Write the Coordinates of the Vertices After a Reflection Over X Axis Calculator

The point that bisects the line segment formed by two points, ( x 1 , y 1 ) and ( x 2 , y 2 ) , is called the midpointGiven two points, ( x 1 , y 1 ) and ( x 2 , y 2 ) , the midpoint is an ordered pair given by ( x 1 + x 2 2 , y 1 + y 2 2 ) . and is given by the following formula:

The midpoint is an ordered pair formed by finding the average of the x-values and the average of the y-values of the given points.

Example 8: Calculate the midpoint between (−1, −2) and (7, 4).

Solution: First, calculate the average of the x- and y-values of the given points.

Next, form the midpoint as an ordered pair using the averaged coordinates.

To verify that this is indeed the midpoint, calculate the distance between the two given points and verify that the result is equal to the sum of the two equal distances from the endpoints to this midpoint. This verification is left to the reader as an exercise.

Try this! Find the midpoint between (−6, 5) and (6, −11).

Topic Exercises

Part A: Ordered Pairs

Give the coordinates of points A , B , C , D , and E .

1.

2.

3.

4.

5.

6.

Graph the given set of ordered pairs.

7. {(−4, 5), (−1, 1), (−3, −2), (5, −1)}

8. {(−15, −10), (−5, 10), (15, 10), (5, −10)}

9. {(−2, 5), (10, 0), (2, −5), (6, −10)}

10. {(−8, 3), (−4, 6), (0, −6), (6, 9)}

11. {(−10, 5), (20,  −10), (30, 15), (50, 0)}

12. { ( − 5 3 , − 1 2 ) , ( − 1 3 , 1 2 ) , ( 2 3 , − 1 ) , ( 5 3 , 1 ) }

13. { ( − 3 5 , − 4 3 ) , ( 2 5 , 4 3 ) , ( 1 , − 2 3 ) , ( 0 , 1 ) }

14. {(−3.5, 0), (−1.5, 2), (0, 1.5), (2.5, −1.5)}

15. {(−0.8, 0.2), (−0.2, −0.4), (0, −1), (0.6, −0.4)}

16. {(−1.2, −1.2), (−0.3, −0.3), (0, 0), (0.6, 0.6), (1.2, 1.2)}

State the quadrant in which the given point lies.

17. (−3, 2)

18. (5, 7)

19. (−12, −15)

20. (7, −8)

21. (−3.8, 4.6)

22. (17.3, 1.9)

23. ( − 1 8 , − 5 8 )

24. ( 3 4 , − 1 4 )

25. x > 0 a n d y < 0

26. x < 0 a n d y < 0

27. x < 0 a n d y > 0

28. x > 0 a n d y > 0

The average price of a gallon of regular unleaded gasoline in US cities is given in the following line graph. Use the graph to answer the following questions.

Source: Bureau of Labor Statistics.

29. What was the average price of a gallon of unleaded gasoline in 2004?

30. What was the average price of a gallon of unleaded gasoline in 1976?

31. In which years were the average price of a gallon of unleaded gasoline $1.20?

32. What is the price increase of a gallon of gasoline from 1980 to 2008?

33. What was the percentage increase in the price of a gallon of unleaded gasoline from 1976 to 1980?

34. What was the percentage increase in the price of a gallon of unleaded gasoline from 2000 to 2008?

The average price of all-purpose white flour in US cities from 1980 to 2008 is given in the following line graph. Use the graph to answer the questions that follow.

Source: Bureau of Labor Statistics.

35. What was the average price per pound of all-purpose white flour in 2000?

36. What was the average price per pound of all-purpose white flour in 2008?

37. In which year did the price of flour average $0.25 per pound?

38. In which years did the price of flour average $0.20 per pound?

39. What was the percentage increase in flour from the year 2000 to 2008?

40. What was the percentage increase in flour from the year 1992 to 2000?

Given the following data, create a line graph.

41. The percentage of total high school graduates who enrolled in college.

Year	Percentage
1969	36%
1979	40%
1989	47%
1999	42%

Source: Digest of Education Statistics.

42. The average daily temperature given in degrees Fahrenheit in May.

Exam	Temperature
8:00 am	60
12:00 pm	72
4:00 pm	75
8:00 pm	67
12:00 am	60
4:00 am	55

Calculate the area of the shape formed by connecting the following set of vertices.

43. {(0, 0), (0, 3), (5, 0), (5, 3)}

44. {(−1, −1), (−1, 1), (1, −1), (1, 1)}

45. {(−2, −1), (−2, 3), (5, 3), (5, −1)}

46. {(−5, −4), (−5, 5), (3, 5), (3, −4)}

47. {(0, 0), (4, 0), (2, 2)}

48. {(−2, −2), (2, −2), (0, 2)}

49. {(0, 0), (0, 6), (3, 4)}

50. {(−2, 0), (5, 0), (3, −3)}

Part B: Distance Formula

Calculate the distance between the given two points.

51. (−5, 3) and (−1, 6)

52. (6, −2) and (−2, 4)

53. (0, 0) and (5, 12)

54. (−6, −8) and (0, 0)

55. (−7, 8) and (5, −1)

56. (−1, −2) and (9, 22)

57. (−1, 2) and (−7/2, −4)

58. ( − 1 2 , 1 3 ) and ( 5 2 , − 11 3 )

59. ( − 1 3 , 2 3 ) and ( 1 , − 1 3 )

60. ( 1 2 , − 3 4 ) and ( 3 2 , 1 4 )

61. (1, 2) and (4, 3)

62. (2, −4) and (−3, −2)

63. (−1, 5) and (1, −3)

64. (1, −7) and (5, −1)

65. (−7, −3) and (−1, 6)

66. (0, 1) and (1, 0)

67. (−0.2, −0.2) and (1.8, 1.8)

68. (1.2, −3.3) and (2.2, −1.7)

For each problem, show that the three points form a right triangle.

69. (−3, −2), (0, −2), and (0, 4)

70. (7, 12), (7, −13), and (−5, −4)

71. (−1.4, 0.2), (1, 2), and (1, −3)

72. (2, −1), (−1, 2), and (6, 3)

73. (−5, 2), (−1, −2), and (−2, 5)

74. (1, −2), (2, 3), and (−3, 4)

Isosceles triangles have two legs of equal length. For each problem, show that the following points form an isosceles triangle.

75. (1, 6), (−1, 1), and (3, 1)

76. (−6, −2), (−3, −5), and (−9, −5)

77. (−3, 0), (0, 3), and (3, 0)

78. (0, −1), (0, 1), and (1, 0)

Calculate the area and the perimeter of the triangles formed by the following set of vertices.

79. {(−4, −5), (−4, 3), (2, 3)}

80. {(−1, 1), (3, 1), (3, −2)}

81. {(−3, 1), (−3, 5), (1, 5)}

82. {(−3, −1), (−3, 7), (1, −1)}

Part C: Midpoint Formula

Find the midpoint between the given two points.

83. (−1, 6) and (−7, −2)

84. (8, 0) and (4, −3)

85. (−10, 0) and (10, 0)

86. (−3, −6) and (−3, 6)

87. (−10, 5) and (14, −5)

88. (0, 1) and (2, 2)

89. (5, −3) and (4, −5)

90. (0, 0) and (1, 1)

91. (−1, −1) and (4, 4)

92. (3, −5) and (3, 5)

93. ( − 1 2 , − 1 3 ) and ( 3 2 , 7 3 )

94. ( 3 4 , − 2 3 ) and ( 1 8 , − 1 2 )

95. ( 5 3 , 1 4 ) and ( − 1 6 , − 3 2 )

96. ( − 1 5 , − 5 2 ) and ( 7 10 , − 1 4 )

97. Given the right triangle formed by the vertices (0, 0), (6, 0), and (6, 8), show that the midpoints of the sides form a right triangle.

98. Given the isosceles triangle formed by the vertices (−10, −12), (0, 12), and (10, −12), show that the midpoints of the sides also form an isosceles triangle.

99. Calculate the area of the triangle formed by the vertices (−4, −3), (−1, 1), and (2, −3). (Hint: The vertices form an isosceles triangle.)

100. Calculate the area of the triangle formed by the vertices (−2, 1), (4, 1), and (1, −5).

Part D: Discussion Board Topics

101. Research and discuss the life and contributions to mathematics of René Descartes.

102. Research and discuss the history of the right triangle and the Pythagorean theorem.

103. What is a Pythagorean triple? Provide some examples.

104. Explain why you cannot use a ruler to calculate distance on a graph.

105. How do you bisect a line segment with only a compass and a straightedge?

Answers

1: A: (3, 5); B: (−2, 3); C: (−5, 0); D: (1, −3); E: (−3, −4)

3: A: (0, 6); B: (−4, 3); C: (−8, 0); D: (−6, −6); E: (8, −9)

5: A: (−10, 25); B: (30, 20); C: (0, 10); D: (15, 0); E: (25, −10)

7:

9:

11:

13:

15:

17: QII

19: QIII

21: QII

23: QIII

25: QIV

27: QII

29: $1.80

31: 1980 to 1984, 1996

33: 100%

35: $0.30

37: 1992

39: 67%

41:

43: 15 square units

45: 28 square units

47: 4 square units

49: 9 square units

51: 5 units

53: 13 units

55: 15 units

57: 13/2 units

59: 5/3 units

61: 10 units

63: 2 17 units

65: 3 13 units

67: 2.8 units

69: Proof

71: Proof

73: Proof

75: Proof

77: Proof

79: Perimeter: 24 units; area: 24 square units

81: Perimeter: 8 + 4 2 units; area: 8 square units

83: (−4, 2)

85: (0, 0)

87: (2, 0)

89: (9/2, −4)

91: (3/2, 3/2)

93: (1/2, 1)

95: (3/4, −5/8)

99: 12 square units

Write the Coordinates of the Vertices After a Reflection Over X Axis Calculator

Source: https://saylordotorg.github.io/text_elementary-algebra/s06-01-rectangular-coordinate-system.html

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