Square Roots and Irrational Numbers PREALGEBRA LESSON 11
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 (For help, go to Lesson 4 -2. ) Write the numbers in each list without exponents. 1. 12, 22, 32, . . . , 122 2. 102, 202, 302, . . . , 1202 11 -1
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 Solutions 1. 12, 22, 32, . . . , 122 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 2. 102, 202, 302, . . . , 1202 100, 400, 900, 1, 600, 2, 500, 3, 600, 4, 900, 6, 400, 8, 100, 10, 000, 12, 100, 14, 400 11 -1
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 Simplify each square root. a. 144 = 12 b. – – 81 81 = – 9 11 -1
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 You can use the formula d = 1. 5 h to estimate the distance d, in miles, to a horizon line when your eyes are h feet above the ground. Estimate the distance to the horizon seen by a lifeguard whose eyes are 20 feet above the ground. d= 1. 5 h Use the formula. d= 1. 5(20) Replace h with 20. d= 30 Multiply. 25 < 25 = 5 30 < 36 Find perfect squares close to 30. Find the square root of the closest perfect square. The lifeguard can see about 5 miles to the horizon. 11 -1
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 Identify each number as rational or irrational. Explain. a. 49 rational, because 49 is a perfect square b. 0. 16 rational, because it is a terminating decimal c. 3 irrational, because 3 is not a perfect square d. 0. 3333. . . rational, because it is a repeating decimal e. – 15 irrational, because 15 is not a perfect square f. 12. 69 rational, because it is a terminating decimal g. 0. 1234567. . . irrational, because it neither terminates nor repeats 11 -1
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 Pages 582– 583 Exercises 1. 2 9. 3 2. – 6 10. 4 3. 1 11. – 6 4. 5 12. – 7 5. – 7 13. 27 mi 6. 9 16. rational; because 16 is a perfect square 17. rational; because it is a repeating decimal 18. irrational; because 5 is not a perfect square 14. irrational; it neither terminates nor repeats 19. rational; it can be expressed as a ratio of two numbers 15. irrational; because 87 is not a perfect square 20. rational; because 144 is a perfect square 7. – 3 8. – 13 11 -1
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 21. irrational; it neither 29. terminates nor repeats 30. 22. 14 31. 2 23. 3 32. 5 24. 7 33. 25. 3 4 34. 26. 3 27. 1 – 9 36. irrational; it neither terminates nor repeats 8 37. x – 10 4 10 rational; can a be expressed as b 35. rational; repeating decimal 28. 6 11 -1 38. Answers may vary. Sample: Think of the perfect squares closest to 30, one less than 30 and one greater than 30. Take the square root of the one closest to 30.
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 39. a. In each repetition of the pattern there is one more zero than in the previous one. b. Answers may vary. Sample: 9. 010010001. . . , 9. 121121112. . . , 9. 565665666. . . 43. 2, – 2 52. – 11 44. 9 cm 53. 25 45. 2 in. 54. A 46. – 2 55. H 47. 3 56. [2] He can use the equation d = 1. 5 h where d is 6. Since 36 is 6, he can conclude that 1. 5 h = 36; 24 ft. 48. – 3 40. 3, – 3 49. 7 41. 5, – 5 50. – 2 42. 10, – 10 51. 5 11 -1 [1] minor error OR answer only
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 57. about 113 cm 3 64. 1, 2, 5, 10, 25, 50 58. about 167 cm 3 65. 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 59. Yes; 17 is 0. 68, or 68%. 25 Since 68% 65%, > Shannon passed the – test. 60. 1, 2, 3, 6, 9, 18 61. 1, 2, 11, 22 62. 1, 3, 11, 33 63. 1, 3, 5, 9, 15, 45 11 -1
Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11 -1 Simplify each square root or estimate to the nearest integer. 1. – 100 2. – 10 57 8 Identify each number as rational or irrational. 3. 48 irrational 4. 0. 0125 rational 5. The formula d = 1. 5 h , where h equals the height, in feet, of the viewer’s eyes, estimates the distance d, in miles, to the horizon from the viewer. Find the distance to the horizon for a person whose eyes are 6 ft above the ground. 3 mi 11 -1
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 (For help, go to Lesson 4 -2. ) Simplify. 1. 42 + 62 2. 52 + 82 3. 72 + 92 4. 92 + 32 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 Solutions 1. 42 + 62 16 + 36 = 52 2. 52 + 8 2 25 + 64 = 89 3. 72 + 92 49 + 81 = 130 4. 92 + 3 2 81 + 9 = 90 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 Find c, the length of the hypotenuse. c 2 = a 2 + b 2 Use the Pythagorean Theorem. c 2 = 282 + 212 Replace a with 28, and b with 21. c 2 = 1, 225 Simplify. c= 1, 225 = 35 Find the positive square root of each side. The length of the hypotenuse is 35 cm. 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 Find the value of x in the triangle. Round to the nearest tenth. a 2 + b 2 = c 2 Use the Pythagorean Theorem. 72 + x 2 = 142 Replace a with 7, b with x, and c with 14. 49 + x 2 = 196 Simplify. x 2 = 147 x= 147 Subtract 49 from each side. Find the positive square root of each side. 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 (continued) Then use one of the two methods below to approximate 147. Method 1 Use a calculator. A calculator value for x 12. 1 147 is 12. 124356. Round to the nearest tenth. Method 2 Use a table of square roots. Use the table on page 778. Find the number closest to 147 in the N 2 column. Then find the corresponding value in the N column. It is a little over 12. x 12. 1 Estimate the nearest tenth. The value of x is about 12. 1 in. 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 The carpentry terms span, rise, and rafter length are illustrated in the diagram. A carpenter wants to make a roof that has a span of 20 ft and a rise of 10 ft. What should the rafter length be? c 2 = a 2 + b 2 Use the Pythagorean Theorem. c 2 = 102 + 102 Replace a with 10 (half the span), and b with 10. c 2 = 100 + 100 Square 10. c 2 = 200 Add. c= c 200 14. 1 Find the positive square root. Round to the nearest tenth. The rafter length should be about 14. 1 ft. 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 Is a triangle with sides 10 cm, 24 cm, and 26 cm a right triangle? a 2 + b 2 = c 2 Write the equation to check. 102 + 242 262 Replace a and b with the shorter lengths and c with the longest length. 100 + 576 676 Simplify. 676 = 676 The triangle is a right triangle. 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 Pages 587– 589 Exercises 1. 13 cm 9. about 15. 2 ft; about 30. 4 ft 2. 11. 3 m 17. yes; 52 + 122 = 132 18. 4. 5 in 10. about 10. 6 ft 3. 8 in. 19. 3. 1 m 11. 6. 7 km 4. 5. 2 mm 20. 4 ft 12. no; 42 + 62 = 72 5. 20 in. 21. 2. 1 km 13. no; 42 + 52 = 62 6. 35 ft 14. yes; 72 + 242 = 252 22. yes; ( 5 )2 + ( = ( 12 )2 7 )2 15. yes; 62 + 72 = 85 23. yes; 32 + 42 = 52 16. no; 82 + 102 =/ 122 24. yes; 72 + 242 = 252 7. 14 ft 8. 44 ft 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 31. about 131. 9 in. 35. 50 26. yes; 52 + 122 = 132 36. 34 27. yes, 62 + 82 = 102; yes, 142 + 482 = 502; yes, 102 + 242 = 262 37. 52 25. no; 102 + 242 =/ 252 28. 6. 4 ft 29. yes; ( 2 )2 + ( = ( 5 )2 38. D 32. Yes; (3 p)2 + (4 p)2 = (5 p)2 for 39. F any value of p. 3 )2 30. no; 0. 562 + 0. 542 =/ 12 33. a. 10 in. b. about 10. 8 in. c. about 15. 8 in. 34. 32 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 40. [4] a. Yes; since 32 + 42 = 52, by the Converse of the Pythagorean Theorem, the sides can form a right triangle. b. You place 3 segments of rope along the wall to one side of the corner. Place 4 segments along the other wall. If you can form a triangle with the remaining rope, the triangle is a right triangle with the right angle in the corner of the room. c. The 12 -segment rope can be used to check if corners are right angles because a triangle with side lengths 3, 4, and 5 (totaling 12) is a right triangle. [3] one minor error in explanation [2] two minor errors in explanation 41. rational; because 36 is a perfect square 42. rational; because it is a repeating decimal 43. irrational; because 12 is not a perfect square 44. rational; because it is a terminating decimal 45. rational; because it is a repeating decimal 46. b 5 c 5 [1] one part only with no explanation 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 47. 16 x 8 48. – 27 b 3 49. a 20 b 8 9 m 2 50. 25 51. 2. 1756 106 km 2 11 -2
The Pythagorean Theorem PRE-ALGEBRA LESSON 11 -2 Find the missing length. Round to the nearest tenth. 1. a = 7, b = 8, c = 10. 6 2. a = 9, c = 17, b = 14. 4 3. Is a triangle with sides 6. 9 ft, 9. 2 ft, and 11. 5 ft a right triangle? Explain. yes; 6. 92 + 9. 22 = 11. 52 4. What is the rise of a roof if the span is 30 ft and the rafter length is 16 ft? Refer to the diagram on page 586. about 5. 6 ft 11 -2
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 (For help, go to Lesson 1 -10. ) Write the coordinates of each point. 1. A 2. D 3. G 11 -3 4. J
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 Solutions 1. A (– 3, 4) 2. D (0, 3) 11 -3 3. G (– 4, – 2) 4. J (3, – 1)
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 Find the distance between T(3, – 2) and V(8, 3). d= (x 2 – x 1)2 + (y 2 – y 1)2 Use the Distance Formula. d= (8 – 3)2 + (3 – (– 2 ))2 Replace (x 2, y 2) with (8, 3) and (x 1, y 1) with (3, – 2). d= 52 + 5 2 Simplify. d= 50 Find the exact distance. d 7. 1 Round to the nearest tenth. The distance between T and V is about 7. 1 units. 11 -3
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 Find the perimeter of WXYZ. The points are W (– 3, 2), X (– 2, – 1), Y (4, 0), Z (1, 5). Use the Distance Formula to find the side lengths. WX = = XY = = (– 2 – (– 3))2 + (– 1 – 2)2 Replace (x 2, y 2) with (– 2, – 1) and (x 1, y 1) with (– 3, 2). 1+9= Simplify. 10 (4 – (– 2))2 + (0 – (– 1)2 Replace (x 2, y 2) with (4, 0) and (x 1, y 1) with (– 2, – 1). 36 + 1 = Simplify. 37 11 -3
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 (continued) YZ = = ZW = = (1 – 4)2 + (5 – 0)2 Replace (x 2, y 2) with (1, 5) and (x 1, y 1) with (4, 0). 9 + 25 = Simplify. 34 (– 3 – 1)2 + (2 – 5)2 Replace (x 2, y 2) with (– 3, 2) and (x 1, y 1) with (1, 5). 16 + 9 = Simplify. 25 = 5 11 -3
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 (continued) perimeter = 10 + 37 + 34 + 5 The perimeter is about 20. 1 units. 11 -3 20. 1
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 Find the midpoint of TV. x 1 + x 2 y 1 + y 2 , 2 2 Use the Midpoint Formula. = 4 + 9 – 3 + 2 , 2 2 Replace (x 1, y 1) with (4, – 3) and (x 2, y 2) with (9, 2). = 13 – 1 , 2 2 Simplify the numerators. 1 2 = 6 , – 1 2 Write the fractions in simplest form. 1 2 The coordinates of the midpoint of TV are 6 , – 11 -3 1. 2
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 Pages 595– 596 Exercises 1. 5 2. 13 15. No; addition is commutative. 9. (1. 5, 2) 10. (1, 0) 3. 23. 3 11. 30; (0, 0) 4. 19. 2 12. 27. 7; (10. 5, 10) 5. 17. 7 13. The student used subtraction in the numerators instead of addition. 6. 24. 0 7. 19. 3 14. (12, 9) 8. 20 11 -3 16. No; the square of a number (x 1 – x 2) and the square of its opposite (x 2 – x 1) are equal.
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 17. a. (– 0. 5, 3) b. AM = (– 3 – (– 0. 5))2 + (5 – 3)2 = 10. 25 MB = (2 – (– 0. 5))2 + (1 – 3)2 = 10. 25 26. 18. Scalene; check students’ work 19. Each coordinate of the midpoint is the average of the corresponding coordinates of the endpoints. 20. 18 27. 9 21. 17. 0 22. 14. 9 28. 12 23. yes; 82 + 152 = 172 29. 12. 5 24. no; 52 + 52 = 82 25. yes; 122 + 162 = 30. 1. 4 202 11 -3
Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11 -3 Find the length (to the nearest tenth) and midpoint of each segment with the given endpoints. 1. A(– 2, – 5) and B(– 3, 4) 2. D(– 4, 6) and E(7, – 2) 9. 1; (– 2 1, – 1 ) 2 13. 6; (11 , 2) 2 3. Find the perimeter of and C(3, 0). 2 ABC, with coordinates A(– 3, 0), B(0, 4), 16 11 -3
Write a Proportion PRE-ALGEBRA LESSON 11 -4 (For help, go to Lesson 6 -2. ) Solve each proportion. 1. a 1 = 12 3 2. 5 = 25 h 3. 1 8 = 4 x 4. 11 -4 20 2 c = 7 35
Write a Proportion PRE-ALGEBRA LESSON 11 -4 Solutions 1. a 1 = 12 3 2. 1 8 = 4 x 4. 3 • a = 1 • 12 3 a = 12 a=4 3. 20 h = 25 5 25 • h = 5 • 20 25 h = 100 h=4 1 • x=4 • 8 x = 32 2 c = 7 35 7 • c = 2 • 35 7 c = 70 c = 10 11 -4
Write a Proportion PRE-ALGEBRA LESSON 11 -4 At a given time of day, a building of unknown height casts a shadow that is 24 feet long. At the same time of day, a post that is 8 feet tall casts a shadow that is 4 feet long. What is the height x of the building? Since the triangles are similar, and you know three lengths, writing and solving a proportion is a good strategy to use. It is helpful to draw the triangles as separate figures. 11 -4
Write a Proportion PRE-ALGEBRA LESSON 11 -4 (continued) Write a proportion using the legs of the similar right triangles. 8 4 = x 24 Write a proportion. 4 x = 24(8) Write cross products. 4 x = 192 Simplify. x = 48 Divide each side by 4. The height of the building is 48 ft. 11 -4
Write a Proportion PRE-ALGEBRA LESSON 11 -4 Pages 600– 601 Exercises 13 20 = 21 ; 12. 3 m x + 20 60 x 2. = 25 ; 36 yd 15 1. 9. – 3 x 64 = x + 1. 5 72 15. [2] 10. 1, 312 ft 3. 1. 2 mi 11. 6. 3 ft 2 4. 10. 5 yd 12. 19. 5 ft 5. 28 ft 13. B 6. 13 students 14. H 72 • x = 64(x + 1. 5) 72 x = 64 x + 96 72 x – 64 x = 64 x – 64 x + 96 8 x = 96 8 x 96 = 8 8 x = 12 [1] minor error OR answer only 7. $20 16. (3, 5) 8. 7: 43 A. M. 17. (0. 5, 4) 11 -4
Write a Proportion PRE-ALGEBRA LESSON 11 -4 18. 19. 20. Keith $80, Lucy $40 11 -4
Write a Proportion PRE-ALGEBRA LESSON 11 -4 Write a proportion and solve. 1. On the blueprints for a rectangular floor, the width of the floor is 6 in. The diagonal distance across the floor is 10 in. If the width of the actual floor is 32 ft, what is the actual diagonal distance across the floor? about 53 ft 2. A right triangle with side lengths 3 cm, 4 cm, and 5 cm is similar to a right triangle with a 20 -cm hypotenuse. Find the perimeter of the larger triangle. 48 cm 3. A 6 -ft-tall man standing near a geyser has a shadow 4. 5 ft long. The geyser has a shadow 15 ft long. What is the height of the geyser? 20 ft 11 -4
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 (For help, go to Lesson 11 -2. ) Find the missing side of each right triangle. 1. legs: 6 m and 8 m 2. leg: 9 m; hypotenuse: 15 m 3. legs: 27 m and 36 m 4. leg: 48 m; hypotenuse: 60 m 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 Solutions 1. c 2 = a 2 + b 2 c 2 = 62 + 82 c 2 = 100 c = 100 = 10 m 2. a 2 + b 2 = c 2 92 + b 2 = 152 81 + b 2 = 225 b 2 = 144 b = 144 = 12 m 3. c 2 = a 2 + b 2 c 2 = 272 + 362 c 2 = 2025 c = 2025 = 45 m 4. 11 -5 a 2 + b 2 = c 2 482 + b 2 = 602 2304 + b 2 = 3600 b 2 = 1296 b = 1296 = 36 m
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 Find the length of the hypotenuse in the triangle. hypotenuse = leg • 2 Use the 45°-90° relationship. y = 10 • 2 The length of the leg is 10. 14. 1 Use a calculator. The length of the hypotenuse is about 14. 1 cm. 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 Patrice folds square napkins diagonally to put on a table. The side length of each napkin is 20 in. How long is the diagonal? hypotenuse = leg • 2 Use the 45°-90° relationship. y = 20 • 2 The length of the leg is 20. 28. 3 Use a calculator. The diagonal length is about 28. 3 in. 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 Find the missing lengths in the triangle. hypotenuse = 2 • shorter leg 14 = 2 • b The length of the hypotenuse is 14. 14 2 b = 2 2 Divide each side by 2. 7=b longer leg = shorter leg • a=7 • 3 a 12. 1 Simplify. 3 The length of the shorter leg is 7. Use a calculator. The length of the shorter leg is 7 ft. The length of the longer leg is about 12. 1 ft. 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 Pages 605– 606 Exercises 1. 4. 2 yd 9. x = 5 yd, y 8. 7 yd 2. 12. 7 cm 10. x = 8 ft, y = 16 ft 3. 7. 6 in 11. x 4. 24. 0 ft 12. Answers may vary. Sample: The student knows that the length of the hypotenuse of an isosceles right triangle with leg length 2 is 2 2. Because of this, the student might assume that a right triangle with hypotenuse length 2 2 must be isosceles. However, a triangle with legs 2 and 6 , for example, has hypotenuse length 2 2. 5. 29. 7 m 6. about 56. 6 ft. 7. 17 in. 8. x = 6 m, y 18. 4 cm, y = 13 cm 5. 2 m 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 13. Answers may vary. Sample: The hypotenuse is twice the shorter leg. Since the shorter leg is 10 ft, the hypotenuse is 20 ft. The longer leg is 3 times the shorter leg, which is 10 3 , or about 17. 3 ft. 14. about 28. 3 ft 19. a. 4 in. b. 2 3 in. , or about 3. 5 in. c. 4 3 in. 2, or about 6. 9 in. 2 d. 24 3 in. 2, or about 41. 6 in. 2 15. 9 20. a. Answers may vary. Sample: 16. 10 17. 12 18. No, the ratio of the shorter leg to the longer leg is 1 : 3. The ratio of the shorter leg to the hypotenuse is 1 : 2. 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 20. (continued) b. Answers may vary. Sample: If the diameter of the circle is 2 in. , one side of each triangle is 1 in. The height of one triangle is 0. 5 • 3 or about 0. 87 in. The area of one triangle is 1 • 0. 87, or about 0. 44 in. 2. The area of the 2 hexagon is 6 • 0. 44, or about 2. 6 in. 2. 24. [2] 45°-90°; this is a 45°-90° triangle because two sides are congruent and the hypotenuse is 2 times a leg. [1] minor error OR answer only 25. [2] 30°-60°-90°; this is a 30°-60°-90° triangle because the 22. H hypotenuse is twice 23. [2] Neither; in a 45°-90° triangle two sides are the shorter leg and the congruent. In a 30°-60°-90° triangle the longer leg is 3 times hypotenuse is twice the shorter leg. [1] minor error OR answer only 21. C 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 26. 33 m 27. about 25. 1 in. 28. about 18. 8 m 29. about 15. 7 ft 30. Yes; for each domain value there is only one range value. 31. Yes; for each domain value there is only one range value. 11 -5
Special Right Triangles PRE-ALGEBRA LESSON 11 -5 Find each missing length. 1. Find the length of the legs of a 45°-90° triangle with a hypotenuse of 4 2 cm. 4 cm 2. Find the length of the longer leg of a 30°-60°-90° triangle with a hypotenuse of 6 in. 3 3 in. 3. Kit folds a bandana diagonally before tying it around her head. The side length of the bandana is 16 in. About how long is the diagonal? about 22. 6 in. 11 -5
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11 -6 (For help, go to Lesson 6 -3. ) Solve each problem. 1. A 6 -ft man casts an 8 -ft shadow while a nearby flagpole casts a 20 -ft shadow. How tall is the flagpole? 2. When a 12 -ft tall building casts a 22 -ft shadow, how long is the shadow of a nearby 14 -ft tree? 11 -6
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11– 6 Solutions 1. 6 x = 8 20 2. 12 14 = 22 x 6 • 20 = 8 • x 22 • 14 = 12 • x 120 = 8 x 308 = 12 x 120 8 x = 8 8 308 12 x = 12 12 x = 25 2 ft 3 x = 15 ft 11 -6
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11 -6 Find the sine, cosine, and tangent of opposite 12 3 adjacent 16 4 12 3 sin A = hypotenuse = 20 = 5 cos A = hypotenuse = = 5 20 tan A = adjacent opposite = 16 = 4 11 -6 A.
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11 -6 Find the trigonometric ratios of 18° using a scientific calculator or the table on page 779. Round to four decimal places. sin 18° 0. 3090 Scientific calculator: Enter 18 and press the key labeled SIN, COS, or TAN. cos 18° 0. 9511 tan 18° Table: Find 18° in the first column. Look 0. 3249 across to find the appropriate ratio. 11 -6
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11 -6 The diagram shows a doorstop in the shape of a wedge. What is the length of the hypotenuse of the doorstop? You know the angle and the side opposite the angle. You want to find w, the length of the hypotenuse. opposite sin A = hypotenuse 10 sin 40° = w w(sin 40°) = 10 10 w = sin 40° w 15. 6 Use the sine ratio. Substitute 40° for the angle, 10 for the height, and w for the hypotenuse. Multiply each side by w. Divide each side by sin 40°. Use a calculator. The hypotenuse is about 15. 6 cm long. 11 -6
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11 -6 Pages 611– 612 Exercises 1. 4 7. 57. 2900 15. 19. 3 ft 2. 3 8. 0. 5000 16. 4. 5 ft 3. 5 9. 0. 9703 17. 1 4 3 4 4. 5 , 5, 3 9 3 or , 15 5 9 3 or 12 4 6. 15 or , 5 12 4 or 9 3 5. 10. 0. 9205 12 or 4 , 15 5 3 9 or , 5 15 18. 1 11. 0. 5317 2 12. 0. 0175 19. 1 2 13. 0. 1944 20. 14. 0. 9925 11 -6 3 2
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11 -6 21. Both are correct. The 25. C ratios include h, a known side length, and 26. F a known angle measure. 27. 5. 0° 22. 10, 000 ft 33. 5, 2, – 7 34. 21, 20, 17 35. – 9, – 6, 3 28. 4. 9 m 29. 4 2 m, or about 5. 7 m 30. 14 ft 23. The sine of an angle equals the cosine of its 31. 13 3 in. , complement. or about 22. 5 in. 24. a. 69 ft b. about 27° 32. RE EA 11 -6
Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11 -6 Solve 1. In ABC, AB = 5, AC = 12, and BC = 13. If find the sine, cosine, and tangent of B. A is a right angle, 12 5 12 , , 13 13 5 2. One angle of a right triangle is 35°, and the adjacent leg is 15. a. What is the length of the opposite leg? about 10. 5 b. What is the length of the hypotenuse? about 18. 3 3. Find the sine, cosine, and tangent of 72° using a calculator or a table. sin 72° 0. 9511; cos 72° 0. 3090; tan 72° 3. 0777 11 -6
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 (For help, go to Lesson 2 -3. ) Find each trigonometric ratio. 1. sin 45° 2. cos 32° 3. tan 18° 4. sin 68° 5. cos 88° 6. tan 84° 11 -7
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11– 7 Solutions 1. sin 45° 0. 7071 2. cos 32° 0. 8480 3. tan 18° 0. 3249 4. sin 68° 0. 9272 5. cos 88° 0. 0349 6. tan 84° 9. 5144 11 -7
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 Janine is flying a kite. She lets out 30 yd of string and anchors it to the ground. She determines that the angle of elevation of the kite is 52°. What is the height h of the kite from the ground? Draw a picture. sin A = opposite hypotenuse h sin 52° = 30 30(sin 52°) = h 24 h The kite is about 24 yd from the ground. 11 -7 Choose an appropriate trigonometric ratio. Substitute. Multiply each side by 30. Simplify.
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 Greg wants to find the height of a tree. From his position 30 ft from the base of the tree, he sees the top of the tree at an angle of elevation of 61°. Greg’s eyes are 6 ft from the ground. How tall is the tree, to the nearest foot? Draw a picture. opposite Choose an appropriate trigonometric ratio. h Substitute 61 for the angle measure and 30 for the adjacent side. tan A = adjacent tan 61° = 30 30(tan 61°) = h 54 + 6 = 60 The tree is about 60 ft tall. 11 -7 Multiply each side by 30. Use a calculator or a table. Add 6 to account for the height of Greg’s eyes from the ground.
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 An airplane is flying 1. 5 mi above the ground. If the pilot must begin a 3° descent to an airport runway at that altitude, how far is the airplane from the beginning of the runway (in ground distance)? Draw a picture (not to scale). tan 3° = 1. 5 d d • tan 3° = 1. 5 Choose an appropriate trigonometric ratio. Multiply each side by d. 11 -7
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 (continued) 1. 5 d • tan 3° = tan 3° 1. 5 Divide each side by tan 3°. d = tan 3° Simplify. d Use a calculator. 28. 6 The airplane is about 28. 6 mi from the airport. 11 -7
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 Pages 617– 618 Exercises 1. about 30 yd 8. angle of elevation = QRS, angle of depression = PQR 9. about 17. 4 yd 2. about 48 ft 3. 26. 7 m 4. 14. 5 ft 5. about 71. 6 mi 10. about 715 m 6. 0. 2 mi 11. Answers may vary. Sample: You need to use a ratio that involves the side length you are looking for and a known side length. 7. angle of elevation = ADC, angle of depression = BAD 12. about 502. 4 m 11 -7
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 27. 100 mm 2; 314. 0 mm 2 13. about 1, 293. 8 m 19. G 14. The angle of depression is between the line of sight and the horizontal. 20. 0. 5543 15. about 2, 932. 6 ft 23. 0. 9703 16. a. about 0. 64 km b. about 0. 43 km 24. 0. 7071 17. about 0. 51 km 18. C 21. 0. 9848 28. 20. 25 in. 2; 63. 6 in. 2 22. 0. 8290 29. 3 25. 64 in. 2; 201. 0 in. 2 26. 3. 61 cm 2; 11. 3 cm 2 11 -7 3 c 4
Angles of Elevation and Depression PRE-ALGEBRA LESSON 11 -7 Solve. Round answers to the nearest unit. 1. The angle of elevation from a boat to the top of a lighthouse is 35°. The lighthouse is 96 ft tall. How far from the base of the lighthouse is the boat? 137 ft 2. Ming launched a model rocket from 20 m away. The rocket traveled straight up. Ming saw it peak at an angle of 70°. If she is 1. 5 m tall, how high did the rocket fly? 57 m 3. An airplane is flying 2. 5 mi above the ground. If the pilot must begin a 3° descent to an airport runway at that altitude, how far is the airplane from the beginning of the runway (in ground distance)? 48 mi 11 -7
Right Triangles in Algebra PRE-ALGEBRA CHAPTER 11 1. 5 10. 9 19. 15. 2 ft 2. – 9 11. 9 20. 7. 5 m 3. 10 12. 11 21. 7. 2 4. – 2 13. rational 22. 1 5. 4 14. irrational 23. 14. 2 6. 7 15. rational 24. 5 7. 2 16. rational 25. (4, 3) 8. 3 17. 5. 3 mm 26. (5. 5, 2) 9. 7 18. 8. 1 mi 27. 16 ft 11 -A
Right Triangles in Algebra PRE-ALGEBRA CHAPTER 11 28. c 15. 6 cm, b = 11 cm 29. t = 18 m n 15. 6 m 30. 0. 7314 31. 3. 7321 35. 0. 9205 36. Answers may vary. Sample: If an unknown measurement can be placed into a trigonometric ratio with two known measurements, you can solve for the unknown measurement. 32. 0. 0698 37. about 113 m 33. 0. 5543 34. 0. 5000 11 -A
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