Practice problems for the chapter 8 exam. Chapter 8 Review Practice problems for the chapter 8 exam.

1. Find the length of the missing side 1. Find the length of the missing side. The triangle is not drawn to scale. 48 10 28 100 8-1

1. Find the length of the missing side 1. Find the length of the missing side. The triangle is not drawn to scale. 𝑎 2 + 𝑏 2 = 𝑐 2 6 2 + 8 2 = 𝑐 2 36+64= 𝑐 2 100= 𝑐 2 100 =c 10=𝑐 Correct answer is B. 8-1

2. Triangle ABC has side lengths 3, 4, and 5 2. Triangle ABC has side lengths 3, 4, and 5. Do the side lengths form a Pythagorean triple? Explain. 8-1 No, they do not form a Pythagorean triple; although , the side lengths do not meet the other requirements of a Pythagorean triple. Yes; they can form a right triangle, so they form a Pythagorean triple. Yes, they form a Pythagorean triple; 3 2 + 4 2 = 5 2 and 3, 4, and 5 are all nonzero whole numbers. No, they do not form a Pythagorean triple; 3 2 + 4 2 ≠ 5 2 .

Since 3, 4, and 5 are all nonzero whole numbers 2. Triangle ABC has side lengths 3, 4, and 5. Do the side lengths form a Pythagorean triple? Explain. 8-1 Since 3, 4, and 5 are all nonzero whole numbers AND 3 2 + 4 2 = 5 2 Correct answer is C. Yes, they form a Pythagorean triple; 3 2 + 4 2 = 5 2 and 3, 4, and 5 are all nonzero whole numbers.

3. Find the length of the missing side 3. Find the length of the missing side. Leave your answer in simplest radical form. 69 ft 269 ft 23 ft 3 ft 8-1

3. Find the length of the missing side 3. Find the length of the missing side. Leave your answer in simplest radical form. 𝑎 2 + 𝑏 2 = 𝑐 2 10 2 + 𝑏 2 = 13 2 100+ 𝑏 2 =169 𝑏 2 =69 𝑏= 69 Correct answer is A. 69 ft 8-1

4. A triangle has sides of lengths 27, 79, and 84 4. A triangle has sides of lengths 27, 79, and 84. Is it a right triangle? Explain. yes; 27 2 + 79 2 = 84 2 no; 27 2 + 79 2 = 84 2 yes; 27 2 + 79 2 ≠ 84 2 no; 27 2 + 79 2 ≠ 84 2 8-1

4. A triangle has sides of lengths 27, 79, and 84 4. A triangle has sides of lengths 27, 79, and 84. Is it a right triangle? Explain. 𝑎 2 + 𝑏 2 ⎕ 𝑐 2 If it is equal then the triangle is a right triangle. 27 2 + 79 2 ⎕ 84 2 729+6241⎕7056 6970⎕7056 6970≠7056 Correct answer is D. no; 27 2 + 79 2 ≠ 84 2 8-1

5. A triangle has side lengths of 23 in, 6 in, and 28 in 5. A triangle has side lengths of 23 in, 6 in, and 28 in. Classify it as acute, obtuse, or right. obtuse right acute 8-1

5. A triangle has side lengths of 23 in, 6 in, and 28 in 5. A triangle has side lengths of 23 in, 6 in, and 28 in. Classify it as acute, obtuse, or right. 𝑐 2 ⎕ 𝑎 2 + 𝑏 2 If it is less then the triangle is an acute triangle. If it is more then the triangle is an obtuse triangle. If it is equal then the triangle is a right triangle. 28 2 ⎕ 23 2 + 6 2 784 ⎕ 529+36 784⎕565 784>565 Correct answer A. Obtuse 8-1

6. In triangle ABC, ∠𝐴 is a right angle and ∠𝐵= 45°. Find BC 6. In triangle ABC, ∠𝐴 is a right angle and ∠𝐵= 45°. Find BC. If your answer is not an integer, leave it in simplest radical form. 10 2 ft 20 2 ft 10 ft 20 ft 8-2

In a 45-45-90 both legs are the same value 6. In triangle ABC, ∠𝐴 is a right angle and ∠𝐵= 45°. Find BC. If your answer is not an integer, leave it in simplest radical form. In a 45-45-90 both legs are the same value the hypotenuse is leg * 2 Since the leg is 10 The hypotenuse is 10 2 The correct answer is A. 10 2 8-2

7. Find the value of the variable 7. Find the value of the variable. If your answer is not an integer, leave it in simplest radical form. 5 3 2 5 3 5 2 5 2 2 8-2

7. Find the value of the variable 7. Find the value of the variable. If your answer is not an integer, leave it in simplest radical form. The get back to the leg of a 45-45-90 given the hypotenuse you must divide by 2 5 2 Will simplify to 5 2 2 Correct answer is D. 5 2 2 8-2

8. Find the value of the variable(s) 8. Find the value of the variable(s). If your answer is not an integer, leave it in simplest radical form. 5 3 1 2 10 3 2 8-2

8. Find the value of the variable(s) 8. Find the value of the variable(s). If your answer is not an integer, leave it in simplest radical form. 8-2 In a 30-60-90 right triangle everything is based off the short leg. Hypotenuse is twice the short leg and long leg is short leg * 3 The short leg is 5 so the long leg is 5 3 Correct answer is A. 5 3

9. Find the value of the variable(s) 9. Find the value of the variable(s). If your answer is not an integer, leave it in simplest radical form. x = 22 3 , y = 11 x = 11 3 , y = 22 x = 22, y = 11 3 x = 11, y = 22 3 8-2

9. Find the value of the variable(s) 9. Find the value of the variable(s). If your answer is not an integer, leave it in simplest radical form. 8-2 In a 30-60-90 right triangle everything is based off the short leg. Hypotenuse is twice the short leg and long leg is short leg * 3 The short leg is 11 so the long leg is 11 3 and the hypotenuse is 22. Correct answer is B. x = 11 3 , y = 22

10. The length of the hypotenuse of a 30°–60°–90° triangle is 9 10. The length of the hypotenuse of a 30°–60°–90° triangle is 9. Find the perimeter. 27+9 3 27 2 + 9 2 3 9 2 + 27 2 3 9+27 3 8-2

8-2 10. The length of the hypotenuse of a 30°–60°–90° triangle is 9. Find the perimeter. In a 30-60-90 right triangle everything is based off the short leg. Since they give you the hypotenuse and the hypotenuse is twice the short leg the short leg is 9 2 . The long leg is short leg * 3 so it is 9 2 3 . The perimeter is the sum of all three sides 9+ 9 2 + 9 2 3 = 27 2 + 9 2 3 . Correct answer is B. 27 2 + 9 2 3

11. Find the missing value to the nearest hundredth. 8-3

11. Find the missing value to the nearest hundredth. 8-3 To find degrees we use the inverse functions. 𝑐𝑜𝑠 −1 2 5 The correct answer is B. 66.42o

12. Use a trigonometric ratio to find the value of x 12. Use a trigonometric ratio to find the value of x. Round your answer to the nearest tenth. 2.6 4.8 3.4 3.1 8-3

12. Use a trigonometric ratio to find the value of x 12. Use a trigonometric ratio to find the value of x. Round your answer to the nearest tenth. Figure out which trigonometric ratio you need. Tangent we have opposite leg and adjacent leg 𝑡𝑎𝑛 40 𝑜 = 4 𝑥 Variable is in the bottom so we divide by the trig function. 𝑥= 4 𝑡𝑎𝑛 40 𝑜 =4.8 The correct answer is B. 4.8 8-3

13. Find the value of x. Round to the nearest tenth. 14.5 10.7 10.2 14.2 8-3

13. Find the value of x. Round to the nearest tenth. 8-3 13. Find the value of x. Round to the nearest tenth. Figure out which trigonometric ratio you need. Cosine we have adjacent leg and hypotenuse 𝑐𝑜𝑠 32 𝑜 = 12 𝑥 Variable is in the bottom so we divide by the trig function. 𝑥= 12 𝑐𝑜𝑠 32 𝑜 =14.15014084 The correct answer is D. 14.2

14. Find the value of x. Round to the nearest tenth. 55.6 55.8 6.7 6.5 8-3

14. Find the value of x. Round to the nearest tenth. Figure out which trigonometric ratio you need. Sine we have opposite leg and hypotenuse 𝑠𝑖𝑛 20 𝑜 = 19 𝑥 Variable is in the bottom so we divide by the trig function. 𝑥= 19 𝑠𝑖𝑛 20 𝑜 =55.5522836 The correct answer is A. 55.6 8-3

15. Find the value of x. Round to the nearest degree. 60 57 29 33 8-3

15. Find the value of x. Round to the nearest degree. Figure out which trigonometric ratio you need. Cosine we have adjacent leg and hypotenuse 𝑐𝑜𝑠 𝑥 𝑜 = 11 20 We are looking for degrees so we use the inverse function. 𝑐𝑜𝑠 −1 11 20 =56.63298703 The correct answer is B. 57 degrees. 8-3

16. What is the description of ∠2 as it relates to the situation shown? ∠2 is the angle of depression from the airplane to the radar tower. ∠2 is the angle of elevation from the airplane to the radar tower. ∠2 is the angle of elevation from the radar tower to the airplane. ∠2 is the angle of depression from the radar tower to the airplane. 8-4

and it goes from the radar tower to the airplane 16. What is the description of ∠2 as it relates to the situation shown? 8-4 Since Angle 2 goes up from the radar tower, it is an angle of elevation and it goes from the radar tower to the airplane The correct answer is C. ∠2 is the angle of elevation from the radar tower to the airplane.

17. Find the value of x. Round the length to the nearest tenth. 6.5 ft 4.7 ft 9.9 ft 5.8 ft 8-4

17. Find the value of x. Round the length to the nearest tenth. Figure out which trigonometric ratio you need. Cosine we have adjacent leg and hypotenuse 𝑐𝑜𝑠 36 𝑜 = 𝑥 8 Variable is in the top so we multiply by the trig function. 𝑥=8∗𝑐𝑜𝑠 36 𝑜 =6.472135955 The correct answer is A. 6.5 ft 8-4

18. Find the value of x. Round the length to the nearest tenth. 777.9 m 595.9 m 321.4 m 652.7 m 8-4

18. Find the value of x. Round the length to the nearest tenth. Move the angle and figure out which trigonometric ratio you need. Sine we have opposite leg and the hypotenuse 𝑠𝑖𝑛 40 𝑜 = 500 𝑥 Variable is in the bottom so we divide by the trig function. 𝑥= 500 sin 40 𝑜 =777.8619134 The correct answer is A. 777.9 m 8-4

19. Find the value of x. Round the length to the nearest tenth. 4.6 yd 10 yd 23.6 yd 5.1 yd 8-4

19. Find the value of x. Round the length to the nearest tenth. Move the angle and figure out which trigonometric ratio you need. Tangent we have opposite leg and adjacent leg 𝑡𝑎𝑛 25 𝑜 = 𝑥 11 Variable is in the top so we multiply by the trig function. 𝑥=11∗𝑡𝑎𝑛 25 𝑜 =5.12938424 The correct answer is D. 5.1 yd 8-4

20. To approach the runway, a pilot of a small plane must begin a 10o descent starting from a height of 1983 feet above the ground. To the nearest tenth of a mile, how many miles from the runway is the airplane at the start of this approach? 2.2 mi 0.4 mi 11,419.6 mi 2.1 mi 8-4

Move the angle and figure out which trigonometric ratio you need. 20. To approach the runway, a pilot of a small plane must begin a 10o descent starting from a height of 1983 feet above the ground. To the nearest tenth of a mile, how many miles from the runway is the airplane at the start of this approach? Move the angle and figure out which trigonometric ratio you need. Sine we have opposite leg and the hypotenuse 𝑠𝑖𝑛 10 𝑜 = 1983 𝑥 Variable is in the bottom so we divide by the trig function. 𝑥= 1983 𝑠𝑖𝑛 10 𝑜 =11419.64187 𝑓𝑒𝑒𝑡 To convert to miles divide by 5280 = 2.1628 The correct answer is A. 2.2 mi 8-4