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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...?

3 years ago

Arun
25768 Points

Let 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.
3 years ago
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• Mind Map
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