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Slopes and Equations of Lines

Chap 8. Slopes and Equations of Lines. Eleanor Roosevelt High School Geometry Chin-Sung Lin. ERHS Math Geometry. Distance Formula Midpoint Formula Slope Formula Parallel Lines Perpendicular Lines. Basic Geometry Formulas. Mr. Chin-Sung Lin. ERHS Math Geometry. Distance Formula.

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Slopes and Equations of Lines

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  1. Chap8 Slopes and Equations of Lines Eleanor Roosevelt High School Geometry Chin-Sung Lin

  2. ERHS Math Geometry • Distance Formula • Midpoint Formula • Slope Formula • Parallel Lines • Perpendicular Lines Basic Geometry Formulas Mr. Chin-Sung Lin

  3. ERHS Math Geometry Distance Formula Mr. Chin-Sung Lin

  4. A (x1, y1) B (x2, y2) ERHS Math Geometry Distance between two pointsA (x1, y1) andB (x2, y2) is given by distance formula d(A, B) =√(x2 − x1)2+ (y2 − y1)2 Distance Formula Mr. Chin-Sung Lin

  5. ERHS Math Geometry Distance Formula - Example Calculate the distance betweenA (4, 5) andB (1, 1) Mr. Chin-Sung Lin

  6. ERHS Math Geometry Distance Formula - Example Calculate the length of AB if the coordinates of A and B are(4, 15) and(-1, 3) respectively Mr. Chin-Sung Lin

  7. ERHS Math Geometry Distance Formula - Example Calculate the distance betweenA (9, 5) andB (1, 5) Mr. Chin-Sung Lin

  8. ERHS Math Geometry Midpoint Formula Mr. Chin-Sung Lin

  9. A (x1, y1) M (x, y) B (x2, y2) ERHS Math Geometry If the coordinates of A and B are ( x1, y1) and ( x2, y2) respectively, then the midpoint, M, of AB is given by the midpoint formula x1 +x2, y1+y2 22 Midpoint Formula M = ( ) Mr. Chin-Sung Lin

  10. ERHS Math Geometry Midpoint Formula - Example Calculate the midpoint of AB if the coordinates of A and B are(2, 7) and(-6, 5) respectively Mr. Chin-Sung Lin

  11. ERHS Math Geometry Midpoint Formula - Example M(1, -2) is the midpoint of AB and the coordinates of A are (-3, 2). Find the coordinates of B Mr. Chin-Sung Lin

  12. ERHS Math Geometry Slope Formula Mr. Chin-Sung Lin

  13. A (x1, y1) B (x2, y2) ERHS Math Geometry If the coordinates of A and B are (x1, y1) and (x2, y2) respectively, then the slope, m, of AB is given by the slope formula y2 -y1 x2 -x1 Slope Formula m = Mr. Chin-Sung Lin

  14. ERHS Math Geometry Slope Formula - Example Calculate the slope of AB, where A (4, 5) andB (2, 1) Mr. Chin-Sung Lin

  15. ERHS Math Geometry Slope Formula - Example Calculate the slope of AB, where A (4, 5) andB (2, 1) 5 - 1 4 - 2 = 2 m = Mr. Chin-Sung Lin

  16. ERHS Math Geometry Slope of Lines in the Coordinate Planes Positive slope Mr. Chin-Sung Lin

  17. ERHS Math Geometry Slope of Lines in the Coordinate Planes Negative slope Mr. Chin-Sung Lin

  18. ERHS Math Geometry Slope of Lines in the Coordinate Planes Zero slope Mr. Chin-Sung Lin

  19. ERHS Math Geometry Slope of Lines in the Coordinate Planes Undefined slope Mr. Chin-Sung Lin

  20. ERHS Math Geometry The straight lines with slopes (m) and (n) are parallel to each other if and only if m = n Slope and Parallel Lines m n Mr. Chin-Sung Lin

  21. ERHS Math Geometry Slope and Parallel Lines - Example If AB is parallel to CD where A (2, 3) andB (4, 9), calculate the slope of CD Mr. Chin-Sung Lin

  22. ERHS Math Geometry Slope and Parallel Lines - Example If AB is parallel to CD where A (2, 3) andB (4, 9), calculate the slope of CD 9 - 3 4 - 2 = 3 m = n = Mr. Chin-Sung Lin

  23. ERHS Math Geometry The straight lines with slopes (m) and (n) are mutually perpendicular if and only if m · n = -1 Slope and Perpendicular Lines n m Mr. Chin-Sung Lin

  24. ERHS Math Geometry If AB is perpendicular to CD where A (1, 2) andB (3, 6), calculate the slope of CD Slope and Perpendicular Lines - Example Mr. Chin-Sung Lin

  25. ERHS Math Geometry If AB is perpendicular to CD where A (1, 2) andB (3, 6), calculate the slope of CD 6 - 2 3 - 1 = 2 since m · n = -1, 2 · n = -1, so, n = -1/2 Slope and Perpendicular Lines - Example m = Mr. Chin-Sung Lin

  26. ERHS Math Geometry Group Work Mr. Chin-Sung Lin

  27. ERHS Math Geometry There are four points A (2, 6),B(6, 4), C(4, 0) and D(0, 2) on the coordinate plane. Identify the pairs of parallel and perpendicular lines Parallel and Perpendicular Lines Mr. Chin-Sung Lin

  28. ERHS Math Geometry Equations of Lines Mr. Chin-Sung Lin

  29. ERHS Math Geometry Linear equation can be written in slope-intercept form: y = mx + b where m is the slope b is the y-intercept Slope Intercept Form b slope: m Mr. Chin-Sung Lin

  30. ERHS Math Geometry Given: If the slope of a line is 3 and it passes through(0, 2), write the equation of the line in slope-intercept form Write Slope Intercept Form Mr. Chin-Sung Lin

  31. ERHS Math Geometry Given: If the slope of a line is 3 and it passes through(0, 2), write the equation of the line in slope-intercept form m = 3, b = 2 y = 3x + 2 Write Slope Intercept Form Mr. Chin-Sung Lin

  32. ERHS Math Geometry Given: y-intercept b and a point (x1, y1) Write Slope Intercept Form (0, b) (x1, y1) Mr. Chin-Sung Lin

  33. ERHS Math Geometry Given: y-intercept b and a point (x1, y1) Step 1: Find the slope m by choosing two points (0, b) and (x1, y1) on the graph of the line Step 2: Find the y-intercept b Step 3: Write the equation y = mx + b Write Slope Intercept Form (0, b) (x1, y1) Mr. Chin-Sung Lin

  34. ERHS Math Geometry Given: Two points (0, 4) and (2, 0) Write Slope Intercept Form (0, 4) (2, 0) Mr. Chin-Sung Lin

  35. ERHS Math Geometry Given: Two points (0, 4) and (2, 0) Step 1: Find the slope by choosing two points on the graph of the line: m = (0-4)/(2-0) = -2 Step 2: Find the y-intercept: b = 4 Step 3: Write the equation: y = -2x + 4 Write Slope Intercept Form (0, 4) (2, 0) Mr. Chin-Sung Lin

  36. ERHS Math Geometry Write Slope Intercept Form - Example A line passing through (2, 3) and the y-intercept is -5. Write the equation Mr. Chin-Sung Lin

  37. ERHS Math Geometry Linear equation can be written in point-slope form: y – y1 = m(x – x1) where m is the slope (x1, y1) is a point on the line Point-Slope Form (x1, y1) slope: m Mr. Chin-Sung Lin

  38. ERHS Math Geometry Given: If the slope of a line is 3 and it passes through(5, 2), write the equation of the line in slope-intercept form Write Point-Slope Form Mr. Chin-Sung Lin

  39. ERHS Math Geometry Given: If the slope of a line is 3 and it passes through(5, 2), write the equation of the line in slope-intercept form m = 3, (x1, y1) = (5, 2) y - 2 = 3(x – 5) Write Point-Slope Form Mr. Chin-Sung Lin

  40. ERHS Math Geometry Given: Two points (x1, y1) and (x2, y2) Write Point-Slope Form (x1, y1) (x2, y2) Mr. Chin-Sung Lin

  41. ERHS Math Geometry Given: Two points (x1, y1) and (x2, y2) Step 1: Find the slope m by plugging two points (x1, y1) and (x2, y2) into the slop formula m = (y2 – y1)/(x2 – x1) Step 2: Write the equation using slope m and any point y – y1 = m(x – x1) Write Point-Slope Form (x1, y1) (x2, y2) Mr. Chin-Sung Lin

  42. ERHS Math Geometry Given: Two points (3, 1) and (1, 4) Write Point-Slope Form Example (1, 4) (3, 1) Mr. Chin-Sung Lin

  43. ERHS Math Geometry Given: Two points (3, 1) and (1, 4) Step 1: Find the slope m by plugging two points (3, 1) and (1, 4) into the slop formula m = (4 – 1)/(1 – 3) = -3/2 Step 2: Write the equation y – 1 = (-3/2)(x – 3) Write Point-Slope Form Example (1, 4) (3, 1) Mr. Chin-Sung Lin

  44. ERHS Math Geometry Given: Two points (-2, 7) and (2, 3) Write Point-Slope Form Example Mr. Chin-Sung Lin

  45. ERHS Math Geometry Equations of Parallel & Perpendicular Lines Mr. Chin-Sung Lin

  46. ERHS Math Geometry Write an equation of the line passing through the point (-1, 1) that is parallel to the line y = 2x – 3 Equation of a Parallel Line Mr. Chin-Sung Lin

  47. ERHS Math Geometry Write an equation of the line passing through the point (-1, 1) that is parallel to the line y = 2x - 3 Step 1: Find the slope m from the given equation: since two lines are parallel, the slopes are the same, so: m = 2 Step 2: Find the y-intercept b by using the m = 2 and the given point (-1, 1): 1 = 2(-1) + b, so, b = 3 Step 3: Write the equation: y = 2x + 3 Equation of a Parallel Line Mr. Chin-Sung Lin

  48. ERHS Math Geometry Write an equation of the line passing through the point (2, 3) that is parallel to the line y = x – 5 Equation of a Parallel Line - Example Mr. Chin-Sung Lin

  49. ERHS Math Geometry Write an equation of the line passing through the point (2, 0) that is parallel to the line y = x - 2 Equation of a Parallel Line Mr. Chin-Sung Lin

  50. ERHS Math Geometry Write an equation of the line passing through the point (2, 3) that is perpendicular to the line y = -2x + 2 Equation of a Perpendicular Line Mr. Chin-Sung Lin

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