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Geometry: Similarity, Right Triangles, and Trigonometry — G-SRT ELG.MA.HS.G.7: Define trigonometric ratios and solve problems involving right triangles. G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ Geometry: Similarity, Right Triangles, and Trigonometry — G-SRT ELG.MA.HS.G.8: Apply trigonometry to general triangles. G-SRT.D.9 (+) Derive the formula A = ½ ab sin (C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces.) Students will demonstrate command of the ELG by: 1 Deriving the formula 𝐴 = 𝑎𝑏𝑠𝑖𝑛𝐶 for the area of a triangle. Proving the Laws of Sines and Cosines. Applying the Laws of Sines and Cosines to solve problems. Applying the Laws of Sines and Cosines to find unknown measurements in triangles. 2 Vocabulary: Auxiliary line Law of Cosines Law of Sines Sample Assessment Questions: 1) Standard(s): G-SRT.D.11 Source: Key Data Systems Item Prompt: Three buoys, X, Y, and Z, are placed in the ocean to mark a sailboat race. Contestants start at X, sail to buoys Y and Z, and then return to buoy X. The location of the buoys and the racecourse are shown below. Part A: What is the distance between buoy X and buoy Z to the nearest tenth of a mile? Part B: Explain how you determined your answer. Correct Answer(s): Part A: 3.1 miles Part B: 𝑠𝑖𝑛 40° From the Law of Sines, = 2.4 2) Standard(s): G-SRT.D.11 𝑠𝑖𝑛 56° 𝑦 . Rearranging leads to y= 2.4 (𝑠𝑖𝑛56°) sin 40° ≈ 3.1. Therefore, the distance between buoys X and Z is approximately 3.1 miles. Source: Key Data Systems Item Prompt: A pasture is in the shape of a triangle. Two sides of this pasture measure 3.4 miles and 2.6 miles, and the angle between these two sides is 40°. Part A: To the nearest tenth of a mile, determine the length of the third side of the pasture. Part B: Explain how you determined your answer. Correct Answer(s): Part A: 2.2 miles Part B: The Law of Cosines states that c2 = a2 + b2 – 2ab · cos C, where C is the angle between side a and side b. Letting a = 3.4 and b = 2.6, substitution yields c2 = 3.42 + 2.62 – 2(3.4)(2.6)(cos 40). Simplifying leads to c2 = 11.56 + 6.76 – 17.68(cos 40) ≈ 4.776. Taking the square root yields c ≈ 2.2 miles.