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three sides of triangle are consecutive integers and the largest angle is twice the smallest angle.the perimeter of the triangle is...?
Please find the answer to your questionLet the sides of ?be n, n + 1, n + 2 where n? N.Let a = n, b = n + 1, c = n + 2 Let the smallest, angle ? A = ? then the greatest ?C = 2? In ?ABC by applying Sine Law we get, Sin ?/n = sin 2 ?/n + 2 ? sin ?/n = 2 sin ? cos ?/ n + 2 ? 1/n = 2 cos ?/n + 2 (as sin ? ? 0) ? cos ? = n + 2/2n . . . . . . . . . . . . . . . . . . . . . . . . . (1) In ? ABC by Cosine Law, we get Cos ? = (n + 1)2 + (n + 2)2 – n2/2(n + 1) (n + 2) . . . . . . . . . . . . . . . . . . . (2) Comparing the values of cos ? from (1) and (2), we get (n + 1)2 + (n + 2)2 – n2/2(n + 1) (n + 2) = n + 2/2n ? (n + 2)2 (n + 1) = n(n + 2)2 + n(n + 1)2 – n3 ? n (n + 2)2 (n + 2)2 = n(n + 2)2 + n(n + 1)2 – n3 ? n2 + 4n + 4 = n3 + 2n2 + n – n3 ? n2 – 3n – 4 = 0 ? (n + 1) (n – 4) = 0 ? n = 4 (as n ? - 1) ? Sides of ? are 4, 4 + 1, 4 + 2, i.e. 4, 5, 6.
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