. Hojnowski © 201438 5. Write an equation in slope-intercept form for the line that passes through (-8, 0) and is perpendicular to y = (-1/2)x - 4 a.a. y = (-1/2)x - 4b. y = 2x + 16c. y = -2x - 16 d. y = (1/2)x + 4b. c. d.

40. Try Again… 04/2014L. Hojnowski © 201439 Careful the questions asked for a line that was perpendicular to the given line. This equations had a parallel slope. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6

41. Correct!! 04/2014L. Hojnowski © 201440 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 You plugged in and solved correctly! Awesome job plugging in the negative reciprocal slope!

42. Try Again… 04/2014L. Hojnowski © 201441 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 The slope in the equation is the reciprocal of the slope given in the problem. Perpendicular slopes are the negative reciprocals.

43. Try Again… 04/2014L. Hojnowski © 201442 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 The slope in the equation is the negative of the slope given in the problem. Perpendicular slopes are the negative reciprocals. Also, be careful of your signs when multiplying.

44. Quiz Question # 6 6. Determine whether 2x + 7y = -35 and 4x + 14y = -42 are parallel, perpendicular, or neither. a.a. neither b. parallel c. perpendicular b. c. 04/2014L. Hojnowski © 201443

45. Try Again… 04/2014L. Hojnowski © 201444 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Take your time and try to find your mistake.

46. Correct!! 04/2014L. Hojnowski © 201445 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 You solved both of the equations correctly for y!

47. Try Again… 04/2014L. Hojnowski © 201446 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Perpendicular lines have negative reciprocal slopes.

48. Quiz Question # 7 6. Determine whether 3x + 5y = 10 and 5x – 3y= -6 are parallel, perpendicular, or neither. a.a. neither b. parallel c. perpendicular b. c. 04/2014L. Hojnowski © 201447

49. Try Again… 04/2014L. Hojnowski © 201448 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Take your time and try to find your mistake.

50. Try Again… 04/2014L. Hojnowski © 201449 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Parallel lines have the same slope.

51. Correct!! 04/2014L. Hojnowski © 201450 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 You solved both of the equations correctly for y!

52. References McGraw-Hill Companies. (2014). Glencoe Algebra 1 Common Core Edition. New York: McGraw Hill. Seminars.usb.ac.ir. (2011). Hitting the objectives, Retrieved on September 14 th, 2012, from http://www.teambuildinggames.org/role-of-the-team-building- facilitator. http://www.teambuildinggames.org/role-of-the-team-building- facilitator Smiley Face, Retrieved on September 14 th, 2012, from http://ed101.bu.edu/StudentDoc/current/ED101fa10/rajensen/ima ges/happy-face1.png. http://ed101.bu.edu/StudentDoc/current/ED101fa10/rajensen/ima ges/happy-face1.png Wee, E. (2011). Try again, Retrieved on September 15 th, 2012, from http://radionjournals.blogspot.com/2011/04/try-again-part- 3-caring-for-children.html.http://radionjournals.blogspot.com/2011/04/try-again-part- 3-caring-for-children.html 04/2014L. Hojnowski © 201451 Reference from the dictionary