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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Ili kupata urefu wa MN na AC, tutatumia Kanuni ya Kosine (Law of Cosines) kwa kila pembetatu. Kwa pembetatu LMN: Tunapewa: • Urefu wa LM = 2 cm • Urefu wa LN = 4 cm • Pembe L = 50^ Step 1: Tumia Kanuni ya Kosine kupata urefu wa MN. Kanuni ya Kosine inasema c^2 = a^2 + b^2 - 2ab(C). Katika kesi hii, c = MN, a = LM, b = LN, na C = L. MN^2 = LM^2 + LN^2 - 2(LM)(LN)( L) MN^2 = (2\, cm)^2 + (4\, cm)^2 - 2(2\, cm)(4\, cm)(50^) Step 2: Kokotoa thamani ya MN^2. MN^2 = 4\, cm^2 + 16\, cm^2 - 16\, cm^2 × (50^) MN^2 = 20\, cm^2 - 16\, cm^2 × 0.6428 MN^2 = 20\, cm^2 - 10.2848\, cm^2 MN^2 = 9.7152\, cm^2 Step 3: Pata mzizi wa mraba wa MN^2 ili kupata MN. MN = sqrt(9.7152\, cm)^2 MN ≈ 3.12 cm Kwa pembetatu ABC: Tunapewa: • Urefu wa AB = 5 cm • Urefu wa BC = 7.5 cm • Pembe B = 75^ Step 4: Tumia Kanuni ya Kosine kupata urefu wa AC. Katika kesi hii, c = AC, a = AB, b = BC, na C = B. AC^2 = AB^2 + BC^2 - 2(AB)(BC)( B) AC^2 = (5\, cm)^2 + (7.5\, cm)^2 - 2(5\, cm)(7.5\, cm)(75^) Step 5: Kokotoa thamani ya AC^2. AC^2 = 25\, cm^2 + 56.25\, cm^2 - 75\, cm^2 × (75^) AC^2 = 81.25\, cm^2 - 75\, cm^2 × 0.2588 AC^2 = 81.25\, cm^2 - 19.41\, cm^2 AC^2 = 61.84\, cm^2 Step 6: Pata mzizi wa mraba wa AC^2 ili kupata AC. AC = sqrt(61.84\, cm)^2 AC ≈ 7.86 cm Tuma swali linalofuata 📸