With usual notations, if in a triangle `A B C(b+c)/(11)=(c+a)/(12)=(a+ – InterviewSolution

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1.	With usual notations, if in a triangle `A B C(b+c)/(11)=(c+a)/(12)=(a+b)/(13)`, then prove that:`(cosA)/7=(cosB)/(19)=(cosC)/(25)`
Answer» Let `(b+c)/11 = (c+a)/12 = (a+b)/13 = k` `=>b+c = 11k->(1)` `=>c+a = 12k->(2)` `=>a+b = 13k->(3)` Solving (1),(2) and (3), we get, `a = 7k, b = 6k, c = 5k` Now, `cosA = (b^2+c^2-a^2)/(2bc) = (36k^2+25k^2-49k^2)/(60k^2) = 1/5` `cosB = (a^2+c^2-b^2)/(2ac) = (49k^2+25k^2-36k^2)/(70k^2) = 38/70 = 19/35` `cos C = (a^2+b^2-c^2)/(2ab) = (49k^2+36k^2-25k^2)/(84k^2) = 60/84 = 5/7` `:. cosA:cosB:cosC = 1/5:19/35:5/7 = 7:19:25` `:. cosA/7 = cosB/19 = cosC/25.`

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