Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Multilateration wikipedia , lookup
Euler angles wikipedia , lookup
Integer triangle wikipedia , lookup
Rational trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Perceived visual angle wikipedia , lookup
Euclidean geometry wikipedia , lookup
Math 11 A monster of a unit 2 review Short Answer 1. Determine the measure of ∠ABF. 2. Determine the values of a, b, and c. 3. Determine the values of a, b, and c. 4. Determine the measure of ∠PQR. 5. Given QP || MR, determine the measure of ∠QMO. 6. Determine the value of x. 7. Each interior angle of a regular convex polygon measures 156°. How many sides does the polygon have? 8. Gareth is measuring the exterior angles of a convex hexagon. So far, he has measured 60°, 60°, 60°, 30°, and 30°. What is the measure of the last exterior angle? Show your calculation. 9. Sketch a triangle that corresponds to the equation. Then, determine the third angle measure and the third side length. 10. Determine the length of d to the nearest tenth of a centimetre. 11. Determine the measure of θ to the nearest degree. 12. Determine the length of w to the nearest tenth of a centimetre. 13. Determine the measure of θ to the nearest degree. 14. A kayak leaves a dock on Lake Athabasca, and heads due north for 2.8 km. At the same time, a second kayak travels in a direction N70°E from the dock for 3.0 km. How you can determine the distance between the kayaks? 15. A radar operator on a ship discovers a large sunken vessel lying parallel to the ocean surface, 180 m directly below the ship. The length of the vessel is a clue to which wreck has been found. The radar operator measures the angles of depression to the front and back of the sunken vessel to be 52° and 67°. How long, to the nearest tenth of a metre, is the sunken vessel? 16. Determine the unknown angle measure to the nearest degree. 17. In ∆ABC, ∠A = 26°, a = 8.5 cm, and b = 5.0 cm. Determine the number of triangles (zero, one, or two) that are possible for these measurements. Draw the triangle(s) to support your answer. 18. In ∆ABC, ∠A = 45°, a = 6.0 cm, and b = 7.5 cm. Determine the number of triangles (zero, one, or two) that are possible for these measurements. Draw the triangle(s) to support your answer. 19. In ∆UVW, ∠V = 73° and VW = 18.6 cm. Calculate the height of the triangle from base VU to the nearest tenth of a centimetre. 20. Determine the indicated angle measure to the nearest degree. Problem 21. Do you need to know QP || MR to determine the measure of ∠QMO? Explain. 22. Prove: FG || HI 23. A floor tiler designs custom floors using tiles in the shape of regular polygons. The tiler uses three different tile shapes to cover a floor, all with the same side length. At each corner, there is one square and one hexagon. What is the third tile shape? Draw part of the tiling. 24. A radio tower is supported by two wires on opposite sides. On the ground, the ends of the wire are 280 m apart. One wire makes a 60° angle with the ground. The other makes a 66° angle with the ground. Draw a diagram of the situation. Then, determine the length of each wire to the nearest metre. Show your work. 25. Stella decided to ski to a friend’s cabin. She skied 8.0 km in the direction N40°E. She rested, then skied S30°E and arrived at the cabin. The cabin is 9.5 km from her home, as the crow flies. Determine, to the nearest tenth of a kilometre, the distance she travelled on the second leg of her trip. Show your work. 26. The pendulum of a grandfather clock is 85.0 cm long. When the pendulum swings from one side to the other side, it travels a horizontal distance of 10.5 cm. Determine the angle through which the pendulum swings. Round your answer to the nearest tenth of a degree. 27. While golfing, Beth hits a tee shot from point T toward a hole at H. However, the ball veers 34° and lands at B. The scorecard says that H is 250 m from T. Beth walks 120 m to her ball. Sketch a diagram of this situation. How far, to the nearest metre, is her ball from the hole? Show your work. 28. The posts of a hockey goal are 2.0 m apart. A player is standing at a point 4.5 m from one post and 6.0 m from the other post. Within what angle must the player shoot the puck to score a goal? Express your answer to the nearest degree. Show your work. 29. A building is observed from two points, P and Q, that are 94.0 m apart. The angle of elevation is 42° at P and 33° at Q. Sketch the situation. Determine the height of the building to the nearest tenth of a metre. U2 review Answer Section SHORT ANSWER 1. ANS: ∠ABF = 66° 2. ANS: ∠a = 18°, ∠b = 54°, ∠c = 27° 3. ANS: ∠a = 15°, ∠b = 30°, ∠c = 10° 4. ANS: ∠PQR = 122° 5. ANS: ∠QMO = 23° 6. ANS: x = 48° 7. ANS: 15 8. ANS: 360° – 60° – 60° – 60° – 30° – 30° = 120° 9. ANS: 70°, 18.8 10. ANS: d = 6.2 cm 11. ANS: θ = 57° 12. ANS: w = 27.3 cm 13. ANS: θ = 57° 14. ANS: Since the measures of two sides and a contained angle are given, I would use the cosine law. 15. ANS: 217.0 m 16. ANS: 42° 17. ANS: one triangle: 18. ANS: two triangles: 19. ANS: 17.8 cm 20. ANS: 49° or 131° PROBLEM 21. ANS: No. Use alternate interior angles and complementary angles to determine ∠MNO. Use the sum of angles in a triangle to determine ∠NMO. Then use the sum of angles on a straight line to solve for ∠QMO. 22. ANS: ∠FHG + ∠GHI + ∠IHJ 94° + ∠GHI + 73° ∠GHI ∠GHI ∠GHI Therefore, FG = 180° = 180° = 180° – 94° – 73° = 13° = ∠FGH || HI Sum of angles in triangle is 180° Substitute known values. Determine ∠GHI. equal alternate interior angles 23. ANS: The measure of an interior angle of a square is 90°. The measure of an interior angle of a regular hexagon is 120°. This leaves a gap of 360° – 90° – 120° = 150°. Determine the number of sides, n, of a regular polygon with 150°-angles: The measure of the interior angles in a regular 12-sided polygon (dodecagon) is 150°. The tiling is made with a square, a regular hexagon, and a regular dodecagon: 24. ANS: Let the x and y be the length of the wires. The third angle is 180° – 66° – 60° = 54°. Use the sine law to determine the length of each wire: The wires are 316 m and 300 m long. 25. ANS: Because the lines are parallel, the angle beside the 30° angle is also 40°. The entire angle is 70°. x + 70° + z = 180° x + 70° + 52.309...° = 180° x = 57.690...° Stella travelled 8.5 km. 26. ANS: a2 10.52 110.25 –14 339.75 = b2 + c2 – 2bc cos A = 85.02 + 85.02 – 2(85.0)(85.0) cos A = 7225.00 + 7225.00 – 14 450.00 cos A = –14 450.00 cos A = cos A ∠A = cos–1(0.9923...) ∠A = 7.082...° The pendulum swings through an angle of 7.1°. 27. ANS: By the cosine law, t2 = h2 + b2 – 2hb cos T t2 = 1202 + 2502 – 2(120)(250) cos 34° t2 = 27 157.745... t = 164.796... Beth's ball is 165 m from the hole. 28. ANS: Draw a diagram of the situation. By the cosine law, a2 2.02 2 2 2.0 – 4.5 – 6.02 –52.25 = b2 + c2 – 2bc cos C = 4.52 + 6.02 – 2(4.5)(6.0) cos C = – 2(4.5)(6.0) cos C = –54 cos C = cos C cos–1 = ∠C 14.6264…° = ∠C The player must shoot within a 15° angle. 29. ANS: The measures of ∠PRQ, PQ, and ∠PQR are known. Use the sine law to determine PR. PR is 327.3 m. The measures of ∠RSP, PR, and ∠SPR are known. Use the sine law to determine RS, or h, the height of the building. The height of the building is 219.0 m.