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Geo - CH8 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. ____ ____ ____ ____ ____ 1. Write a similarity statement comparing the three triangles in the diagram. a. ΔGFJ ∼ ΔGHF ∼ ΔJHF c. ΔGFJ ∼ ΔFHG ∼ ΔFJH b. ΔGFJ ∼ ΔGFH ∼ JFH d. ΔGFJ ∼ ΔGHF ∼ ΔFHJ 2. Use your calculator to find trigonometric ratios sin 79°, cos 47°, and tan 77°. Round to the nearest hundredth. a. sin 79° = –0.99, cos 47° = –0.44, tan 77° = –32.27 b. sin 79° = –0.44, cos 47° = –0.99, tan 77° = –32.27 c. sin 79°= 0.68, cos 47° = 0.98, tan 77° = 4.33 d. sin 79° = 0.98, cos 47° = 0.68, tan 77° = 4.33 3. Jessie is building a ramp for loading motorcycles onto a trailer. The trailer is 2.8 feet off of the ground. To avoid making it too difficult to push a motorcycle up the ramp, Jessie decides to make the angle between the ramp and the ground 15°. To the nearest hundredth of a foot, find the length of the ramp. a. 10.82 feet c. 0.72 feet b. 2.90 feet d. 10.45 feet −1 −1 −1 4. Use your calculator to find the angle measures sin (0.7), cos (0.3), and tan (38.4)to the nearest tenth of a degree. a. sin−1(0.7) = 44.4º, cos−1(0.3) = 72.5º, tan−1(38.4) = 88.5º b. sin−1(0.7) = 0.8º, cos−1(0.3) = 1.3º, tan−1(38.4) = 1.5º c. sin−1(0.7) = 1.3º, cos−1(0.3) = 0.8º, tan−1(38.4) = 1.5º d. sin−1(0.7) = 72.5º, cos−1(0.3) = 44.4º, tan−1(38.4) = 88.5º 5. Some mountains in the Alps are very steep and have a grade of 42.7%. To the nearest degree, what angle do these mountains make with a horizontal line? a. 23° c. 47° ° b. 67 d. 32° ____ ____ ____ ____ 6. The largest Egyptian pyramid is 146.5 m high. When Rowena stands far away from the pyramid, her line of sight to the top of the pyramid forms an angle of elevation of 20° with the ground. What is the horizontal distance between the center of the pyramid and Rowena? Round to the nearest meter. a. 402 m c. 156 m b. 427 m d. 65 m 7. An eagle 300 feet in the air spots its prey on the ground. The angle of depression to its prey is 15°. What is the horizontal distance between the eagle and its prey? Round to the nearest foot. a. 1,120 ft c. 310 ft b. 1,159 ft d. 723 ft 8. Use a calculator to find the trigonometric ratios sin 123°, cos 95°, and tan 125°. Round to the nearest hundredth. a. sin 123° = –0.09, cos 95° = 0.84, tan 125° = –1.43 b. sin 123° = –0.46, cos 95° = 0.73, tan 125° = –0.78 c. sin 123° = 0.84, cos 95° = –0.09, tan 125° = –1.43 d. sin 123° = 0.84, cos 95° = 0.996194698092, tan 125° = –1.43 → 9. Write the vector AB in component form. a. b. ____ 〈 5, − 3 〉 〈 −3, 5 〉 c. d. 〈 −5, 3 〉 〈 3, − 5 〉 10. Sketch a 160° angle on the coordinate plane. Find the measure of its reference angle. a. Reference angle: 110° c. Reference angle: 340° b. Reference angle: 20° d. Reference angle: 70° Numeric Response 11. A hiking trail has a slope of 327 . What is the measure of the angle that the trail makes with a horizontal line? Round to the nearest degree. 12. Mike and Lani stand 21.2 meters apart. From Mike’s position, the angle of elevation to the top of the Eiffel Tower is 40°. From Lani’s position, the angle of elevation to the top of the Eiffel Tower is 38.5°. How many meters high is the Eiffel Tower? Round to the nearest meter. → → 13. TM has an initial point of (−5, 5) and a terminal point of (−7, − 3). Find the magnitude of TM to the nearest tenth. Matching Match each vocabulary term with its definition. a. cosine b. adjacent leg c. sine d. tangent e. trigonometric ratio f. angle of depression g. opposite leg h. angle of elevation i. geometric mean ____ 14. the angle formed by a horizontal line and a line of sight to a point above ____ 15. for positive numbers a and b, the positive number x such that a = x ____ ____ ____ ____ x b 16. in a right triangle, the ratio of the length of the leg opposite the angle to the length of the hypotenuse 17. in a right triangle, the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse 18. in a right triangle, the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle 19. a ratio of two sides of a right triangle Match each vocabulary term with its definition. a. parallel vectors b. resultant vector c. additive vector d. direction e. equal vectors f. magnitude g. multiplicative vector h. component form i. vector ____ ____ 20. a quantity that has both magnitude and direction 21. two vectors that have the same magnitude and the same direction ____ 22. the length of a vector, written AB or v | | | | ____ 23. the orientation of a vector, which is determined by the angle the vector makes with a horizontal line | → | | → | Geo - CH8 Practice Test Answer Section MULTIPLE CHOICE 1. ANS: D Sketch the three right triangles with the angles of the triangles in corresponding positions. The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle, so ΔGFJ ∼ ΔGHF ∼ ΔFHJ. Feedback A B C D All corresponding long legs must match up. All right angles must align. Sketch the triangles in similar orientations to help you align corresponding parts. Correct! PTS: 1 DIF: Average REF: Page 518 OBJ: 8-1.1 Identifying Similar Right Triangles TOP: 8-1 Similarity in Right Triangles 2. ANS: D Make sure your calculator is in degree mode. sin 79° = 0.98, cos 47° = 0.68, tan 77° = 4.33 NAT: 12.3.2.e Feedback A B C D Change your calculator to degree mode. Change your calculator to degree mode. Switch the first and second answers. Correct! PTS: OBJ: TOP: 3. ANS: 1 DIF: Basic REF: Page 526 8-2.3 Calculating Trigonometric Ratios 8-2 Trigonometric Ratios A NAT: 12.2.1.m sin A = BC AB sin 15° = 2.8 AB AB = 2.8 sin 15° AB ≈ 10.82 feet Write a trigonometric ratio. Substitute the given values. Multiply both sides by AB and divide by sin 15°. Simplify the expression. Feedback A B C D Correct! Since the hypotenuse and opposite side are involved, use the sine ratio. The ramp should be substantially larger than the opposite leg, because the opposite leg is across from the smallest possible angle in the triangle. Draw a picture to help determine which trigonometric ratio to use. PTS: 1 DIF: Average REF: Page 528 OBJ: 8-2.5 Problem-Solving Application NAT: 12.2.1.m TOP: 8-2 Trigonometric Ratios 4. ANS: A Change your calculator to degree mode. Use the inverse trigonometric functions on your calculator to find each angle measure. sin−1 ( 0.7 ) = 44.4°, cos−1 ( 0.3 ) = 72.5°, tan−1 ( 38.4 ) = 88.5° Feedback A B C D Correct! Change your calculator to degree mode. Change your calculator to degree mode. Switch the first and second answer. PTS: OBJ: NAT: 5. ANS: 1 DIF: Basic REF: Page 535 8-3.2 Calculating Angle Measures from Trigonometric Ratios 12.2.1.m TOP: 8-3 Solving Right Triangles A 42.7% = 42.7 100 Change the percent grade to a fraction. A 42.7% grade means the mountain rises 42.7 ft for every 100 ft of horizontal distance. ∠A is the angle the mountain makes with a horizontal line. ˆ˜ m∠A = tan−1 ÊÁË 42.7 100 ¯ ≈ 23° Feedback A B C D Correct! Find the complement of this angle. Convert 42.7% to a fraction. Take the inverse tangent of this fraction. Convert 42.7% to a fraction. Take the inverse tangent of this fraction. PTS: 1 NAT: 12.2.1.m 6. ANS: A tan 20° = 146.2 x 146.2 x= tan 20° x ≈ 402 m DIF: Average REF: Page 536 TOP: 8-3 Solving Right Triangles OBJ: 8-3.5 Application Use the side opposite ∠A and x, and the side adjacent to ∠A to write the tangent ratio. Multiply both sides by x and divide both sides by tan 20°. Simplify. Feedback A B C D Correct! Divide 146.2 by tan 20. Divide 146.2 by tan 20. Change your calculator to degree mode. PTS: OBJ: TOP: 7. ANS: 1 DIF: Average REF: Page 545 8-4.2 Finding Distance by Using Angle of Elevation 8-4 Angles of Elevation and Depression A NAT: 12.2.1.k By the Alternate Interior Angles Theorem, m∠S = 15°. From the sketch, tan 15° = 300 . So x x = 300 ≈ 1, 120 ft. tan 15° Feedback A B C D Correct! Divide 300 by tan 15. Divide 300 by tan 15. Divide 300 by tan 15. PTS: 1 DIF: Average REF: Page 545 OBJ: 8-4.3 Finding Distance by Using Angle of Depression NAT: 12.2.1.k TOP: 8-4 Angles of Elevation and Depression 8. ANS: C Change the calculator mode to degrees.sin 123° = 0.84, cos 95° = −0.09, tan 125° = −1.43 Feedback A B C D The first question uses the sine function. Change the calculator mode to degrees. Correct! The second question uses the cosine function. PTS: 1 DIF: Basic REF: Page 551 OBJ: 8-5.1 Finding Trigonometric Ratios for Obtuse Angles TOP: 8-5 Law of Sines and Law of Cosines 9. ANS: A The horizontal change from A to B is 5 units. The vertical change from A to B is –3 units. → So the component of AB is 〈 5, − 3 〉. Feedback A B C D Correct! The first number is the horizontal component. The horizontal component is 5. The first number is the horizontal component. PTS: 1 DIF: Average REF: Page 559 OBJ: 8-6.1 Writing Vectors in Component Form TOP: 8-6 Vectors 10. ANS: B NAT: 12.3.4.e The reference angle is the acute angle formed between the ray shown in the sketch, and the negative x-axis. That angle is 180° − 160° = 20°. Feedback A B C D The reference angle is an acute angle. Correct! The reference angle is an acute angle. The x-axis (not the y-axis) is one side of a reference angle. PTS: 1 DIF: Basic OBJ: 8-Ext.1 Finding Reference Angles KEY: unit circle | reference angle REF: Page 570 TOP: 8-Ext Trigonometry and the Unit Circle NUMERIC RESPONSE 11. ANS: 12 PTS: 1 DIF: Basic TOP: 8-3 Solving Right Triangles 12. ANS: 324 NAT: 12.2.1.m PTS: 1 DIF: Advanced NAT: 12.2.1.m TOP: 8-4 Angles of Elevation and Depression 13. ANS: 8.2 PTS: 1 DIF: Advanced TOP: 8-6 Vectors 14. ANS: H PTS: 1 DIF: MATCHING Basic REF: Page 544 15. 16. 17. 18. 19. TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: 20. ANS: TOP: 21. ANS: TOP: 22. ANS: TOP: 23. ANS: TOP: 8-4 Angles of Elevation and Depression I PTS: 1 DIF: 8-1 Similarity in Right Triangles C PTS: 1 DIF: 8-2 Trigonometric Ratios A PTS: 1 DIF: 8-2 Trigonometric Ratios D PTS: 1 DIF: 8-2 Trigonometric Ratios E PTS: 1 DIF: 8-2 Trigonometric Ratios I 8-6 Vectors E 8-6 Vectors F 8-6 Vectors D 8-6 Vectors Basic REF: Page 519 Basic REF: Page 525 Basic REF: Page 525 Basic REF: Page 525 Basic REF: Page 525 PTS: 1 DIF: Basic REF: Page 559 PTS: 1 DIF: Basic REF: Page 561 PTS: 1 DIF: Basic REF: Page 560 PTS: 1 DIF: Basic REF: Page 560