1. Chapter 10 Geometry © 2010 Pearson Education, Inc. All rights reserved.

2. 10.1 Basic Geometric Terms Objectives Slide 8.1- 2 1. Identify and name lines, line segments, and rays. 2. Identify parallel and intersecting lines. 3. Identify and name angles. 4. Classify angles as right, acute, straight, or obtuse. 5. Identify perpendicular lines. Copyright © 2010 Pearson Education, Inc. All rights reserved.

3. Slide 8.1- 3 Geometry starts with the idea of a point. A point is a location in space. It has no length or width. A point is represented by a dot and is named by writing a capital letter next to the dot. Point R Copyright © 2010 Pearson Education, Inc. All rights reserved.

4. Slide 8.1- 4 A line is a straight row of points that goes on forever in both directions. A line is named using the letters of any two points on the line. A piece of line that has two endpoints is called a line segment. Copyright © 2010 Pearson Education, Inc. All rights reserved.

5. Slide 8.1- 5 A ray is a part of a line that has only one endpoint and goes on forever in one direction. The endpoint is always written first when naming a ray. Copyright © 2010 Pearson Education, Inc. All rights reserved.

6. Identify each figure below as a line, line segment, or ray, and name it using the appropriate symbol. a.b.c. Parallel Example 1 Identifying and Naming Lines, Rays, and Line Segments Slide 8.1- 6 Copyright © 2010 Pearson Education, Inc. All rights reserved.

7. Slide 8.1- 7 A plane is an infinitely large flat surface. A floor or a wall is part of a plane. Lines that are in the same plane, but that never intersect (never cross), are called parallel lines, while lines that cross are called intersecting lines. Copyright © 2010 Pearson Education, Inc. All rights reserved.

8. Label each pair of lines as appearing to be parallel or as intersecting. a.b.c. Parallel Example 2 Identifying Parallel and Intersecting Lines Slide 8.1- 8 Copyright © 2010 Pearson Education, Inc. All rights reserved.

9. Slide 8.1- 9 An angle is made up of two rays that start at a common endpoint. This common endpoint is called the vertex. Copyright © 2010 Pearson Education, Inc. All rights reserved.

10. Slide 8.1- 10 Copyright © 2010 Pearson Education, Inc. All rights reserved.

11. Name the highlighted angle (pink) in three different ways. Parallel Example 3 Identifying and Naming an Angle Slide 8.1- 11 E F G H J 23 4 Copyright © 2010 Pearson Education, Inc. All rights reserved.

12. Slide 8.1- 12 Angles can be measured in degrees. The symbol for degrees is a small raised circle . An angle of 180  is called a straight angle. An angle of 90  is called a right angle. Copyright © 2010 Pearson Education, Inc. All rights reserved.

13. Slide 8.1- 13 Some other terms used to describe angles are shown below. Acute angles measure less than 90  Obtuse angles measure more than 90  but less than 180 . Copyright © 2010 Pearson Education, Inc. All rights reserved.

14. Slide 8.1- 14 Copyright © 2010 Pearson Education, Inc. All rights reserved.

15. Label each angle as acute, right, obtuse, or straight. a.b. c.d. Parallel Example 4 Classifying Angles Slide 8.1- 15 straight angle acute angle obtuse angle right angle Copyright © 2010 Pearson Education, Inc. All rights reserved.

16. Slide 8.1- 16 Two lines are called perpendicular lines if they intersect to form a right angle. are perpendicular lines because they intersect at right angles. Copyright © 2010 Pearson Education, Inc. All rights reserved.

17. Which pairs of lines are perpendicular? a.b. Parallel Example 5 Identifying Perpendicular Lines Slide 8.1- 17 PerpendicularIntersecting but not perpendicular A B C D E A B C D Copyright © 2010 Pearson Education, Inc. All rights reserved.

18. 10.2 Rectangles and Squares Objectives Slide 8.3- 18 1.Find the perimeter and area of a rectangle. 2.Find the perimeter and area of a square. 3.Find the perimeter and area of a composite figure. Copyright © 2010 Pearson Education, Inc. All rights reserved.

19. A rectangle has four sides that meet to form 90 ° angles. Each set of opposite sides is parallel and congruent (has the same length). 5 cm 9 cm 5 cm 9 cm In a rectangle, if one right angle is shown, the other three are also right angles. 90° angles Each longer side of a rectangle is called the length (l) and each shorter side is called the width (w). Slide 8.3- 19 Copyright © 2010 Pearson Education, Inc. All rights reserved.

20. Slide 8.3- 20 Copyright © 2010 Pearson Education, Inc. All rights reserved.

21. Parallel Example 1 Finding the Perimeter of a Rectangle Slide 8.3- 21 Find the perimeter of each rectangle. a. 6 m 16 m 6 m 16 m P = 2 l + 2 w P = 2 16 m + 2 6 m P = 32 m + 12 m P = 44 m The perimeter of the rectangle is 44 m. Copyright © 2010 Pearson Education, Inc. All rights reserved.

22. Parallel Example 1 continued Finding the Perimeter of a Rectangle Slide 8.3- 22 Find the perimeter of each rectangle. b. A rectangle 7.8 ft by 12.3 ft P = 2 l + 2 w Either method will give you the same result. P = 2 12.3 ft + 2 7.8 ft P = 24.6 ft + 15.6 ft P = 40.2 ft Or, you can add up the lengths of the four sides. P = 12.3 ft + 12.3 ft + 7.8 ft + 7.8 ft P = 40.2 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

23. Slide 8.3- 23 The perimeter of a rectangle is the distance around the outside edges. The area of a rectangle is the amount of surface inside the rectangle. 8 m 5 m 1 m We have five rows of eight square meters for a total of 40 square meters. 1 square meter or (m) 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

24. Slide 8.3- 24 Copyright © 2010 Pearson Education, Inc. All rights reserved.

25. Squares of many sizes can be used to measure area. For smaller areas, you might use the ones shown below. Slide 8.3- 25 1 square inch (1 in. 2 ) 1 in. 1 square centimeter (1 cm 2 ) 1 cm 1 square millimeter (1 mm 2 ) 1 mm (Approximate-size drawings) Other sizes of squares that are often used to measure area: 1 square meter (1 m 2 )1 square foot (1 ft 2 ) 1 square kilometer (1 km 2 )1 square yard (1 yd 2 ) 1 square mile (1 mi 2 ) Copyright © 2010 Pearson Education, Inc. All rights reserved.

26. Parallel Example 2 Finding the Area of a Rectangle Slide 8.3- 26 Find the area of each rectangle. a. 7 yd 15 yd A = l w A = 15 yd 7 yd A = 105 yd 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

27. Parallel Example 2 continued Finding the Area of a Rectangle Slide 8.3- 27 Find the area of each rectangle. b. A = l w A = 18 cm 3 cm A = 54 cm 2 18 cm 3 cm Copyright © 2010 Pearson Education, Inc. All rights reserved.

28. Slide 8.3- 28 Copyright © 2010 Pearson Education, Inc. All rights reserved.

29. Parallel Example 3 Finding the Perimeter and Area of a Square Slide 8.3- 29 a. Find the perimeter of a square where each side measures 7 m. Use the formula. P = 4 s P = 4 7 m P = 28 m Or add up the four sides. P = 7 m + 7 m + 7 m + 7 m P = 28 m Same answer Copyright © 2010 Pearson Education, Inc. All rights reserved.

30. Parallel Example 3 continued Finding the Perimeter and Area of a Square Slide 8.3- 30 b. Find the area of a square where each side measures 7 m. A = s s A = 7 m 7 m A = s 2 A = 49 m 2 Square units for area. Copyright © 2010 Pearson Education, Inc. All rights reserved.

31. Parallel Example 4 Finding the Perimeter and Area of a Composite Figure Slide 8.3- 31 a. The floor of a room has the shape shown. 6 ft 30 ft 21 ft 24 ft 15 ft Suppose you want to put new wallpaper border along the top of the walls. How much material do you need? Find the perimeter of the room by adding up the length of the sides. P = 30 ft + 21 ft + 24 ft + 15 ft + 6 ft + 6 ft= 102 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

32. Parallel Example 4 continued Finding the Perimeter and Area of a Composite Figure Slide 8.3- 32 b. The carpet you like cost $24.25 per square yard. How much will it cost to carpet the room? First change the measurements from feet to yards. 2 yd 10 yd 7 yd 8 yd 5 yd 6 ft 30 ft 21 ft 24 ft 15 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

33. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parallel Example 4 continued Finding the Perimeter and Area of a Composite Figure Slide 8.3- 33 b. Next break the room into two pieces. Use just the measurements for the length and width of each piece. 2 yd 7 yd 8 yd Area of rectangle = l w A = 8 yd 7 yd A = 56 yd 2 Area of square = s 2 A = s s A = 2 yd 2 yd A = 4 yd 2 Total area = 56 yd 2 + 4 yd 2 = 60 yd 2

34. Parallel Example 4 continued Finding the Perimeter and Area of a Composite Figure Slide 8.3- 34 b. Finally, multiply to find the cost of the carpet. 2 yd 7 yd 8 yd $24.25 1 yd 2 60 yd 2 1 = $1455 It will cost $1455 to carpet the room. Copyright © 2010 Pearson Education, Inc. All rights reserved.

35. 10.1 (cont.) Angles and Their Relationships Objectives Slide 8.2- 35 1. Identify complementary angles and supplementary angles and find the measure of a complement or supplement of an angle. 2. Identify congruent angles and vertical angles and use this knowledge to find the measures of angles. Copyright © 2010 Pearson Education, Inc. All rights reserved.

36. Slide 8.2- 36 Two angles are called complementary angles if the sum of their measures is 90 . If two angles are complementary, each angle is the complement of the other. Copyright © 2010 Pearson Education, Inc. All rights reserved.

37. Identify each pair of complementary angles. Parallel Example 1 Identifying Complementary Angles Slide 8.2- 37 1 2 3 4 15  75  Copyright © 2010 Pearson Education, Inc. All rights reserved.

38. Find the complement of each angle. a. 32  b. 65  Parallel Example 2 Finding the Complement of Angles Slide 8.2- 38 Copyright © 2010 Pearson Education, Inc. All rights reserved.

39. Slide 8.2- 39 Two angles are called supplementary angles if the sum of their measures is 180 . If two angles are supplementary, each angle is the supplement of the other. Copyright © 2010 Pearson Education, Inc. All rights reserved.

40. Identify each pair of supplementary angles. Parallel Example 3 Identifying Supplementary Angles Slide 8.2- 40 2 15  165  1 3 4 15  Copyright © 2010 Pearson Education, Inc. All rights reserved.

41. Find the supplement of each angle. a. 84  b. 135  Parallel Example 4 Finding the Supplement of Angles Slide 8.2- 41 Copyright © 2010 Pearson Education, Inc. All rights reserved.

42. Slide 8.2- 42 Two angles are called congruent angles if they measure the same number of degrees. Copyright © 2010 Pearson Education, Inc. All rights reserved.

43. Identify the angles that are congruent. Parallel Example 5 Identifying Congruent Angles Slide 8.2- 43 95  85  95  J K L M N Copyright © 2010 Pearson Education, Inc. All rights reserved.

44. Slide 8.2- 44 Copyright © 2010 Pearson Education, Inc. All rights reserved.

45. Identify the vertical angles in the figure. Parallel Example 6 Identifying Vertical Angles Slide 8.2- 45 A B C D E Copyright © 2010 Pearson Education, Inc. All rights reserved.

46. In the figure below, find the measure of each unlabeled angle. Parallel Example 7 Finding the Measure of Vertical Angles Slide 8.2- 46 Copyright © 2010 Pearson Education, Inc. All rights reserved.

47. 10.2/10.3 Rectangles and Squares Objectives Slide 8.3- 47 1.Find the perimeter and area of a rectangle. 2.Find the perimeter and area of a square. 3.Find the perimeter and area of a composite figure. Copyright © 2010 Pearson Education, Inc. All rights reserved.

48. A rectangle has four sides that meet to form 90 ° angles. Each set of opposite sides is parallel and congruent (has the same length). 5 cm 9 cm 5 cm 9 cm In a rectangle, if one right angle is shown, the other three are also right angles. 90° angles Each longer side of a rectangle is called the length (l) and each shorter side is called the width (w). Slide 8.3- 48 Copyright © 2010 Pearson Education, Inc. All rights reserved.

49. Slide 8.3- 49 Copyright © 2010 Pearson Education, Inc. All rights reserved.

50. Parallel Example 1 Finding the Perimeter of a Rectangle Slide 8.3- 50 Find the perimeter of each rectangle. a. 6 m 16 m 6 m 16 m P = 2 l + 2 w P = 2 16 m + 2 6 m P = 32 m + 12 m P = 44 m The perimeter of the rectangle is 44 m. Copyright © 2010 Pearson Education, Inc. All rights reserved.

51. Parallel Example 1 continued Finding the Perimeter of a Rectangle Slide 8.3- 51 Find the perimeter of each rectangle. b. A rectangle 7.8 ft by 12.3 ft P = 2 l + 2 w Either method will give you the same result. P = 2 12.3 ft + 2 7.8 ft P = 24.6 ft + 15.6 ft P = 40.2 ft Or, you can add up the lengths of the four sides. P = 12.3 ft + 12.3 ft + 7.8 ft + 7.8 ft P = 40.2 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

52. Slide 8.3- 52 The perimeter of a rectangle is the distance around the outside edges. The area of a rectangle is the amount of surface inside the rectangle. 8 m 5 m 1 m We have five rows of eight square meters for a total of 40 square meters. 1 square meter or (m) 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

53. Slide 8.3- 53 Copyright © 2010 Pearson Education, Inc. All rights reserved.

54. Squares of many sizes can be used to measure area. For smaller areas, you might use the ones shown below. Slide 8.3- 54 1 square inch (1 in. 2 ) 1 in. 1 square centimeter (1 cm 2 ) 1 cm 1 square millimeter (1 mm 2 ) 1 mm (Approximate-size drawings) Other sizes of squares that are often used to measure area: 1 square meter (1 m 2 )1 square foot (1 ft 2 ) 1 square kilometer (1 km 2 )1 square yard (1 yd 2 ) 1 square mile (1 mi 2 ) Copyright © 2010 Pearson Education, Inc. All rights reserved.

55. Parallel Example 2 Finding the Area of a Rectangle Slide 8.3- 55 Find the area of each rectangle. a. 7 yd 15 yd A = l w A = 15 yd 7 yd A = 105 yd 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

56. Parallel Example 2 continued Finding the Area of a Rectangle Slide 8.3- 56 Find the area of each rectangle. b. A = l w A = 18 cm 3 cm A = 54 cm 2 18 cm 3 cm Copyright © 2010 Pearson Education, Inc. All rights reserved.

57. Slide 8.3- 57 Copyright © 2010 Pearson Education, Inc. All rights reserved.

58. Parallel Example 3 Finding the Perimeter and Area of a Square Slide 8.3- 58 a. Find the perimeter of a square where each side measures 7 m. Use the formula. P = 4 s P = 4 7 m P = 28 m Or add up the four sides. P = 7 m + 7 m + 7 m + 7 m P = 28 m Same answer Copyright © 2010 Pearson Education, Inc. All rights reserved.

59. Parallel Example 3 continued Finding the Perimeter and Area of a Square Slide 8.3- 59 b. Find the area of a square where each side measures 7 m. A = s s A = 7 m 7 m A = s 2 A = 49 m 2 Square units for area. Copyright © 2010 Pearson Education, Inc. All rights reserved.

60. Parallel Example 4 Finding the Perimeter and Area of a Composite Figure Slide 8.3- 60 a. The floor of a room has the shape shown. 6 ft 30 ft 21 ft 24 ft 15 ft Suppose you want to put new wallpaper border along the top of the walls. How much material do you need? Find the perimeter of the room by adding up the length of the sides. P = 30 ft + 21 ft + 24 ft + 15 ft + 6 ft + 6 ft= 102 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

61. Parallel Example 4 continued Finding the Perimeter and Area of a Composite Figure Slide 8.3- 61 b. The carpet you like cost $24.25 per square yard. How much will it cost to carpet the room? First change the measurements from feet to yards. 2 yd 10 yd 7 yd 8 yd 5 yd 6 ft 30 ft 21 ft 24 ft 15 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

62. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parallel Example 4 continued Finding the Perimeter and Area of a Composite Figure Slide 8.3- 62 b. Next break the room into two pieces. Use just the measurements for the length and width of each piece. 2 yd 7 yd 8 yd Area of rectangle = l w A = 8 yd 7 yd A = 56 yd 2 Area of square = s 2 A = s s A = 2 yd 2 yd A = 4 yd 2 Total area = 56 yd 2 + 4 yd 2 = 60 yd 2

63. Parallel Example 4 continued Finding the Perimeter and Area of a Composite Figure Slide 8.3- 63 b. Finally, multiply to find the cost of the carpet. 2 yd 7 yd 8 yd $24.25 1 yd 2 60 yd 2 1 = $1455 It will cost $1455 to carpet the room. Copyright © 2010 Pearson Education, Inc. All rights reserved.

64. 10.2/10.3 Parallelograms and Trapezoids Objectives Slide 8.4- 64 1.Find the perimeter and area of a parallelogram. 2.Find the perimeter and area of a trapezoid. Copyright © 2010 Pearson Education, Inc. All rights reserved.

65. A parallelogram is a four-sided figure with opposite sides parallel, such as the ones below. Notice that the opposite sides have the same length. Slide 8.4- 65 Copyright © 2010 Pearson Education, Inc. All rights reserved.

66. Parallel Example 1 Finding the Perimeter of a Parallelogram Slide 8.4- 66 Find the perimeter of a the parallelogram. P = 15 cm + 9 cm + 15 cm + 9 cm 15 cm 9 cm = 48 cm Copyright © 2010 Pearson Education, Inc. All rights reserved.

67. Slide 8.4- 67 Copyright © 2010 Pearson Education, Inc. All rights reserved.

68. Parallel Example 2 Finding the Area of a Parallelogram Slide 8.4- 68 Find the area of the parallelogram. The base is 10 m and the height is 3 m. Use the formula to solve. 10 m 4 m 3 m A = b ∙ h A = 10 m ∙ 3 m A = 30 m 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

69. A trapezoid is a four-sided figure with exactly one pair of parallel sides, such as the figures shown below. Unlike the parallelogram, opposite sides of a trapezoid might not have the same length. Slide 8.4- 69 Copyright © 2010 Pearson Education, Inc. All rights reserved.

70. Parallel Example 3 Finding the Perimeter of a Trapezoid Slide 8.4- 70 Find the perimeter of a the trapezoid. P = 10 m + 13 m + 10 m + 7 m= 40 m 7 m 13 m 8 m 10 m Notice the height (8 m) is not part of the perimeter, because the height is not one of the outside edges of the shape. Copyright © 2010 Pearson Education, Inc. All rights reserved.

71. Slide 8.4- 71 Copyright © 2010 Pearson Education, Inc. All rights reserved.

72. Parallel Example 4 Finding the Area of a Trapezoid Slide 8.4- 72 Find the area of this trapezoid. The short and long bases are the parallel sides. 7 m 13 m 8 m 10 m 1 10 Note: You can also use 0.5, the decimal equivalent for ½ in the formula. Copyright © 2010 Pearson Education, Inc. All rights reserved.

73. Parallel Example 6 Applying Knowledge of Area Slide 8.4- 73 Suppose the figure in Example 4 represents the floor plan of a hospital lobby. What is the cost to tile the area if tile costs $16.75 per square meter? The floor area is 80 m 2. To find the cost to tile the floor, multiply the number of square meters times the cost of the tile per square meter. The cost of tile for the lobby is $1340. Copyright © 2010 Pearson Education, Inc. All rights reserved.

74. 10.4 Triangles Objectives Slide 8.5- 74 1. Find the perimeter of a triangle. 2. Find the area of a triangle. 3. Given the measures of two angles in a triangle, find the measure of the third angle. Copyright © 2010 Pearson Education, Inc. All rights reserved.

75. Slide 8.5- 75 A triangle is a figure with exactly three sides. To find the perimeter of a triangle, add the lengths of the three sides. Copyright © 2010 Pearson Education, Inc. All rights reserved.

76. Find the perimeter of the triangle. P = 12 ft + 16 ft + 20 ft = 48 ft Parallel Example 1 Finding the Perimeter of a Triangle Slide 8.5- 76 12 ft 16 ft 20 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

77. Slide 8.5- 77 The height of a triangle is the distance from one vertex of the triangle to the opposite side (base). The height line must be perpendicular to the base; that is, it must form a right angle with the base. Copyright © 2010 Pearson Education, Inc. All rights reserved.

78. Slide 8.5- 78 Copyright © 2010 Pearson Education, Inc. All rights reserved.

79. Find the area of each triangle. a. Parallel Example 2 Find the Area of a Triangle Slide 8.5- 79 Copyright © 2010 Pearson Education, Inc. All rights reserved.

80. Find the area of each triangle. b. Parallel Example 2 continued Find the Area of a Triangle Slide 8.5- 80 45.5 cm 34.2 cm 14.6 cm Copyright © 2010 Pearson Education, Inc. All rights reserved.

81. Find the area of each triangle. c. Parallel Example 2 continued Find the Area of a Triangle Slide 8.5- 81 Copyright © 2010 Pearson Education, Inc. All rights reserved.

82. Find the area of the shaded part of the figure. Parallel Example 3 Using the Concept of Area Slide 8.5- 82 The entire figure is a rectangle. Find the area. The unshaded part is a triangle. Find the area of the triangle. Subtract to find the area of the shaded part. Copyright © 2010 Pearson Education, Inc. All rights reserved.

83. Slide 8.5- 83 Copyright © 2010 Pearson Education, Inc. All rights reserved.

84. Find the number of degrees in Angle C. Parallel Example 5 Finding an Angle Measurement in Triangles Slide 8.5- 84 Step 1 Add the two angle measurements you are given. Step 2 Subtract the sum from 180 . Copyright © 2010 Pearson Education, Inc. All rights reserved.

85. Find the number of degrees in Angle D. Parallel Example 5 Slide 8.5- 85 Step 1 Step 2 Finding an Angle Measurement in Triangles Copyright © 2010 Pearson Education, Inc. All rights reserved.

86. 10.4 (cont.) Pythagorean Theorem Objectives Slide 8.8- 86 1.Find square roots using the square root key on a calculator. 2.Find the unknown length in a right triangle. 3.Solve application problems involving right triangles. Copyright © 2010 Pearson Education, Inc. All rights reserved.

87. A number that has a whole number as its square root is called a perfect square. The first few perfect squares are listed below. Slide 8.8- 87 Copyright © 2010 Pearson Education, Inc. All rights reserved.

88. Parallel Example 1 Find the Square Root of Numbers Slide 8.8- 88 Use a calculator to find each square root. Round answers to the nearest thousandth. a. The calculator shows 6.782329983; round to 6.782 b. The calculator shows 11.66190379; round to 11.662 c. The calculator shows 16.1245155; round to 16.125 Copyright © 2010 Pearson Education, Inc. All rights reserved.

89. Slide 8.8- 89 One place you will use square roots is when working with the Pythagorean Theorem. This theorem applies only to right triangles. Recall that a right triangle is a triangle that has one 90° angle. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called legs. Copyright © 2010 Pearson Education, Inc. All rights reserved.

90. Slide 8.8- 90 Copyright © 2010 Pearson Education, Inc. All rights reserved.

91. Slide 8.8- 91 Copyright © 2010 Pearson Education, Inc. All rights reserved.

92. Parallel Example 2 Find the Unknown Length in Right Triangles Slide 8.8- 92 Find the unknown length in each right triangle. Round answers to the nearest tenth if necessary. a. The unknown length is the side opposite the right angle. Use the formula for finding the hypotenuse. 8 cm 15 cm The length is 17 cm. long Copyright © 2010 Pearson Education, Inc. All rights reserved.

93. Parallel Example 2 continued Find the Unknown Length in Right Triangles Slide 8.8- 93 Find the unknown length in each right triangle. Round answers to the nearest tenth if necessary. b. Use the formula for finding the leg. 15 ft 40 ft The length is approximately 37.1 ft long. Copyright © 2010 Pearson Education, Inc. All rights reserved.

94. Parallel Example 3 Using the Pythagorean Theorem Slide 8.8- 94 An electrical pole is shown below. Find the length of the guy wire. Round your answer to the nearest tenth of a foot if necessary. The length of the guy wire is approximately 69.5 ft. 35ft 60 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

95. 10.5 Circles Objectives Slide 8.6- 95 1.Find the radius and diameter of a circle. 2.Find the circumference of a circle. 3.Find the area of a circle. 4.Become familiar with Latin and Greek prefixes used in math terminology. Copyright © 2010 Pearson Education, Inc. All rights reserved.

96. Slide 8.6- 96 Suppose you start with one dot on a piece of paper. Then place many dots that are each 3 cm away from the first dot. If we place enough dots (points) we’ll end up with a circle. The 3cm is the radius of the circle. The distance across is the diameter. Each line below is 3 cm long. 3 cm diameter radius center r r d Copyright © 2010 Pearson Education, Inc. All rights reserved.

97. Slide 8.6- 97 r r d Copyright © 2010 Pearson Education, Inc. All rights reserved.

98. Slide 8.6- 98 r r d Copyright © 2010 Pearson Education, Inc. All rights reserved.

99. Parallel Example 1 Finding the Diameter and Radius of a Circle Slide 8.6- 99 Find the unknown length of the diameter or radius in each circle. a. r = 12 in. d = ? Because the radius is 12 in., the diameter is twice as long. d = 2 r d = 2 12 in. d = 24 in. Copyright © 2010 Pearson Education, Inc. All rights reserved.

100. Parallel Example 1 continued Finding the Diameter and Radius of a Circle Slide 8.6- 100 Find the unknown length of the diameter or radius in each circle. b. r = ? d = 7 m The radius is half the diameter. r = d 2 7 m 2 r = 3.5 m or 3 m 1 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

101. Slide 8.6- 101 The perimeter of a circle is called its circumference. Circumference is the distance around the edge of a circle. Copyright © 2010 Pearson Education, Inc. All rights reserved.

102. Slide 8.6- 102 Dividing the circumference of any circle by its diameter always gives an answer close to 3.14. This means that going around the edge of any circle is a little more than 3 times as far as going straight across the circle. This ratio of circumference to diameter is called Copyright © 2010 Pearson Education, Inc. All rights reserved.

103. Slide 8.6- 103 Copyright © 2010 Pearson Education, Inc. All rights reserved.

104. Parallel Example 2 Finding the Circumference of Circles Slide 8.6- 104 Find the circumference of each circle. Use 3.14 as the approximate value for. Round answers to the nearest tenth. a. 24 m The diameter is 24 m, so use the formula with d in it. C = d C = 3.14 24 m C ≈ 75.4 m Rounded Copyright © 2010 Pearson Education, Inc. All rights reserved.

105. Parallel Example 2 Finding the Circumference of Circles Slide 8.6- 105 Find the circumference of each circle. Use 3.14 as the approximate value for. Round answers to the nearest tenth. b. 6.5 cm In this example, the radius is labeled, so it is easier to use the formula with r in it. C = 2 r C = 2 3.14 6.5 cm C ≈ 40.8 cm Rounded Copyright © 2010 Pearson Education, Inc. All rights reserved.

106. Slide 8.6- 106 Finding the Area of a Circle C = 2 π r Unfold each circle.Now put them together. 2 π r rr Copyright © 2010 Pearson Education, Inc. All rights reserved.

107. Slide 8.6- 107 The figure is approximately a parallelogram. Area = b h Area = 2 r r 2 r r Area = Note: This is the area for 2 circles. The area for one circle is found by using Copyright © 2010 Pearson Education, Inc. All rights reserved.

108. Slide 8.6- 108 Copyright © 2010 Pearson Education, Inc. All rights reserved.

109. Parallel Example 3 Finding the Area of Circles Slide 8.6- 109 Find the area of each circle. Use 3.14 for. Round answers to the nearest tenth. a. A circle with a radius of 14.2 cm. Rounded; square units for area A = r r A ≈ 3.14 14.2 cm 14.2 cm A ≈ 633.1 cm 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

110. Parallel Example 3 continued Finding the Area of Circles Slide 8.6- 110 Find the area of each circle. Use 3.14 for. Round answers to the nearest tenth. b. Now find the area. 24 ft First find the radius. r = d 2 r = = 12 ft 24 ft 2 A ≈ 3.14 12 ft 12 ft A ≈ 452.2 ft 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

111. Parallel Example 4 Finding the Area of a Semicircle Slide 8.6- 111 Find the area of the semicircle. Use 3.14 for. Round your answer to the nearest tenth. A = r r 9 ft First, find the area of the whole circle with the radius of 9 ft. A ≈ 3.14 9 ft 9 ft  Do not round yet. A ≈ 254.34 ft 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

112. Parallel Example 4 continued Finding the Area of a Semicircle Slide 8.6- 112 9 ft Now, divide the area of the whole circle by 2. 2 254.34 ft 2 127.17 ft 2 = The last step is to round it the nearest tenth. The area of the semicircle is approximately 127.2 ft 2. Copyright © 2010 Pearson Education, Inc. All rights reserved.

113. Parallel Example 5 Applying the Concept of Circumference Slide 8.6- 113 A circular rug is 10 feet in diameter. The cost of fringe for the edge is $3.20 per foot. What will it cost to add fringe to the rug? Use 3.14 for. C = 3.14 10 ft C ≈ 31.4 ft cost = cost per foot circumference cost = $3.20 31.4 ft 1 ft1 cost = $100.48 The cost of adding fringe to the rug is $100.48. Copyright © 2010 Pearson Education, Inc. All rights reserved.

114. Parallel Example 7 Using Prefixes to Understand Math Terms Slide 8.6- 114 Listed below are some Latin and Greek root words and prefixes with their meanings in parentheses. List at least one math term and one nonmathematical word that use each prefix or root word. cent- (100):centigram;centipede circum- (around): de- (down): dec- (10): circumference;circumvent decrease;defame decagon;decathlon Copyright © 2010 Pearson Education, Inc. All rights reserved.

115. 10.6/10.7 Volume Objectives Slide 8.7- 115 Find the volume of a 1. rectangular solid; 2. sphere; 3. cylinder; 4. cone and pyramid. Copyright © 2010 Pearson Education, Inc. All rights reserved.

116. Slide 8.7- 116 A shoe box and a cereal box are examples of three-dimensional (or solid) figures. The three dimensions are length, width, and height. If you want to know how much a shoe box will hold, you find its volume. Copyright © 2010 Pearson Education, Inc. All rights reserved.

117. Slide 8.7- 117 Three sizes of cubic units Copyright © 2010 Pearson Education, Inc. All rights reserved.

118. Slide 8.7- 118 Copyright © 2010 Pearson Education, Inc. All rights reserved.

119. Find the volume of each box. a. The figure is made up of 3 layers of 20 cubes each, so its volume is 60 cubic centimeters (cm 3 ). Parallel Example 1 Finding the Volume of Rectangular Solids Slide 8.7- 119 5 cm 4 cm 3 cm Copyright © 2010 Pearson Education, Inc. All rights reserved.

120. Find the volume of each box. b. Parallel Example 1 Finding the Volume of Rectangular Solids Slide 8.7- 120 10 in. 6 in. 5 in. Copyright © 2010 Pearson Education, Inc. All rights reserved.

121. Slide 8.7- 121 A sphere is shown below. Examples of spheres include baseballs, oranges, and Earth. The radius of a sphere is the distance from the center to the edge of the sphere. Copyright © 2010 Pearson Education, Inc. All rights reserved.

122. Find the volume of the sphere with the help of calculator. Use 3.14 as the approximate value of . Round to the nearest tenth. Parallel Example 2 Finding the Volume of a Sphere Slide 8.7- 122 8 in. Copyright © 2010 Pearson Education, Inc. All rights reserved.

123. Slide 8.7- 123 Half a sphere is called a hemisphere. The volume of a hemisphere is half the volume of a sphere. Copyright © 2010 Pearson Education, Inc. All rights reserved.

124. Find the volume of the hemisphere with the help of calculator. Use 3.14 for . Round to the nearest tenth. Parallel Example 3 Finding the Volume of a Hemisphere Slide 8.7- 124 6 in. Copyright © 2010 Pearson Education, Inc. All rights reserved.

125. Slide 8.7- 125 Several cylinders are shown below. The height must be perpendicular to the circular top and bottom of the cylinder. Copyright © 2010 Pearson Education, Inc. All rights reserved.

126. Find the volume of each cylinder. Use 3.14 for . Round to the nearest tenth. a. Parallel Example 4 Finding the Volume of Cylinders Slide 8.7- 126 16 m 7 m The diameter is 16 m so the radius is half: 16 ÷ 2 = 8 m. Copyright © 2010 Pearson Education, Inc. All rights reserved.

127. Find the volume of each cylinder. Use 3.14 for . Round to the nearest tenth. b. Parallel Example 4 Finding the Volume of Cylinders Slide 8.7- 127 16 ft 3 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.

128. Slide 8.7- 128 A cone and pyramid are shown below. Notice that the height line is perpendicular to the base in both solids. Copyright © 2010 Pearson Education, Inc. All rights reserved.

129. Find the volume of the cone. Use 3.14 for . Round to the nearest tenth. Parallel Example 5 Finding the Volume of a Cone Slide 8.7- 129 First find the value of B in the formula, which is the area of the circular base. 12 cm 5 cm Now find the volume. Copyright © 2010 Pearson Education, Inc. All rights reserved.

130. Slide 8.7- 130 Copyright © 2010 Pearson Education, Inc. All rights reserved.

131. Find the volume of the pyramid. Round to the nearest tenth. Parallel Example 6 Finding the Volume of a Pyramid Slide 8.7- 131 First find the value of B in the formula, which is the area of a rectangular base. Now find the volume. 12 cm 6 cm 5 cm Copyright © 2010 Pearson Education, Inc. All rights reserved.

132. 10.4 (cont.) Pythagorean Theorem Objectives Slide 8.8- 132 1.Find square roots using the square root key on a calculator. 2.Find the unknown length in a right triangle. 3.Solve application problems involving right triangles. Copyright © 2010 Pearson Education, Inc. All rights reserved.

133. A number that has a whole number as its square root is called a perfect square. The first few perfect squares are listed below. Slide 8.8- 133 Copyright © 2010 Pearson Education, Inc. All rights reserved.

134. Parallel Example 1 Find the Square Root of Numbers Slide 8.8- 134 Use a calculator to find each square root. Round answers to the nearest thousandth. a. The calculator shows 6.782329983; round to 6.782 b. The calculator shows 11.66190379; round to 11.662 c. The calculator shows 16.1245155; round to 16.125 Copyright © 2010 Pearson Education, Inc. All rights reserved.

135. Slide 8.8- 135 One place you will use square roots is when working with the Pythagorean Theorem. This theorem applies only to right triangles. Recall that a right triangle is a triangle that has one 90° angle. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called legs. Copyright © 2010 Pearson Education, Inc. All rights reserved.

136. Slide 8.8- 136 Copyright © 2010 Pearson Education, Inc. All rights reserved.

137. Slide 8.8- 137 Copyright © 2010 Pearson Education, Inc. All rights reserved.

138. Parallel Example 2 Find the Unknown Length in Right Triangles Slide 8.8- 138 Find the unknown length in each right triangle. Round answers to the nearest tenth if necessary. a. The unknown length is the side opposite the right angle. Use the formula for finding the hypotenuse. 8 cm 15 cm The length is 17 cm. long Copyright © 2010 Pearson Education, Inc. All rights reserved.

139. Parallel Example 2 continued Find the Unknown Length in Right Triangles Slide 8.8- 139 Find the unknown length in each right triangle. Round answers to the nearest tenth if necessary. b. Use the formula for finding the leg. 15 ft 40 ft The length is approximately 37.1 ft long. Copyright © 2010 Pearson Education, Inc. All rights reserved.

140. Parallel Example 3 Using the Pythagorean Theorem Slide 8.8- 140 An electrical pole is shown below. Find the length of the guy wire. Round your answer to the nearest tenth of a foot if necessary. The length of the guy wire is approximately 69.5 ft. 35ft 60 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.