This document discusses finding distances between lines and points. It begins by defining key terms like equidistant and the distance between a point and line. It then provides steps to find the distance between a point and line, which include finding the equations of the original line and a perpendicular line through the point, solving the system of equations, and using the distance formula. An example problem demonstrates finding the distance between a point and line. It also provides steps to find the distance between parallel lines by finding the equation of a perpendicular line and the intersection using a system of equations.

1. Section 2-10
Perpendiculars and Distance

2. Essential Questions
n How do you find the distance between a point and
a line?
n How do you find the distance between parallel
lines?

3. Vocabulary
1. Equidistant:
2. Distance Between a Point and a Line:
3. Distance Between Parallel Lines:

4. Vocabulary
1. Equidistant: The distance between any two lines as
measured along a perpendicular is the same; this
occurs with parallel lines
2. Distance Between a Point and a Line:
3. Distance Between Parallel Lines:

5. Vocabulary
1. Equidistant: The distance between any two lines as
measured along a perpendicular is the same; this
occurs with parallel lines
2. Distance Between a Point and a Line: The length
of the segment perpendicular to the line, with the
endpoints being that point and a point on the line
3. Distance Between Parallel Lines:

6. Vocabulary
1. Equidistant: The distance between any two lines as
measured along a perpendicular is the same; this
occurs with parallel lines
2. Distance Between a Point and a Line: The length
of the segment perpendicular to the line, with the
endpoints being that point and a point on the line
3. Distance Between Parallel Lines: The length of the
segment perpendicular to the two parallel lines with
the endpoints on either of the parallel lines

7. Postulates & Theorems
1. Perpendicular Postulate:
2. Two Lines Equidistant from a Third:

8. Postulates & Theorems
1. Perpendicular Postulate: If given a line and a point
not on the line, then there exists exactly one line
through the point that is perpendicular to the given
line
2. Two Lines Equidistant from a Third:

9. Postulates & Theorems
1. Perpendicular Postulate: If given a line and a point
not on the line, then there exists exactly one line
through the point that is perpendicular to the given
line
2. Two Lines Equidistant from a Third: In a plane, if
two lines are each equidistant from a third line,
then the two lines are parallel to each other

10. Steps to find the Distance
from a Point to a Line

11. Steps to find the Distance
from a Point to a Line
1. Find the equation of the original line

12. Steps to find the Distance
from a Point to a Line
1. Find the equation of the original line
2. Find the equation of the perpendicular line through the
other point

13. Steps to find the Distance
from a Point to a Line
1. Find the equation of the original line
2. Find the equation of the perpendicular line through the
other point
3. Solve the system of these two equations.

14. Steps to find the Distance
from a Point to a Line
1. Find the equation of the original line
2. Find the equation of the perpendicular line through the
other point
3. Solve the system of these two equations.
4. Use the distance formula utilizing this point on the line
and the point not on the line.

15. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).

16. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line

17. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5

18. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5

19. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1

20. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)

21. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = mx + b

22. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
y = mx + b

23. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
2. Find the equation of the perpendicular line through the
other point
y = mx + b

24. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
2. Find the equation of the perpendicular line through the
other point
m = 1
y = mx + b

25. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
2. Find the equation of the perpendicular line through the
other point
m = 1
V(1, 5)
y = mx + b

26. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
2. Find the equation of the perpendicular line through the
other point
m = 1
V(1, 5)
y − y1
= m(x − x1
)
y = mx + b

27. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
2. Find the equation of the perpendicular line through the
other point
m = 1
V(1, 5)
y − y1
= m(x − x1
)
y = mx + b
y − 5 = 1(x −1)

28. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
2. Find the equation of the perpendicular line through the
other point
m = 1
V(1, 5)
y − y1
= m(x − x1
)
y = mx + b
y − 5 = 1(x −1)
y − 5 = x −1

29. Example 1
The line a contains the points T(0, 0) and U(−5, 5).
Find the distance between line a and the point V(1, 5).
1. Find the equation of the original line
m =
0 − 5
0 + 5
=
−5
5
= −1 T(0, 0)
y = −x
2. Find the equation of the perpendicular line through the
other point
m = 1
V(1, 5)
y − y1
= m(x − x1
)
y = mx + b
y − 5 = 1(x −1)
y − 5 = x −1
y = x + 4

30. Example 1
3. Solve the system of these two equations.

31. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩

32. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩
−x = x + 4

33. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩
−x = x + 4
−2x = 4

34. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩
−x = x + 4
−2x = 4
x = −2

35. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩
−x = x + 4
−2x = 4
x = −2
y = −(−2)

36. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩
−x = x + 4
−2x = 4
x = −2
y = −(−2) = 2

37. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩
−x = x + 4
−2x = 4
x = −2
y = −(−2) = 2
2 = −2+ 4

38. Example 1
3. Solve the system of these two equations.
y = −x
y = x + 4
⎧
⎨
⎩
−x = x + 4
−2x = 4
x = −2
y = −(−2) = 2
2 = −2+ 4
(−2,2)

39. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.

40. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)

41. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)
d = (x2
− x1
)2
+(y2
− y1
)2

42. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (−2−1)2
+(2− 5)2

43. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (−2−1)2
+(2− 5)2
= (−3)2
+(−3)2

44. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (−2−1)2
+(2− 5)2
= (−3)2
+(−3)2
= 9+ 9

45. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (−2−1)2
+(2− 5)2
= (−3)2
+(−3)2
= 9+ 9 = 18

46. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (−2−1)2
+(2− 5)2
= (−3)2
+(−3)2
= 9+ 9 = 18 ≈ 4.24

47. Example 1
4. Use the distance formula utilizing this point on the line
and the point not on the line.
(1, 5), (−2, 2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (−2−1)2
+(2− 5)2
= (−3)2
+(−3)2
= 9+ 9 = 18 ≈ 4.24 units

48. Example 2
Find the distance between the parallel lines m and n with
the following equations.
y = 2x + 3 y = 2x −1

49. Example 2
Find the distance between the parallel lines m and n with
the following equations.
y = 2x + 3 y = 2x −1
1. Find the equation of the perpendicular line.

50. Example 2
Find the distance between the parallel lines m and n with
the following equations.
y = 2x + 3 y = 2x −1
1. Find the equation of the perpendicular line.
y = mx + b

51. Example 2
Find the distance between the parallel lines m and n with
the following equations.
y = 2x + 3 y = 2x −1
1. Find the equation of the perpendicular line.
y = mx + b
m = −
1
2
,(0,3)

52. Example 2
Find the distance between the parallel lines m and n with
the following equations.
y = 2x + 3 y = 2x −1
1. Find the equation of the perpendicular line.
y = mx + b
y = −
1
2
x + 3
m = −
1
2
,(0,3)

53. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.

54. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪

55. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3

56. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3
5
2
x = 4

57. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3
5
2
x = 4 x = 1.6

58. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3
5
2
x = 4 x = 1.6
y = 2(1.6)−1

59. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3
5
2
x = 4 x = 1.6
y = 2(1.6)−1
y = 2.2

60. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3
5
2
x = 4 x = 1.6
y = 2(1.6)−1
y = 2.2
y = −
1
2
(1.6)−1

61. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3
5
2
x = 4 x = 1.6
y = 2(1.6)−1
y = 2.2
y = −
1
2
(1.6)−1
y = 2.2

62. Example 2
2. Find the intersection of the perpendicular line and the
other parallel line using a system.
y = 2x −1
y = −
1
2
x + 3
⎧
⎨
⎪
⎩
⎪
2x −1= −
1
2
x + 3
5
2
x = 4 x = 1.6
y = 2(1.6)−1
y = 2.2
y = −
1
2
(1.6)−1
y = 2.2
(1.6, 2.2)

63. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.

64. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)

65. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)
d = (x2
− x1
)2
+(y2
− y1
)2

66. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (1.6 − 0)2
+(2.2− 3)2

67. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (1.6 − 0)2
+(2.2− 3)2
= (1.6)2
+(−0.8)2

68. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (1.6 − 0)2
+(2.2− 3)2
= (1.6)2
+(−0.8)2
= 2.56 +.64

69. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (1.6 − 0)2
+(2.2− 3)2
= (1.6)2
+(−0.8)2
= 2.56 +.64
= 3.2

70. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (1.6 − 0)2
+(2.2− 3)2
= (1.6)2
+(−0.8)2
= 2.56 +.64
= 3.2 ≈1.79

71. Example 2
3. Use the new point and original y-intercept you chose in
step 2 in the distance formula.
(0, 3), (1.6, 2.2)
d = (x2
− x1
)2
+(y2
− y1
)2
= (1.6 − 0)2
+(2.2− 3)2
= (1.6)2
+(−0.8)2
= 2.56 +.64
= 3.2 ≈1.79 units

72. Example 3
Line h contains the points E(2, 4) and F(5, 1). Find
the distance between line h and the point G(1, 1).

73. Example 3
Line h contains the points E(2, 4) and F(5, 1). Find
the distance between line h and the point G(1, 1).
Solution:

74. Example 3
Line h contains the points E(2, 4) and F(5, 1). Find
the distance between line h and the point G(1, 1).
Solution:
d = 8

75. Example 3
Line h contains the points E(2, 4) and F(5, 1). Find
the distance between line h and the point G(1, 1).
Solution:
d = 8 ≈ 2.83

76. Example 3
Line h contains the points E(2, 4) and F(5, 1). Find
the distance between line h and the point G(1, 1).
Solution:
d = 8 ≈ 2.83 units