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Integrated Math 2 Section 8-1

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Parallel and Perpendicular Lines

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Integrated Math 2 Section 8-1

  1. 1. Chapter 8 Systems of Equations and Inequalities
  2. 2. Section 8-1 Parallel and Perpendicular Lines
  3. 3. Essential Questions ✤ How do you determine if two lines are parallel or perpendicular? ✤ How do you write equations of parallel and perpendicular lines? ✤ Where you’ll see this: ✤ Sports, travel, safety
  4. 4. Vocabulary 1. Negative Reciprocals:
  5. 5. Vocabulary 1. Negative Reciprocals: Two rational numbers whose product is -1
  6. 6. Vocabulary 1. Negative Reciprocals: Two rational numbers whose product is -1 a b and − are negative reciprocals b a
  7. 7. Vocabulary 1. Negative Reciprocals: Two rational numbers whose product is -1 a b and − are negative reciprocals b a 2 3 and − are negative reciprocals 3 2
  8. 8. Parallel Lines
  9. 9. Parallel Lines ✤ Don’t intersect
  10. 10. Parallel Lines ✤ Don’t intersect ✤ Same slope
  11. 11. Parallel Lines ✤ Don’t intersect ✤ Same slope
  12. 12. Parallel Lines ✤ Don’t intersect ✤ Same slope ✤ If two lines are parallel, then they have the same slope
  13. 13. Parallel Lines ✤ Don’t intersect ✤ Same slope ✤ If two lines are parallel, then they have the same slope ✤ If two lines have the same slope, then they are parallel
  14. 14. Perpendicular Lines
  15. 15. Perpendicular Lines ✤ Intersect at 90 degree angles
  16. 16. Perpendicular Lines ✤ Intersect at 90 degree angles ✤ Slopes are negative reciprocals
  17. 17. Perpendicular Lines ✤ Intersect at 90 degree angles ✤ Slopes are negative reciprocals
  18. 18. Perpendicular Lines ✤ Intersect at 90 degree angles ✤ Slopes are negative reciprocals ✤ If two lines are perpendicular, then the product of their slopes is -1
  19. 19. Perpendicular Lines ✤ Intersect at 90 degree angles ✤ Slopes are negative reciprocals ✤ If two lines are perpendicular, then the product of their slopes is -1 ✤ If the product of the slopes of two lines is -1, then they are perpendicular
  20. 20. Example 1 Suppose you have a line that has a slope of 4. What would be the slope of: a. the perpendicular line? b. the parallel line?
  21. 21. Example 1 Suppose you have a line that has a slope of 4. What would be the slope of: a. the perpendicular line? 1 m=− 4 b. the parallel line?
  22. 22. Example 1 Suppose you have a line that has a slope of 4. What would be the slope of: a. the perpendicular line? 1 m=− 4 b. the parallel line? m=4
  23. 23. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12.
  24. 24. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12
  25. 25. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 +6x +6x
  26. 26. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 +6x +6x 3y = 6x + 12
  27. 27. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 +6x +6x 3y = 6x + 12 3 3
  28. 28. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 +6x +6x 3y = 6x + 12 3 3 y = 2x + 4
  29. 29. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 +6x +6x 3y = 6x + 12 3 3 y = 2x + 4 m=2
  30. 30. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 +6x +6x 3y = 6x + 12 3 3 y = 2x + 4 m=2 (-1, 3)
  31. 31. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 y − y1 = m(x − x1 ) +6x +6x 3y = 6x + 12 3 3 y = 2x + 4 m=2 (-1, 3)
  32. 32. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 y − y1 = m(x − x1 ) +6x +6x y − 3 = 2(x + 1) 3y = 6x + 12 3 3 y = 2x + 4 m=2 (-1, 3)
  33. 33. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 y − y1 = m(x − x1 ) +6x +6x y − 3 = 2(x + 1) 3y = 6x + 12 3 3 y − 3 = 2x + 2 y = 2x + 4 m=2 (-1, 3)
  34. 34. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12. 3y − 6x = 12 y − y1 = m(x − x1 ) +6x +6x y − 3 = 2(x + 1) 3y = 6x + 12 3 3 y − 3 = 2x + 2 y = 2x + 4 y = 2x + 5 m=2 (-1, 3)
  35. 35. Formulas Slope-intercept: Point-slope:
  36. 36. Formulas Slope-intercept: y = mx + b Point-slope:
  37. 37. Formulas Slope-intercept: y = mx + b Point-slope: y − y1 = m(x − x1 )
  38. 38. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4
  39. 39. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4 4 m= 3
  40. 40. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4 4 m= (2, 7) 3
  41. 41. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4 4 m= (2, 7) 3 y − y1 = m(x − x1 )
  42. 42. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4 4 y − 7 = (x − 2) 4 3 m= (2, 7) 3 y − y1 = m(x − x1 )
  43. 43. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4 4 y − 7 = (x − 2) 4 3 m= (2, 7) 3 4 8 y−7= x− y − y1 = m(x − x1 ) 3 3
  44. 44. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4 4 y − 7 = (x − 2) 4 3 m= (2, 7) 3 4 8 y−7= x− y − y1 = m(x − x1 ) 3 3 4 13 y= x+ 3 3
  45. 45. Quick Questions 1. What is true about the slopes of all horizontal lines? 2. If you knew the coordinates of four vertices of a quadrilateral, how could you use slope to determine if the figure is a parallelogram?
  46. 46. Homework
  47. 47. Homework p. 336 #1-39 odd “It is perhaps a more fortunate destiny to have a taste for collecting shells than to be born a millionaire.” - Robert Louis Stevenson

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