Transcript 4.7 Notes
* Objectives: -Solve oblique triangles using Law of Sines and Law of Cosines -Find areas of oblique triangles. * -Triangles that are not right triangles * When the 3 measurements provided fit one of these cases: * Two angles and a nonincluded side (AAS) * Two angles and the included side (ASA) * Two sides and the included angle (SSA) * *choose any 2 ratios to create a proportion with 3 known measurements, and 1 unknown. * A) Solve ΔLMN. Round side lengths to the nearest tenth and angle measures to the nearest degree. * B) Solve for y in ΔXYZ. Round side lengths to the nearest tenth. * You know that the measures of two sides and a nonincluded angle (SSA) do not necessarily define a unique triangle. Consider the angle and side measures given in the figures below. In general, given an SSA case, one of the following will be true: * No triangle exists (no solution) * Exactly 1 triangle exists (one solution) * Two triangles exist (two solutions) SSA It is possible for more than 1 triangle to exist, or NO triangle to exist. It all depends on if the given angle is acute or obtuse. Always look for 2 triangles when finding an angle using Law of Sines. * Examples: Group 1: If < A = 30º, b = 10, & a = 4, find < B. Group 2: If < A = 30º, b = 10, & a = 5, find < B. Group 3: If < A = 30º, b = 10, & a = 6, find < B. * A) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mA = 63°, a = 18, b = 25 * B) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mC = 105°, b = 55, c = 73, * A) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mB = 45°, b = 18, and c = 24. * B) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mC = 24°, c = 13, and a = 15. * When the 3 measurements provided fit one of these cases: * Three sides (SSS) * Two sides and the included angle (SAS) 𝑏2 − 𝑐 2 + 𝑎2 cos 𝐴 = 2𝑏𝑐 * *In SSS, you must find the LARGEST angle first, then use the Law of Sines or Cosines to find the SMALLER of the 2 remaining angles. *In SAS, you must first find the side across from the known angle, then use the Law of Sines or Cosines to find the SMALLER of the 2 remaining angles. * A) Solve ΔMNP if mM = 54o, n = 17, and p = 12. Round side lengths to the nearest tenth and angle measures to the nearest degree. * B) Solve ΔABC. Round angle measures to the nearest degree. * C) Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree. * * A tree is leaning 10° past vertical as shown in the figure. A wire that makes a 42° angle with the ground 10 feet from the base of the tree is attached to the top of the tree. How tall is the tree? * A triangular area of lawn has a sprinkler located at each vertex. If the sides of the lawn are a = 19 feet, b = 24.3 feet, and c = 21.8 feet, what angle of sweep should each sprinkler be set to cover? * The angle of elevation from the top of a building to a hot air balloon is 62º. The angle of elevation to the hot air balloon from the top of a second building that is 650 feet due east is 49º. Find the distance from the hot air balloon to each building. * A triangular lot has sides of 120 feet, 186 feet, and 147 feet. Find the angle across from the shortest side. * Two airplanes leave an airport at the same time on different runways. One flies on a bearing of N66°W at 325 miles per hour. The other airplane flies on a bearing of S26°W at 300 miles per hour. How far apart will the airplanes be after two hours? * Two airplanes leave an airport at the same time on different runways. One flies directly north at 400 miles per hour. The other airplane flies on a bearing of N75°E at 350 miles per hour. How far apart will the airplanes be after two hours? * * A) Find the area of ΔABC to the nearest tenth. * B) Find the area of ΔDEF to the nearest tenth. * 12 10 132° 108° 10 * A) A post is driven in a certain spot. Proceed due east for 300 ft, then proceed S40oE for another 150 ft. Turn direction again S60oW for 400 ft and then back to the post in a straight line. Sketch the figure and find its area. * B) From an iron post, proceed 500 m northeast to the brook, then 300 m east along the brook to the old mill, then 200 m S15°E to a post on the edge of Wiggin’s Road, and finally along Wiggin’s Road back to the iron post. Sketch the figure and find its area.