2. Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

3. Splash Screen

4. Contents Lesson 7-1Geometric Mean Lesson 7-2The Pythagorean Theorem and Its Converse Lesson 7-3Special Right Triangles Lesson 7-4Trigonometry Lesson 7-5Angles of Elevation and Depression Lesson 7-6The Law of Sines Lesson 7-7The Law of Cosines

5. Lesson 1 Contents Example 1Geometric Mean Example 2Altitude and Segments of the Hypotenuse Example 3Altitude and Length of the Hypotenuse Example 4Hypotenuse and Segment of Hypotenuse

6. Example 1-1a Find the geometric mean between 2 and 50. Definition of geometric mean Let x represent the geometric mean. Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is 10.

7. Example 1-1b Find the geometric mean between 25 and 7. Definition of geometric mean Let x represent the geometric mean. Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is about 13.2. Use a calculator.

8. Example 1-1c a. Find the geometric mean between 3 and 12. b. Find the geometric mean between 4 and 20. Answer: 6 Answer: 8.9

9. Example 1-2a

10. Example 1-2b Cross products Take the positive square root of each side. Use a calculator. Answer: CD is about 12.7.

11. Example 1-2c Answer: about 8.5

12. Example 1-3a KITES Ms. Alspach is constructing a kite for her son. She has to arrange perpendicularly two support rods, the shorter of which is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right triangles with the kite fabric, what is the length of the long rod?

13. Example 1-3b Draw a diagram of one of the right triangles formed. Let be the altitude drawn from the right angle of

14. Example 1-3c Cross products Divide each side by 7.25. Answer: The length of the long rod is 7.25 + 25.2, or about 32.4 inches long.

15. AIRPLANES A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, intersects at point E. If AE is 163 feet, what is the length of the aircraft? Answer: about 231.3 ft Example 1-3d

16. Example 1-4a Find c and d in

17. Example 1-4b is the altitude of right triangle JKL. Use Theorem 7.2 to write a proportion. Cross products Divide each side by 5.

18. Example 1-4c is the leg of right triangle JKL. Use the Theorem 7.3 to write a proportion. Answer: Cross products Take the square root. Simplify. Use a calculator.

19. Example 1-4d Find e and f. Answer: f

20. End of Lesson 1

21. Lesson 2 Contents Example 1Find the Length of the Hypotenuse Example 2Find the Length of a Leg Example 3Verify a Triangle is a Right Triangle Example 4Pythagorean Triples

22. Example 2-1a LONGITUDE AND LATITUDE Carson City, Nevada, is located at about 120 degrees longitude and 39 degrees latitude. NASA Ames is located about 122 degrees longitude and 37 degrees latitude. Use the lines of longitude and latitude to find the degree distance to the nearest tenth degree if you were to travel directly from NASA Ames to Carson City.

23. Example 2-1b Use the Pythagorean Theorem to find the distance in degrees from NASA Ames to Carson City, represented by c. The change in latitude is or 2 degrees latitude. Let this distance be b. The change in longitude between NASA Ames and Carson City is or 2 degrees. Let this distance be a.

24. Example 2-1c Pythagorean Theorem Simplify. Add. Take the square root of each side. Use a calculator. Answer: The degree distance between NASA Ames and Carson City is about 2.8 degrees.

25. Example 2-1d LONGITUDE AND LATITUDE Carson City, Nevada, is located at about 120 degrees longitude and 39 degrees latitude. NASA Dryden is located about 117 degrees longitude and 34 degrees latitude. Use the lines of longitude and latitude to find the degree distance to the nearest tenth degree if you were to travel directly from NASA Dryden to Carson City. Answer: about 5.8 degrees

26. Example 2-2a Find d.

27. Example 2-2b Pythagorean Theorem Simplify. Subtract 9 from each side. Take the square root of each side. Use a calculator. Answer:

28. Example 2-2c Find x. Answer:

29. Example 2-3a COORDINATE GEOMETRY Verify that is a right triangle.

30. Example 2-3b Use the Distance Formula to determine the lengths of the sides. Subtract. Simplify. Subtract. Simplify.

31. Example 2-3c Subtract. Simplify. By the converse of the Pythagorean Theorem, if the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

32. Example 2-3d Converse of the Pythagorean Theorem Simplify. Add. Answer: Since the sum of the squares of two sides equals the square of the longest side, is a right triangle.

33. Example 2-3e COORDINATE GEOMETRY Verify that is a right triangle. Answer: is a right triangle because

34. Example 2-4a Determine whether 9, 12, and 5 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Since the measure of the longest side is 15, 15 must be c. Let a and b be 9 and 12. Pythagorean Theorem Simplify. Add.

35. Example 2-4b Answer: These segments form the sides of a right triangle since they satisfy the Pythagorean Theorem. The measures are whole numbers and form a Pythagorean triple.

36. Example 2-4c Determine whether 21, 42, and 54 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Pythagorean Theorem Simplify. Add. Answer: Since, segments with these measures cannot form a right triangle. Therefore, they do not form a Pythagorean triple.

37. Example 2-4d Pythagorean Theorem Simplify. Add. Determine whether 4, and 8 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Answer: Since 64 = 64, segments with these measures form a right triangle. However, is not a whole number. Therefore, they do not form a Pythagorean triple.

38. Example 2-4e Answer: The segments form the sides of a right triangle and the measures form a Pythagorean triple. Answer: The segments do not form the sides of a right triangle, and the measures do not form a Pythagorean triple. Answer: The segments form the sides of a right triangle, but the measures do not form a Pythagorean triple. Determine whether each set of measures are the sides of a right triangle. Then state whether they form a Pythagorean triple. a. 6, 8, 10 b. 5, 8, 9 c.

39. End of Lesson 2

40. Lesson 3 Contents Example 1Find the Measure of the Hypotenuse Example 2Find the Measure of the Legs Example 330°–60°–90° Triangles Example 4Special Triangles in a Coordinate Plan

41. Example 3-1a WALLPAPER TILING The wallpaper in the figure can be divided into four equal square quadrants so that each square contains 8 triangles. What is the area of one of the squares if the hypotenuse of each 40°-45°-90° triangle measures millimeters?

42. Example 3-1b The length of the hypotenuse of one 40°-45°-90° triangle is millimeters. The length of the hypotenuse is times as long as a leg. So, the length of each leg is 7 millimeters. The area of one of these triangles is or 24.5 millimeters. Answer: Since there are 8 of these triangles in one square quadrant, the area of one of these squares is 8(24.5) or 196 mm 2.

43. Example 3-1c WALLPAPER TILING If each 40°-45°-90° triangle in the figure has a hypotenuse of millimeters, what is the perimeter of the entire square? Answer: 80 mm

44. Example 3-2a Find a. The length of the hypotenuse of a 40°-45°-90° triangle is times as long as a leg of the triangle.

45. Example 3-2b Multiply. Divide. Rationalize the denominator. Divide each side by Answer:

46. Example 3-2c Find b. Answer:

47. Example 3-3a Find QR.

48. Example 3-3b is the longer leg, is the shorter leg, and is the hypotenuse. Multiply each side by 2. Answer:

49. Example 3-3c Find BC. Answer: BC = 8 in.

50. Example 3-4a COORDINATE GEOMETRY is a 30°-60°-90° triangle with right angle X and as the longer leg. Graph points X(-2, 7) and Y(-7, 7), and locate point W in Quadrant III.

51. Example 3-4b Graph X and Y. lies on a horizontal gridline of the coordinate plane. Since will be perpendicular to it lies on a vertical gridline. Find the length of

52. Example 3-4b is the shorter leg. is the longer leg. So, Use XY to find WX. Point W has the same x-coordinate as X. W is located units below X. Answer: The coordinates of W are or about

53. Example 3-4d COORDINATE GEOMETRY is at 30°-60°-90° triangle with right angle R and as the longer leg. Graph points T(3, 3) and R(3, 6) and locate point S in Quadrant III. Answer: The coordinates of S are or about

54. End of Lesson 3

55. Lesson 4 Contents Example 1Find Sine, Cosine, and Tangent Ratios Example 2Evaluate Expressions Example 3Use Trigonometric Ratios to Find a Length Example 4Use Trigonometric Ratios to Find an Angle Measure

56. Example 4-1a Find sin L, cos L, tan L, sin N, cos N, and tan N. Express each ratio as a fraction and as a decimal.

57. Example 4-1b

58. Example 4-1c

59. Example 4-1d

60. Example 4-1e Answer:

61. Example 4-1f Find sin A, cos A, tan A, sin B, cos B, and tan B. Express each ratio as a fraction and as a decimal.

62. Example 4-1g Answer:

63. Example 4-2a Use a calculator to find tan to the nearest ten thousandth. KEYSTROKES: 56 1.482560969 TANENTER Answer:

64. Example 4-2b KEYSTROKES: 90 0 COSENTER Answer: Use a calculator to find tan to the nearest ten thousandth.

65. a. Use a calculator to find sin 48° to the nearest ten thousandth. b. Use a calculator to find cos 85° to the nearest ten thousandth. Example 4-2c Answer:

66. Example 4-3a EXERCISING A fitness trainer sets the incline on a treadmill to The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches.

67. Example 4-3b KEYSTROKES: 60 7 7.312160604 SINENTER Multiply each side by 60. Use a calculator to find y. Answer: The treadmill is about 7.3 inches high.

68. Example 4-3c CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about how high does the ramp rise off the ground to the nearest inch? Answer: about 15 in.

69. Example 4-4a COORDINATE GEOMETRY Find m  X in right  XYZ for X(–2, 8), Y(–6, 4), and Z(–3, 1).

70. Example 4-4b PlanUse the Distance Formula to find the measure of each side. Then use one of the trigonometric ratios to write an equation. Use the inverse to find Explore You know the coordinates of the vertices of a right triangle and that is the right angle. You need to find the measures of one of the angles. Solve or

71. Example 4-4c or

72. Example 4-4d Use the cosine ratio. Simplify. Solve for x.

73. Example 4-4e Use a calculator to find KEYSTROKES: 4 5 2ND ENTER ) Examine Use the sine ratio to check the answer. Simplify.

74. Example 4-4f KEYSTROKES: 3 5 2ND ENTER ) Answer: The measure of is about 36.9.

75. Example 4-4g COORDINATE GEOMETRY Answer: about 56.3

76. End of Lesson 4

77. Lesson 5 Contents Example 1Angle of Elevation Example 2Angle of Depression Example 3Indirect Measurement

78. Example 5-1a CIRCUS ACTS At the circus, a person in the audience watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is Make a drawing.

79. Example 5-1b Since QR is 25 feet and RS is 5 feet 6 inches or 5.5 feet, QS is 30.5 feet. Let x represent PQ. Multiply both sides by x. Divide both sides by tan Simplify.

80. Example 5-1c Answer: The audience member is about 60 feet from the base of the platform.

81. Example 5-1d DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree? Answer: about

82. Example 5-2a SHORT-RESPONSE TEST ITEM A wheelchair ramp is 3 meters long and inclines at Find the height of the ramp to the nearest tenth centimeter. Read the Test Item The angle of depression between the ramp and the horizontal is Use trigonometry to find the height of the ramp. Solve the Test Item Method 1 The ground and the horizontal level with the platform to which the ramp extends are parallel. Therefore, since they are alternate interior angles.

83. Example 5-2b Answer: The height of the ramp is about 0.314 meters, Mulitply each side by 3. Simplify. YWYW

84. Example 5-2c Method 2 The horizontal line from the top of the platform to which the wheelchair ramp extends and the segment from the ground to the platform are perpendicular. So, and are complementary angles. Therefore, YWYW

85. Example 5-2d Answer: The height of the ramp is about 0.314 meters, Multiply each side by 3. Simplify.

86. Example 5-2e SHORT-RESPONSE TEST ITEM A roller coaster car is at one of its highest points. It drops at a angle for 320 feet. How high was the roller coaster car to the nearest foot before it began its fall? Answer: The roller coaster car was about 285 feet above the ground.

87. Example 5-3a Vernon is on the top deck of a cruise ship and observes two dolphins following each other directly away from the ship in a straight line. Vernon’s position is 154 meters above sea level, and the angles of depression to the two dolphins are Find the distance between the two dolphins to the nearest meter.

88. Example 5-3b are right triangles. The distance between the dolphins is JK or Use the right triangles to find these two lengths. Because are horizontal lines, they are parallel. Thus, and because they are alternate interior angles. This means that

89. Example 5-3c Multiply each side by JL. Divide each side by tan Use a calculator.

90. Example 5-3d Multiply each side by KL. Use a calculator. Divide each side by tan Answer: The distance between the dolphins is, or about 8 meters.

91. Example 5-3e Madison looks out her second-floor window, which is 15 feet above the ground. She observes two parked cars. One car is parked along the curb directly in front of her window, and the other car is parked directly across the street from the first car. The angles of depression of Madison’s line of sight to the cars are Find the distance between the two cars. Answer: about 24 feet

92. End of Lesson 5

93. Lesson 6 Contents Example 1Use the Law of Sines Example 2Solve Triangles Example 3Indirect Measurement

94. Example 6-1a Find p. Round to the nearest tenth.

95. Example 6-1b Law of Sines Use a calculator. Divide each side by tan Cross products Answer:

96. Example 6-1c Law of Sines Cross products Divide each side by 7. to the nearest degree in,

97. Solve for L. Example 6-1d Use a calculator. Answer:

98. a. Find c. b. Find m  T to the nearest degree in  RST if r = 12, t = 7, and m  T = 76. Example 6-1e Answer:

99. Example 6-2a We know the measures of two angles of the triangle. Use the Angle Sum Theorem to find. Round angle measures to the nearest degree and side measures to the nearest tenth.

100. Example 6-2b Angle Sum Theorem Subtract 120 from each side. Add. Since we know and f, use proportions involving

101. Example 6-2c To find d: Law of Sines Cross products Substitute. Use a calculator. Divide each side by sin 8°.

102. Example 6-2d To find e: Law of Sines Cross products Substitute. Use a calculator. Divide each side by sin 8°. Answer:

103. Example 6-2e We know the measure of two sides and an angle opposite one of the sides. Law of Sines Cross products Round angle measures to the nearest degree and side measures to the nearest tenth.

104. Example 6-2f Solve for L. Angle Sum Theorem Use a calculator. Add. Substitute. Divide each side by 16. Subtract 116 from each side.

105. Example 6-2g Cross products Use a calculator. Law of Sines Divide each side by sin Answer:

106. a. Solve Round angle measures to the nearest degree and side measures to the nearest tenth. b. Round angle measures to the nearest degree and side measures to the nearest tenth. Example 6-2h Answer:

107. Example 6-3a A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is Draw a diagram Draw Then find the

108. Example 6-3b Since you know the measures of two angles of the triangle, and the length of a side opposite one of the angles you can use the Law of Sines to find the length of the shadow.

109. Example 6-3c Cross products Use a calculator. Law of Sines Answer: The length of the shadow is about 75.9 feet. Divide each side by sin

110. Example 6-3d A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water? Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.

111. End of Lesson 6

112. Lesson 7 Contents Example 1Two Sides and the Included Angle Example 2Three Sides Example 3Select a Strategy Example 4Use Law of Cosines to Solve Problems

113. Example 7-1a Use the Law of Cosines since the measures of two sides and the included angle are known.

114. Example 7-1b Simplify. Take the square root of each side. Law of Cosines Use a calculator. Answer:

115. Example 7-1c Answer:

116. Example 7-2a Law of Cosines Simplify.

117. Example 7-2b Solve for L. Use a calculator. Subtract 754 from each side. Divide each side by –270. Answer:

118. Example 7-2c Answer:

119. Example 7-3a Determine whether the Law of Sines or the Law of Cosines should be used first to solve Then solve Round angle measures to the nearest degree and side measures to the nearest tenth. Since we know the measures of two sides and the included angle, use the Law of Cosines.

120. Example 7-3b Take the square root of each side. Use a calculator. Law of Cosines Next, we can find If we decide to find we can use either the Law of Sines or the Law of Cosines to find this value. In this case, we will use the Law of Sines.

121. Example 7-3c Cross products Divide each side by 46.9. Law of Sines Take the inverse of each side. Use a calculator.

122. Example 7-3d Use the Angle Sum Theorem to find Angle Sum Theorem Subtract 168 from each side. Answer:

123. Example 7-3e Determine whether the Law of Sines or the Law of Cosines should be used first to solve Then solve Round angle measures to the nearest degree and side measures to the nearest tenth. Answer:

124. Since is an isosceles triangle, Example 7-4a AIRCRAFT From the diagram of the plane shown, determine the approximate exterior perimeter of each wing. Round to the nearest tenth meter.

125. Example 7-4b Cross products Law of Sines Simplify. Divide each side by sin. Use the Law of Sines to find KJ.

126. Example 7-4c Use the Law of Sines to find. Cross products Law of Sines Solve for H. Divide each side by 9. Use a calculator.

127. Example 7-4d Use the Angle Sum Theorem to find Subtract 95 from each side. Angle Sum Theorem

128. Example 7-4e Use the Law of Sines to find HK. Cross products Law of Sines Use a calculator. Divide each side by sin

129. Example 7-4f Answer: The perimeter is about or about 67.1 meters. The perimeter of the wing is equal to

130. Example 7-4g The rear side window of a station wagon has the shape shown in the figure. Find the perimeter of the window if the length of DB is 31 inches. Answer: about 93.5 in.

131. End of Lesson 7

132. Online Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Glencoe Geometry 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.geometryonline.com/extra_examples.

133. Transparency 1 Click the mouse button or press the Space Bar to display the answers.

134. Transparency 1a

135. Transparency 2 Click the mouse button or press the Space Bar to display the answers.

136. Transparency 2a

137. Transparency 3 Click the mouse button or press the Space Bar to display the answers.

138. Transparency 3a

139. Transparency 4 Click the mouse button or press the Space Bar to display the answers.

140. Transparency 4a

141. Transparency 5 Click the mouse button or press the Space Bar to display the answers.

142. Transparency 5a

143. Transparency 6 Click the mouse button or press the Space Bar to display the answers.

144. Transparency 6a

145. Transparency 7 Click the mouse button or press the Space Bar to display the answers.

146. Transparency 7a

147. End of Custom Show End of Custom Shows WARNING! Do Not Remove This slide is intentionally blank and is set to auto-advance to end custom shows and return to the main presentation.

148. End of Slide Show