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4. Contents Lesson 11-1Squares and Square Roots Lesson 11-2Estimating Square Roots Lesson 11-3The Pythagorean Theorem Lesson 11-4Area of Parallelograms Lesson 11-5Area of Triangles and Trapezoids Lesson 11-6Area of Circles Lesson 11-7Area of Complex Figures Lesson 11-8Area Models and Probability

5. Lesson 1 Contents Example 1Find Squares of Numbers Example 2Find Squares of Numbers Example 3Find a Square to Solve a Problem Example 4Find Square Roots Example 5Find Square Roots Example 6Find Square Roots

6. Find the square of 5. Answer: 25 Example 1-1a

7. Example 1-1b Find the square of 7. Answer: 49

8. Find the square of 19. Answer: 361 Example 1-2a 19 361 ENTER

9. Example 1-2b Find the square of 21. Answer: 441

10. PHYSICAL SCIENCE A 6-kilogram ball is soaring through the air at 5 meters per second. If kinetic energy where m is the mass and v is the speed, what is the ball’s kinetic energy? Example 1-3a

11. Answer: So, the ball’s kinetic energy is Example 1-3b Write the formula. Replace m with 6 and v with 5. Find the square of 5. Simplify.

12. Example 1-3c PHYSICAL SCIENCE A 9-kilogram ball is soaring through the air at 3 meters per second. If kinetic energy where m is the mass and v is the speed, what is the ball’s kinetic energy? Answer:

13. Example 1-4a Find Answer: 6

14. Example 1-4b Find Answer: 8

15. Example 1-5a Find Answer: So, 676 26 ENTER 2nd

16. Example 1-5b Find Answer: 23

17. Example 1-6a GAMES A checkerboard is a square with an area of 1,225 square centimeters. What are the dimensions of the checkerboard? Answer: So, a checkerboard measures 35 centimeters by 35 centimeters. Find the square root of 1,225. 1225 35 ENTER 2nd

18. Example 1-6b GARDENING Kyle is planting a new garden that is a square with an area of 42.25 square feet. What are the dimensions of Kyle’s garden? Answer:

19. End of Lesson 1

20. Lesson 2 Contents Example 1Estimate the Square Root Example 2Use a Calculator to Estimate

21. Example 2-1a Estimate to the nearest whole number. List some perfect squares. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100… 96 96 is between the perfect squares 81 and 100. Find the square root of each number. and

22. Example 2-1b So, is between 9 and 10. Since 96 is closer to 100 than 81, the best whole number estimate is 10. Verify with a calculator. Answer: 10

23. Example 2-1c Estimate Answer: 6

24. Example 2-2a Answer: 6.1 Use a calculator to find the value of to the nearest tenth. Check Since 37 is between 36 and 49, the answer, 6.1, is reasonable. 37 6.08276253 ENTER 2nd

25. Example 2-2b Answer: 8.8 Use a calculator to find the value of to the nearest tenth.

26. End of Lesson 2

27. Lesson 3 Contents Example 1Find the Length of the Hypotenuse Example 2Find the Length of a Leg Example 3Solve a Real-Life Problem Example 4Identify Right Triangles Example 5Identify Right Triangles

28. Example 3-1a GYMNASTICS A gymnastics tumbling floor is in the shape of a square with sides 12 meters long. If a gymnast flips from one corner to the opposite corner, about how far has he flipped?

29. Example 3-1b To solve, find the length of the hypotenuse c. Pythagorean Theorem Replace a with 12 and b with 12. Evaluate Add. Take the square root of each side. Simplify. Answer: The gymnast will have flipped about 17 meters.

30. Example 3-1c SEWING Rose has a rectangular piece of fabric measuring 28 inches in length and 16 inches in width. She wants to decorate the fabric with a piece of lace sewn across both diagonals. How much lace will Rose need to complete the project? Answer: about 64.5 in.

31. Example 3-2a Find the missing measure of the triangle below.

32. Example 3-2b Pythagorean Theorem Replace b with 9 and c with 15. Evaluate Subtract 81 from each side. Simplify. Take the square root of each side. Simplify. Answer: The length of the leg is 12 centimeters.

33. Example 3-2c Find the missing measure of the triangle below. Round to the nearest tenth if necessary. Answer: 18.7 in.

34. Example 3-3a TELEVISION Televisions are measured according to their diagonal measure. If the diagonal of a television is 36 inches, and its height is 21.6 inches, what is its width? The diagonal of the television is the hypotenuse of a right triangle. Write an equation and solve for b.

35. Example 3-3b Pythagorean Theorem Replace a with 21.6 and c with 36. Evaluate Subtract 466.56 from each side. Simplify. Take the square root of each side. Simplify. Answer: The width of the television is 28.8 inches.

36. Example 3-3c SWIMMING The diagonal of a rectangular swimming pool measures 60 feet. Find the length of the pool if the width measures 30 feet. Round to the nearest tenth if necessary. Answer: about 52.0 ft

37. Example 3-4a Determine whether a triangle with the lengths 2.5 centimeters, 6 centimeters, and 6.5 centimeters is a right triangle. Answer: The triangle is a right triangle. Pythagorean Theorem Replace a with 2.5, b with 6, and c with 6.5. Evaluate squares. Simplify.

38. Example 3-4b Determine whether a triangle with the lengths 5 inches, 12 inches, and 13 inches is a right triangle. Answer: right triangle

39. Example 3-5a Determine whether a triangle with the lengths 5 feet, 6 feet, and 8 feet is a right triangle. Answer: The triangle is not a right triangle. Pythagorean Theorem Replace a with 5, b with 6, and c with 8. Evaluate squares. Simplify.

40. Example 3-5b Determine whether a triangle with the lengths 4.5 centimeters, 9 centimeters, and 12.5 centimeters is a right triangle. Answer: not a right triangle

41. End of Lesson 3

42. Lesson 4 Contents Example 1Find the Area of a Parallelogram Example 2Find the Area of a Parallelogram

43. Example 4-1a Answer: The area of the parallelogram is 48 square centimeters. This is the same as the estimate. Find the area of the parallelogram. Estimate A = 8 6 or 48 cm 2 Area of a parallelogram Replace b with 7.5 and h with 6.4. Multiply.

44. Example 4-1b Find the area of the parallelogram. Answer: 52 in 2

45. Example 4-2a Find the area of the parallelogram below. The base is 8 centimeters, and the height is 4.5 centimeters. Estimate A = 8 5 or 40 cm 2

46. Example 4-2b Area of a parallelogram Replace b with 8 and h with 4.5. Multiply. Answer: The area of the parallelogram is 36 square centimeters. This is close to the estimate.

47. Example 4-2c Find the area of the parallelogram to the right. Answer: 10.8 m 2

48. End of Lesson 4

49. Lesson 5 Contents Example 1Find the Area of a Triangle Example 2Find the Area of a Trapezoid Example 3Use a Formula to Estimate Area

50. Example 5-1a Find the area of the triangle below. Estimate

51. Example 5-1b Answer: The area of the triangle is 14.4 square centimeters. This is close to the estimate. Area of a triangle Replace b with 9 and h with 3.2. Multiply.

52. Example 5-1c Find the area of the triangle below. Answer: 13.5 ft 2

53. Example 5-2a Find the area of the trapezoid below. The bases are 4 meters and 7.6 meters. The height is 3 meters.

54. Example 5-2b Answer: The area of the trapezoid is 17.4 square meters. Area of a trapezoid Replace h with 3, b 1 with 4, and b 2 with 7.6. Add 4 and 7.6. Multiply.

55. Example 5-2c Find the area of the trapezoid below. Answer: 61.5 cm 2

56. Example 5-3a GEOGRAPHY The shape of the state of Montana resembles a trapezoid. Estimate its area in square miles.

57. Example 5-3b Answer: The area of Montana is about 145,493 square miles. Area of a trapezoid Replace h with 285, b 1 with 542, and b 2 with 479. Add 542 and 479. Multiply.

58. Example 5-3c PAINTING The diagram below is of a canvas resembling a trapezoid that will be painted. In order to determine how much paint will be needed, estimate the area of the canvas in square feet. Answer: 150 ft 2

59. End of Lesson 5

60. Lesson 6 Contents Example 1Find the Area of Circles Example 2Find the Area of Circles

61. Example 6-1a Find the area of the circle shown below. Area of a circle Replace r with 4. Answer: The area of the circle is approximately 50.3 square centimeters. 4 50.26548246 ENTER

62. Example 6-1b Find the area of the circle shown below. Answer: approximately 346.4 ft 2

63. Example 6-2a Find the area of a circle with a diameter of 11.6 centimeters. Area of a circle Answer: The area of the circle is approximately 105.7 square centimeters. Use a calculator. Replace r with

64. Example 6-2b Find the area of a circle with a diameter of 16.4 inches. Answer: 211.2 in 2

65. End of Lesson 6

66. Lesson 7 Contents Example 1Find the Area of an Irregular Room Example 2Find the Area of a Complex Figure

67. Example 7-1a WINDOWS The diagram below shows the dimensions of a window that is 3.4 feet by 7.2 feet. Find the area of the window. Round to the nearest tenth. The figure can be separated into a semicircle and a rectangle.

68. Example 7-1b Area of Semicircle Area of a semicircle Replace r with Simplify.

69. Example 7-1c Area of Rectangle Area of a rectangle Multiply. Answer: The area of the window is approximately 4.5 + 18.7 or 23.2 square feet.

70. Example 7-1d DRIVEWAY The diagram below shows the dimensions of a new driveway. Find the area of the driveway. Round to the nearest tenth. Answer: about 220.8 ft 2

71. Example 7-2a GRID-IN TEST ITEM Find the area of the figure below in square centimeters.

72. Example 7-2b Read the Test Item Solve the Test Item Area of a rectangle Multiply. The figure can be separated into a rectangle and a triangle. Find the area of each. Area of Rectangle

73. Example 7-2c Area of Triangle Area of a triangle Multiply. The area is 150 + 10 or 160 square centimeters.

74. Example 7-2c Answer:

75. Example 7-2d GRID-IN TEST ITEM Find the area of the figure in square inches. Answer:

76. End of Lesson 7

77. Lesson 8 Contents Example 1Use Area Models to Find Probability Example 2Find the Probability of Winning a Game

78. Example 8-1a PROBABILITY A randomly dropped counter falls somewhere in the squares below. Find the probability that it falls on the shaded squares.

79. Example 8-1b

80. Example 8-1c Area of Shaded Squares Area of a circle Simplify. Area of All Squares Area of a square Simplify. Answer: The probability of a counter falling in the shaded squares is approximately or about

81. Example 8-1d PROBABILITY A randomly dropped counter falls somewhere in the squares below. Find the probability that it falls on the shaded squares. Answer: 12%

82. Example 8-2a GAMES Suppose a dart is equally likely to hit any point on the figure below. What is the probability that it hits in the shaded region?

83. Example 8-2b Area of Shaded Region Area of a semicircle Simplify. Area of Total Region Area of a square Simplify. Replace r with 10.

84. Example 8-2c Use a calculator. Answer: The probability of hitting the shaded region is about 39.3%

85. Example 8-2d GAMES Suppose a dart is equally likely to hit any point on the figure below. What is the probability that it hits in the shaded region? Answer: 50%

86. End of Lesson 8

87. Online Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Mathematics: Applications and Concepts, Course 2 Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.msmath2.net/extra_examples.

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