The document provides an interactive lesson on key concepts relating to lines and formulas in geometry, including: - The formula for calculating the distance between two points on a line is D=(y2-y1)2+(x2-x1)2. - The formula for finding the midpoint between two points is (X1+X2)/2, (Y1+Y2)/2. - The standard formula for a line is y=mx+b, where m is the slope (rise over run), and b is the y-intercept (where the line crosses the y-axis). - Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each

1. LINES

2. What is the formula for the
distance of a line?

3. What is the formula for the
distance of a line?
D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

4. What is the formula for the
distance of a line?
D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

5. What is the formula for the
distance of a line?
D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

6. What is the formula for the
distance of a line?
D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

7. What is the formula for the
distance of a line?
D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

8. Let’s ask that again:

9. Let’s ask that again:
What is the formula for the distance of a line?

10. Let’s ask that again:
What is the formula for the distance of a line?
Say it out loud.

11. D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

12. One more time:

13. One more time:
What is the formula for the distance of a line?

14. D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

15. What is the distance between
point A and Point B?
(8,2)
(4,-1)

16. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

17. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

18. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (𝟒 − 𝑦1)2+(𝑥2 − 𝑥1)2

19. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (𝟒 − 𝑦1)2+(𝑥2 − 𝑥1)2

20. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (𝟒 − 𝑦1)2+(𝑥2 − 𝑥1)2

21. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (−𝟏 − 𝟐)2+(𝑥2 − 𝑥1)2

22. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (−1 − 2)2+(𝟒 − 𝑥1)2

23. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (−1 − 2)2+(4 − 𝟖)2

24. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (−1 − 2)2+(4 − 8)2

25. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = (−𝟑)2+(4 − 8)2

26. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = 𝟗 + (4 − 8)2

27. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = 9 + (−𝟒)2

28. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = 9 + 𝟏𝟔

29. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = 𝟐𝟓

30. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = 𝟓

31. What is the distance between
point A and Point B?
(8,2)
(4,-1)
D = 𝟓 The distance between
Point A and Point B is 5
units

32. Last time:
What is the formula for the distance of a line?

33. D = (𝑦2 − 𝑦1)2+(𝑥2 − 𝑥1)2

34. What is the formula for the
midpoint of a line?

35. 𝑋1+𝑋2
2
,
𝑌1+𝑌2
2

36. See if you can say it before going to
the next slide.

37. Did you get it right?
𝑋1+𝑋2
2
,
𝑌1+𝑌2
2

38. What is midpoint between these
two points?
(8,2)
(4,-1)

39. What is midpoint between these
two points?
(8,2)
(4,-1)
𝑋1+𝑋2
2
,
𝑌1+𝑌2
2

40. What is midpoint between these
two points?
(8,2)
(4,-1)
𝑋1+𝑋2
2
,
𝑌1+𝑌2
2

41. What is midpoint between these
two points?
(8,2)
(4,-1)
𝑋1+𝑋2
2
,
𝑌1+𝑌2
2

42. What is midpoint between these
two points?
(8,2)
(4,-1)
𝑋1+𝑋2
2
,
𝑌1+𝑌2
2

43. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟒 +𝑋2
2
,
𝑌1+𝑌2
2

44. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟒 +𝑋2
2
,
𝑌1+𝑌2
2

45. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟒 +𝟖
2
,
𝑌1+𝑌2
2

46. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟒 +𝟖
2
,
𝑌1+𝑌2
2

47. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟒 +𝟖
2
,
−𝟏+𝑌2
2

48. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟒 +𝟖
2
,
−𝟏+𝑌2
2

49. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟒 +𝟖
2
,
−𝟏+𝟐
2

50. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟏𝟐
2
,
−𝟏+𝟐
2

51. What is midpoint between these
two points?
(8,2)
(4,-1)
𝟏𝟐
2
,
𝟏
2

52. What is midpoint between these
two points?
(8,2)
(4,-1)
6 ,
1
2

53. What is midpoint between these
two points?
(8,2)
(4,-1)
6 ,
1
2

54. What is midpoint between these
two points?
(8,2)
(4,-1)
6 , 0.5

55. What is midpoint between these
two points?
(8,2)
(4,-1)
(𝟔,0.5)
The
Midpoin
t

56. What is the formula for a line?

57. What is the formula for a line?
𝑦 = 𝑚𝑥 + 𝑏

58. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝑏
This is the
slope which
usually is a
fraction

59. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝒃
This is where
the line
crosses the Y
Axix

60. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝟏
If the y
intercept is
1 then the
line goes
through
the Y axis
at 1.

61. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝟏
If the y
intercept is
1 then the
line goes
through
the Y axis
at 1.
1

62. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝟐
If the y
intercept is
2 then the
line goes
through
the Y axis
at 2
2

63. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝟑
If the y
intercept is
3 then the
line goes
through
the Y axis
at 3
3

64. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝟒
If the y
intercept is
4 then the
line goes
through
the Y axis
at 4
4

65. What is the formula for a line?
𝑦 = 𝒎𝑥 + 𝟎
If the y
intercept is
0 then the
line goes
through
the Y axis
at 0
0

66. Let’s go back to this formula

67. Let’s go back to this formula
𝑦 = 𝒎𝑥 + 𝟏
1
The slope is
always a
fraction.

68. Let’s go back to this formula
𝑦 = 𝒎𝑥 + 𝟏
1
Let’s say
the slope is
1/2

69. Let’s go back to this formula
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1
Let’s say
the slope is
1/2

70. Let’s go back to this formula
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1

71. Let’s go back to this formula
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1
This is the rise

72. Let’s go back to this formula
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1
This is the rise
Over the run

73. Let’s go back to this formula
𝑦 =
𝟏
2
𝑥 + 1
1
rise1

74. Let’s go back to this formula
𝑦 =
1
𝟐
𝑥 + 1
1
run
1
2

75. 1
Let’s go back to this formula
𝑦 =
1
𝟐
𝑥 + 1
run
1
2

76. 1
Let’s go back to this formula
𝑦 =
𝟏
𝟐
𝑥 + 1
1
2 1
2

77. Let’s go back to this formula
𝑦 =
𝟏
𝟐
𝑥 + 1

78. Let’s go back to this formula
𝑦 =
1
2
𝑥 + 𝟏
If the y
intercept
goes to 0
watch what
happens

79. Let’s go back to this formula
𝑦 =
1
2
𝑥 + 𝟎
If the y
intercept
goes to 0
watch what
happens

80. Let’s go back to this formula
𝑦 =
1
2
𝑥 + 𝟎
If the y
intercept
goes to 0
watch what
happens
0

81. Let’s go back to this formula
𝑦 =
1
2
𝑥 + 𝟎
If the y
intercept
goes to 2
watch what
happens
0

82. Let’s go back to this formula
𝑦 =
1
2
𝑥 + 𝟐
If the y
intercept
goes to 2
watch what
happens
0

83. Let’s go back to this formula
𝑦 =
1
2
𝑥 + 𝟐
If the y
intercept
goes to 2
watch what
happens
2

84. Without going to the next slide can
you say what the formula for a line
is?

85. Without going to the next slide can
you say what the formula for a line
is?
𝒚 = 𝒎𝒙 + 𝒃
Did you get it right?

86. What does the “m” mean?
𝒚 = 𝒎𝒙 + 𝒃

87. slope
𝒚 =
𝟏
𝟐
𝒙 + 𝒃

88. slope
𝒚 =
𝟏
𝟐
𝒙 + 𝒃
1
1
2

89. slope
𝒚 =
𝟏
𝟐
𝒙 + 𝒃
1
1
2
rise
1

90. slope
𝒚 =
𝟏
𝟐
𝒙 + 𝒃
1
1
2 run
2

91. slope
𝒚 =
𝟏
𝟐
𝒙 + 𝒃
1

92. What does the “b” mean?
𝒚 = 𝑚𝒙 + 𝒃

93. Y - intercept
𝒚 = 𝑚𝒙 + 𝒃
1
1
2 1
2

94. Y - intercept
𝒚 = 𝑚𝒙 + 𝟏
1
1
2 1
2

95. Where the line crosses the Y
axis
𝒚 = 𝑚𝒙 + 𝟏
1
1
2 1
2

96. Where the line crosses the Y
axis
𝒚 = 𝑚𝒙 + 𝟏
1
2 1
2
1

97. What do parallel lines have in
common?

98. Let’s go back to this formula
0
𝑦 =
𝟏
𝟐
𝑥 + 𝟎

99. Let’s go back to this formula
0
𝑦 =
𝟏
𝟐
𝑥 + 𝟎

100. Let’s go back to this formula
0
𝑦 =
𝟏
𝟐
𝑥 + 𝟎
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1

101. Let’s go back to this formula
0
𝑦 =
𝟏
𝟐
𝑥 + 𝟎
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1

102. 0
𝑦 =
𝟏
𝟐
𝑥 + 𝟎
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1
𝑦 =
𝟏
𝟐
𝑥 + 𝟐
2

103. All these slopes are the same
because the lines are parallel
0
𝑦 =
𝟏
𝟐
𝑥 + 𝟎
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1
𝑦 =
𝟏
𝟐
𝑥 + 𝟐
2

104. All these slopes are the same
because the lines are parallel
0
𝑦 =
𝟏
𝟐
𝑥 + 𝟎
𝑦 =
𝟏
𝟐
𝑥 + 𝟏
1
𝑦 =
𝟏
𝟐
𝑥 + 𝟐
2

105. What is the slope of a
perpendicular line to the line
below?
𝑦 =
𝟏
𝟐
𝑥 + 𝟐

106. What is the slope of a
perpendicular line to the line
below?
𝑦 =
𝟐
𝟏
𝑥 + 𝟐
Just flip the slope and
then multiply by -1

107. What is the slope of a
perpendicular line to the line
below?
𝑦 =
𝟐
𝟏
𝑥 + 𝟐
Just flip the slope and then
multiply by -1

108. What is the slope of a
perpendicular line to the line
below?
𝑦 = −
𝟐
𝟏
𝑥 + 𝟐
Just flip the slope and then
multiply by -1

109. What is the slope of a
perpendicular line to the line
below?
𝑦 =
𝟑
𝟒
𝑥 + 𝟐

110. What is the slope of a
perpendicular line to the line
below?
−
𝟒
𝟑
𝑥

111. What is the slope of a
perpendicular line to the line
below?
𝑦 = 𝟐𝑥 + 𝟐

112. What is the slope of a
perpendicular line to the line
below?
−
𝟏
𝟐
𝑥