This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Identify the given information for the triangle. We are given a triangle with sides BC = 15 cm, AC = 25 cm, and the included angle C = 93^. We need to find the length of side AB. Step 2: Apply the Law of Cosines. The Law of Cosines states that for a triangle with sides a, b, c and angles A, B, C opposite those sides, c^2 = a^2 + b^2 - 2ab (C). In this case, a = BC = 15 cm, b = AC = 25 cm, and c = AB. AB^2 = BC^2 + AC^2 - 2(BC)(AC) (C) AB^2 = 15^2 + 25^2 - 2(15)(25) (93^) Step 3: Calculate the values. AB^2 = 225 + 625 - 750 (93^) AB^2 = 850 - 750(-0.052336) AB^2 = 850 + 39.252 AB^2 = 889.252 AB = sqrt(889.252) AB ≈ 29.82 cm The length of AB is 29.82 cm. --- Step 1: Identify the given information for the triangle. We are given a triangle with sides a = 13, b = 11, and c = 16. We need to find all unknown angles (A, B, C). Step 2: Use the Law of Cosines to find angle A. The Law of Cosines states a^2 = b^2 + c^2 - 2bc (A). 13^2 = 11^2 + 16^2 - 2(11)(16) (A) 169 = 121 + 256 - 352 (A) 169 = 377 - 352 (A) 352 (A) = 377 - 169 352 (A) = 208 (A) = (208)/(352) = (13)/(22) A = ((13)/(22)) ≈ 53.80^ Step 3: Use the Law of Cosines to find angle B. The Law of Cosines states b^2 = a^2 + c^2 - 2ac (B). 11^2 = 13^2 + 16^2 - 2(13)(16) (B) 121 = 169 + 256 - 416 (B) 121 = 425 - 416 (B) 416 (B) = 425 - 121 416 (B) = 304 (B) = (304)/(416) = (19)/(26) B = ((19)/(26)) ≈ 43.10^ Step 4: Find angle C using the sum of angles in a triangle. The sum of angles in a triangle is 180^. C = 180^ - A - B C = 180^ - 53.80^ - 43.10^ C = 180^ - 96.90^ C = 83.10^ The angles of the triangle are A ≈ 53.80^, B ≈ 43.10^, C ≈ 83.10^. Send me the next one 📸