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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...?

three sides of triangle are consecutive integers and the largest angle is twice the smallest angle.the perimeter of the triangle is...?

Grade:12th pass

1 Answers

Arun
25750 Points
6 years ago
Please find the answer to your question

Let the sides of ?be n, n + 1, n + 2 where n? N.
Let a = n, b = n + 1, c = n + 2

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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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