1. Chin-Sung Lin

2. Mr. Chin-Sung Lin  Distance Formula  Midpoint Formula  Slope Formula  Parallel Lines  Perpendicular Lines

3. Mr. Chin-Sung Lin

4. Distance between two points A ( x 1, y 1 ) and B ( x 2, y 2 ) is given by distance formula d ( A, B ) = √ ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 A (x 1, y 1 )B (x 2, y 2 ) Mr. Chin-Sung Lin

5. Calculate the distance between A ( 4, 5 ) and B ( 1, 1 ) Mr. Chin-Sung Lin

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

7. Calculate the distance between A ( 9, 5 ) and B ( 1, 5 ) Mr. Chin-Sung Lin

9. If the coordinates of A and B are ( x 1, y 1 ) and ( x 2, y 2 ) respectively, then the midpoint, M, of AB is given by the midpoint formula x 1 + x 2, y 1 + y 2 2 2 M = ( ) A (x 1, y 1 )B (x 2, y 2 )M (x, y) Mr. Chin-Sung Lin

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

11. 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

13. If the coordinates of A and B are (x 1, y 1 ) and (x 2, y 2 ) respectively, then the slope, m, of AB is given by the slope formula y 2 - y 1 x 2 - x 1 m = A (x 1, y 1 ) B (x 2, y 2 ) Mr. Chin-Sung Lin

14. Calculate the slope of AB, where A ( 4, 5 ) and B ( 2, 1 ) Mr. Chin-Sung Lin

15. Calculate the slope of AB, where A ( 4, 5 ) and B ( 2, 1 ) 5 - 1 4 - 2 = 2 m = Mr. Chin-Sung Lin

16. Positive slope Mr. Chin-Sung Lin

17. Negative slope Mr. Chin-Sung Lin

18. Zero slope

19. Undefined slope Mr. Chin-Sung Lin

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

21. If AB is parallel to CD where A ( 2, 3 ) and B ( 4, 9 ), calculate the slope of CD Mr. Chin-Sung Lin

22. If AB is parallel to CD where A ( 2, 3 ) and B ( 4, 9 ), calculate the slope of CD 9 - 3 4 - 2 = 3 m = n = Mr. Chin-Sung Lin

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

24. If AB is perpendicular to CD where A ( 1, 2 ) and B ( 3, 6 ), calculate the slope of CD Mr. Chin-Sung Lin

25. If AB is perpendicular to CD where A ( 1, 2 ) and B ( 3, 6 ), calculate the slope of CD 6 - 2 3 - 1 = 2 since m · n = - 1, 2 · n = -1, so, n = - 1 / 2 m = Mr. Chin-Sung Lin

27. 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 Mr. Chin-Sung Lin

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

30. 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 Mr. Chin-Sung Lin

31. 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 Mr. Chin-Sung Lin

32. Given: y-intercept b and a point (x 1, y 1 ) (0, b) (x 1, y 1 ) Mr. Chin-Sung Lin

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

34. Given: Two points (0, 4) and (2, 0) (0, 4) (2, 0) Mr. Chin-Sung Lin

35. 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 (0, 4) (2, 0) Mr. Chin-Sung Lin

36. A line passing through (2, 3) and the y-intercept is -5. Write the equation Mr. Chin-Sung Lin

37. Linear equation can be written in point-slope form: y – y 1 = m(x – x 1 ) where m is the slope (x 1, y 1 ) is a point on the line slope: m (x 1, y 1 ) Mr. Chin-Sung Lin

38. 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 Mr. Chin-Sung Lin

39. 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, (x 1, y 1 ) = (5, 2) y - 2 = 3(x – 5) Mr. Chin-Sung Lin

40. Given: Two points (x 1, y 1 ) and (x 2, y 2 ) (x 1, y 1 ) (x 2, y 2 ) Mr. Chin-Sung Lin

41. Given: Two points (x 1, y 1 ) and (x 2, y 2 ) Step 1: Find the slope m by plugging two points (x 1, y 1 ) and (x 2, y 2 ) into the slop formula m = (y 2 – y 1 )/(x 2 – x 1 ) Step 2: Write the equation using slope m and any point y – y 1 = m(x – x 1 ) (x 1, y 1 ) (x 2, y 2 ) Mr. Chin-Sung Lin

42. Given: Two points (3, 1) and (1, 4) (1, 4) (3, 1) Mr. Chin-Sung Lin

43. 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) (1, 4) (3, 1) Mr. Chin-Sung Lin

44. Given: Two points (-2, 7) and (2, 3) Mr. Chin-Sung Lin

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

47. 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 Mr. Chin-Sung Lin

48. Write an equation of the line passing through the point (2, 3) that is parallel to the line y = x – 5 Mr. Chin-Sung Lin

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

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

51. Write an equation of the line passing through the point (2, 3) that is perpendicular to the line y = -2x + 2 Step 1: Find the slope m from the given equation: since two lines are perpendicular, the product of the slopes is equal to -1, so: m = 1/2 Step 2: Find the y-intercept b by using the m = 2 and the given point (-1, 1): 3 = (1/2)(2) + b, so, b = 2 Step 3: Write the equation: y = (1/2)x + 2 Mr. Chin-Sung Lin

52. Write an equation of the line passing through the point (1, 2) that is perpendicular to the line y = x + 3 Mr. Chin-Sung Lin

53. Write an equation of the line passing through the point (4, 1) that is perpendicular to the line y = -x + 2 Mr. Chin-Sung Lin

55. Based on the information in the graph, write the equations of line P and line Q in both slope- intercept form and point-slope form Mr. Chin-Sung Lin (4, 3) y = 2x -5 -2 K P Q

56. Mr. Chin-Sung Lin

57. Two types of proofs in coordinate geometry: Special cases Given ordered pairs of numbers, and prove something about a specific segment or polygon General Theorems When the given information is a figure that represents a particular type of polygon, we must state the coordinates of its vertices in general terms using variables Mr. Chin-Sung Lin

58. Two skills of proofs in coordinate geometry: Line segments bisect each other the midpoints of each segment are the same point Two lines are perpendicular to each other the slope of one line is the negative reciprocal of the slope of the other Mr. Chin-Sung Lin

59. If the coordinates of four points are A(-3, 5), B(5, 1), C(-2, -3), and D(4, 9), prove that AB and CD are perpendicular bisector to each other Mr. Chin-Sung Lin

60. The vertices of rhombus ABCD are A(2, -3), B(5, 1), C(10, 1) and D(7, -3). (a) Prove that the diagonals bisect each other. (b) Prove that the diagonals are perpendicular to each other. Mr. Chin-Sung Lin

61. If the coordinates of three points are A(-1, 4), B(4, 7), and C(1, 2), prove that ABC is an isosceles triangle Mr. Chin-Sung Lin

62. If the coordinates of three points are A(-1, 4), B(4, 7), and C(1, 2), prove that ABC is an isosceles triangle Mr. Chin-Sung Lin

63. If the coordinates of three points are A(4, 3), B(6, 7), and C(-4, 7), prove that ΔABC is a right triangle. Which angle is the right angle? Mr. Chin-Sung Lin

64. Vertices definition in coordinate geometry: Any triangle— (a, 0), (0, b), (c, 0) Mr. Chin-Sung Lin (0, b) (a, 0) (c, 0) (0, b) (a, 0) (c, 0)

65. Vertices definition in coordinate geometry: Right triangle— (a, 0), (0, b), (0, 0) Mr. Chin-Sung Lin (0, b) (0, 0) (a, 0)

66. Vertices definition in coordinate geometry: Isosceles triangle— (-a, 0), (0, b), (a, 0) Mr. Chin-Sung Lin (0, b) (-a, 0) (a, 0)

67. Vertices definition in coordinate geometry: Midpoint of segments— (2a, 0), (0, 2b), (2c, 0) Mr. Chin-Sung Lin (0, 2b) (2a, 0) (2c, 0)

68. Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices Mr. Chin-Sung Lin

69. Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices Mr. Chin-Sung Lin B (0, 2b) A (2a, 0) C(0, 0) M Given: Right triangle ABC whose vertices are A(2a, 0), B(0, 2b), and C(0,0). Let M be the midpoint of the hypotenuse AB Prove: AM = BM = CM

70. Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices Mr. Chin-Sung Lin B (0, 2b) A (2a, 0) C(0, 0) M

71. Mr. Chin-Sung Lin

72. Orthocenter: The altitudes of a triangle intersect in one point Acute Triangle Mr. Chin-Sung Lin B(0, b) A(a, 0) C(c, 0)

73. Theorem: The altitudes of a triangle are concurrent (intersecting in one point— orthocenter) Acute Triangle Mr. Chin-Sung Lin B(0, b) A(a, 0) C(c, 0)

74. Orthocenter: The altitudes of a triangle intersect in one point Right Triangle Mr. Chin-Sung Lin B(0, b) A(a, 0) C(c, 0)

75. Theorem: The altitudes of a triangle are concurrent (intersecting in one point— orthocenter) Right Triangle Mr. Chin-Sung Lin B(0, b) A(a, 0) C(c, 0)

76. Orthocenter: The altitudes of a triangle intersect in one point Obtuse Triangle Mr. Chin-Sung Lin B(0, b) A(a, 0) C(c, 0)

77. Theorem: The altitudes of a triangle are concurrent (intersecting in one point— orthocenter) Obtuse Triangle Mr. Chin-Sung Lin B(0, b) A(a, 0) C(c, 0)

78. The coordinates of the vertices of ΔABC are A(0, 0), B(-2, 6), and C(4, 0). Find the coordinates of the orthocenter of the triangle Mr. Chin-Sung Lin

79. The coordinates of the vertices of ΔABC are A(0, 0), B(-2, 6), and C(4, 0). Find the coordinates of the orthocenter of the triangle Answer: (-2, -2) Mr. Chin-Sung Lin

80. The coordinates of the vertices of ΔABC are A(0, 0), B(3, 4), and C(2, 1). Find the coordinates of the orthocenter of the triangle Mr. Chin-Sung Lin

81. The coordinates of the vertices of ΔABC are A(0, 0), B(3, 4), and C(2, 1). Find the coordinates of the orthocenter of the triangle Answer: (6, -2) Mr. Chin-Sung Lin